CONTI N U0 US LINEAR RE PR ESE NTATlONS
NORTH-HOLLAND MATHEMATICS STUDIES 168 (Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN Cen tro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U.S.A.
NORTH-HOLLAND -AMSTERDAM
LONDON
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CONTINUOUS LINEAR R E PR ESE NTAT10NS
Zoltan MAGYAR Mathematical Institute of the Hungarian Academy of Sciences Budapest V Realtanoda u. 13-15 Hungary - 1053
1992
NORTH-HOLLAND - AMSTERDAM
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L i b r a r y o f Congress C a t a l o g i n g - i n - P u b l i c a t i o n
Data
Magyar. Z o l t a n . Continuous l i n e a r r e p r e s e n t a t i o n s / Z o l t a n Magyar. p. cm. - - ( N o r t h - H o l l a n d m a t h e m a t i c s s t u d i e s 168) I n c l u d e s b i b l i o g r a p h i c a l r e f e r e n c e s and indexes. ISBN 0-444-89072-6 2. R e p r e s e n t a t i o n s o f g r o u p s . I. T i t l e . 1. L i e g r o u p s . 11. Series. 0A387. M34 1 9 9 2 512 .55--dc20 91-40659 CIP
.
ISBN: 0 444 89072 6
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V
PREFACE This book is intended t o give access t o the theory of continuous linear representations of general real Lie groups to readers who are already familiar with the rudiments of functional analysis and Lie groups. Nonetheless, a voluminous Appendix is devoted to the list of those definitions and results which are used in the main text. Now if some readers (particularly specialists in other branches or even other disciplines) are willing to believe the contents of the Appendix then they might also use this book with success. The Appendix is the place where we establish a number of notations and I hope that this will not be inconvenient for the readers. It is advisable for readers knowing well the areas discussed in the Appendix that they consult it only when some ambiguity happens to arise. The questions of what we mean by ‘continuous linear representations’ and by the emphasis on general Lie groups are clarified by the Introduction below and, eventually, by the whole book. Let it suffice to say now that we focus on continuous representations over Banach spaces (the duals of such also encompass the important special case of separately weak* continuous representations over von Neumann algebras), and we do not go into the depths of the theory of representations of solvable or those of semi-simple groups. These theories might be the topics for further volumes. I think they are covered by earlier literature more thoroughly than the topic discussed i n the present book. Even though we consider general locally convex spaces whenever the methods allow this, readers interested just in representations over Banach spaces may substitute Banach spaces instead of them and do not bother about such things as ‘barrelled space’, ‘sequentially complete space’, etc. I think that the relevance of group representations t o physics (as it has been emphasized by a number of scientists but perhaps most strongly by E. P. Wigner) is of outstanding importance. The material in Chapters 9 and 10 mainly serves as an illustration concerning the symmetry groups of the Newtonian and of the relativistic mechanics, the so called Galilean groups and PoincarC groups, respectively. The Heisenberg group and its Schrodinger representation really arise as ancillary mathematical concepts when studying the projective representations of the Galilean groups; though this fact is rather classical and gives some “philosophical support” to quantum mechanics, it seems to be far from widely known. I myself when I was a university student just heard of it at a mathematical physics course without any proof; it was very hard to trace the literature for these proofs. T h e corresponding chapters of the present book may now save the readers from similar pains. Also observe the usage of the more modern results of G. W. Mackey and J . Glimm: they are about 30 years of age, “more modern” is meant in contrast t o the classical approaches of H . Weyl, J . von Neumann, E. P. Wigner and others in the period from about 1925 to 50. My major concern was throughout the book that the proofs be as simple and clearly exposed as possible. This required some work even in those cases when eventually the ideas of the proofs i n earlier literature could be applied without alteration, as in parts of Chapters 6...8 (the basic theory of unitary representations: the induced representations of Mackey, representations of commutative and of compact
vi
Preface
groups), in Chapter 1 (the theory of one-parameter semigroups over Banach spaces), or in proving the integrability theorems of infinitesimal representations taken from [RusS] and [BGJR] (Theorems 5.14 and 5.15 in this book). Chapters 2...4 seem more of an invention, though the majority of their results were also known earlier. The book is not self-contained in the traditional sense because I think that it would be superfluous to retell things which are already very well exposed by Rudin, Arveson, Dixmier and others in their respective areas. Thus we also escape the horror of a too thick book. On the other hand, readers with a certain level of background knowledge should consider the book self-contained; since the prospective audience consists of diverse people in this respect, fixing this level caused me a serious dilemma. The Appendix is an attempt to cope with this problem, and I am rather happy about my solution even though I will welcome any criticism. A particular point is the treatment of some delicate results about partial differential equations in the second half of Appendix E : I do not consider myself an expert on PDE and possibly someone who is such an expert has written a book which would make this part superfluous. I treat C*-algebras and von Neumann algebras very vaguely in the Appendix, because these are more closely related to some topics in the main text. In Chapters 6 and 7 I made a number of more or less detailed references, mainly to places in the classics of Dixmier. I have not always endeavored to trace the historical origins of the ideas exposed and do not want t o be the judge of who was the real discoverer of this or that result. Of course, I gave references when the results are not quite classical but I can not tell whether these references are the “best” (also considering the possibly different tastes of the readers). The book also contains results discovered in the course of writing. I included exercises in the end of each chapter of the main text. These are generally not used in the subsequent development but may greatly clarify the picture; examples are mostly found in the exercises. I encourage the readers to read just the chapters they feel interested in. T h e feasibility of such practice depends, of course, on the background knowledge of the particular reader, so it is hard to give general advice. Let me mention, though, that the dependence of Chapters 6...10 on Chapters 1...5 is very little, and several parts (e.g., Chapter 9 or the first half of Chapter 8 ) are also readable quite apart from the rest of the main text. My guiding principle in ordering the chapters was that more specialized representations come lat$. This contradicts sharply t o the historical order: we prove the 60-year-old Stone-von Neumann Theorem in Chapter 10 , while, e.g., some integrability theorems from Chapter 5 are quite recent. During the preparation of this book I was a research associate at the Mathematical Institute of the Hungarian Academy of Sciences and I was also supported in part by the Hungarian National Foundation for Scientific Research (grant no. 1816). Budapest, April 1990
Zoltin Magyar
vii
CONTENTS PREFACE ....................................
.......................
v
0. Introduction .............................
.......................
1
1. The Hille-Yosida Theory ..................................................
.3
2. Convolution and Regularization ..........................................
.25
.............................
.39
.................................
.73
5. The Integrability Problem ................................................
.91
3. Smooth Vectors .......................... 4. Analytic Mollifying . . . . . . . . .
6. Compact Groups . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 113
.......................
7. Commutative Groups . . . . . . . . . . .
.139
8. Induced Representations .................................................
155
9. Projective Representations ..............................................
.171
. . . . . . . . . ..189
10. The Galilean
............................
206
...........................................
,207
B. Measure and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.211
APPENDIX . . . . A. Topology
. . . . . . . . . . . . . . . . . . ..219
C. Functional Analysis . . . . . . . .
.................................
235
E. Manifolds, Distributions, Differential Operators ......................
241
F. Locally Compact Groups, Lie Groups . . . . . .
269
D. Analytic Mappings
.....................
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .. 2 8 3
Index of Notation ............................
............. 291
Index . . . . . . . . . . . . .
...............................................
.293
This Page Intentionally Left Blank
1 0. Introduction
The family of “symmetries” of some object X often forms a group, e.g., think of the congruence group of the ordinary (three dimensional Euclidean) space. In fact, groups themselves usually arise in practice this way. On the other hand, different such realizations might yield the same group, the group structure can perhaps be studied more easily in the abstract image, and one may obtain information about a complicated object X through the study of some symmetry group. In other words, we are dealing with a so called representation of a group. In order t o be able to define the scope of the present book we fix some notation a t once. Let G be a group, X be a set and suppose that we are given a mapping @ : G x X H X sat-isfying the axioms
Then it is said that an action of G over X is given, or that G acts over X (from the left). Of course, it is also possible to consider these axioms if G is just a semigroup with unit; moreover, sometimes actions of semigroups (not necessarily possessing a unit) are talked of (meaning thereby that axiom (0-1) holds). In this book we are not concerned with general semigroups, others than (R+)” are just touched very slightly i n the beginning of Chapter 3 . We always mean by an action a mapping which satisfies (0-2), too. Of course, if the operation is written additively then the unit element is denoted by 0 instead of 1. As regards groups we shall always consider actions of locally compact groups and mostly those of Lie groups (cf. Definition F.l ). It is customary to write simply 93: instead of @(g,x ) , which practice is justified by (0-1). It is also usual to consider an action as a ‘representation’, i.e., the function T which takes the elements of the group to m a p p i n p of the set X into itself by setting T(g)(t) = g x . Then the axioms (0-1,2) will turn into the requirement that T be a homomorphism of groups or unital semigroups, respectively. If G and X are endowed with topology then it is natural to call an action continuous if it is continuous with respect to the product topology (“jointly continuous”), and whenever we consider an action which is just ‘separately’ or ‘partially’ continuous (i.e., continuous in both variables with the other variable fixed), we shall always say so. Now the scope of this book is limited by the following. T h e (semi)group is locally compact and if it is not a group then mostly it is very special as described above; the action is continuous or a t least such that modifying X in a not essential way it becomes continuous (see especially Theorem 3.11 and Exercise 3.5); and we have linearity in the sense that either X is a locally convex space and each T ( g ) is a linear operator or this situation can be achieved by introducing appropriate new objects. This branch of mathematics might be called the theory of continuous linear representations of locally compact (semi)groups. By now human knowledge h a s become so vast in this area that it is generally assumed to be beyond the
2
In troduc t ion
comprehension of a single individual. It is certainly impossible to cover it in a book like this. T h e main theme of this book is the exposition of the methods which make no use of the special structure of the groups; the unifying idea is that of regularization. We also include the classical Cartan-Weyl theory of representations of compact Lie groups (as a deviation from the main theme above). For orientation we outline the contents of the book. We may say that the first five chapters are heading for a thorough understanding of the following problem: how and in what sense can we come and go between a representation of a group and the tangent of this representation? Over finite dimensional spaces the solution is very old and satisfactory. Over infinite dimensional Banach spaces this is rather more complicated even if the group is the real line. In Chapter 1 we also consider the related problems of representations of the half-line and holomorphic representations of complex sectors. We call attention t o the treatment of analytic vectors. Chapter 2 provides a systematic development of the concept of convolution of distributions over Lie groups, and serves as a preparation for application in later chapters. The Regularization Theorem of this chapter is also a n illustration of the smooth mollifying investigated in Chapter 3 . The latter is the longest chapter: it contains the basic facts about Girding subspaces and tangents of general representations and of the regular representations over various spaces, related results about formally weaker but equivalent conditions for continuity of a representation, and about the denseness of algebraic combinations of tangent operators in certain corresponding adjoints. Theorem 3.35 is a new result and it has not been published elsewhere. Chapter 4 is essentially devoted to the denseness of analytic vectors (i.e., vectors a t which the representation can be recovered from its tangent on a neighborhood of 1 by means of a natural power series). Chapter 5 is a partial overview of the very difficult question of going from Lie algebra representation towards group representation. We incorporate here several authors’ very nice recent results. The second five chapters concern with the more common topic of unitary representations and the special goal is the study of projective representations of certain groups which are neither semi-simple nor solvable but of physical interest. The transition from the general theory over locally convex spaces to the unitary theory is somewhat smoothened in Chapter 6 : we present the general facts as exist about representations of compact groups and also introduce some of the von Neumann algebra theory relevant to unitary representations of locally compact groups. The beautiful old theory of highest weights is also explained here. T h e unitary representations of commutative (locally compact) groups are understood well enough (SNAG Theorem); Chapter 7 contains this theory and the related parts of Mackey’s normal subgroup analysis. T h e latter leads us over to the induced representations, which we present in Chapter 8 i n the most modern version. Then we obtain a description of the spectrum of certain semi-direct product groups (to be utilized in Chapter 10). Chapter 9 classifies (in a sense) the ancillary groups for a given Lie group on which certain unitary representations yield the projective representations of the given group. Chapter 10 contains a detailed investigation of the projective representations of the Galilean and PoincarC groups (including the non-connected ones).
3 1. The Hille-Yosida Theory The exponential function
can also be considered if we replace the complex number z by an N x N complex matrix. Now the theory mentioned in the title of this chapter is roughly nothing else than the analysis of the exponential function when z is a n infinite dimensional matrix. More precisely, let z be a (generally not continuous) linear operator in a locally convex space, then one may ask what is the “right” requirement about z to make it possible to define a reasonable exponential of it. If z is a normal operator in a Hilbert space t,hen the problem is solved by spectral theory (cf. Appendix C ) ; here we shall study the other extremity, i.e., we concentrate on locally convex spaces. We do not include Stone’s celebrated theorem because it is a special case of the SNAG (Stone, Naimark, Ambrose, Godement) Theorem what we shall discuss in Chapter 7 below. For numbers (or, more generally, fo, bounded operators) the exponential function can also be defined by the properties that e(‘+’)* = e t Z e s z , e O = l and % e t z I t = o = z . This is also the connection to representation theory. Now the following pair of fundamental definitions is more or less natural.
Definition 1.1. Let X be a sequentially complete locally convex space over C (in our terminology these are necessarily Rausdorff, cf. Appendix C) and Rt be the set of non-negative reals considcred as a semigroup with addition as its operation. Consider the continuous linear actions of R+ over X , i.e., the mappings @ : R+ x X H 2 satisfying the axioms (0-1,2) which are linear in the second variable and (jointly) continuous. We call these actions and also the corresponding represen tations one-parameter semigroups (following the tradition, even though these are not semigroups but homomorphisms between semigroups). The requirement about sequential completeness is not very substantial because even if we drop it the local equicontinuity of the representing operators enables us to extend the action continuously onto the completion of X . The reason for requiring just sequential completeness is that it is the minimal assumption in order to have certain derivatives and integrals inside the space. For the relations between ‘separately continuous representations’ and oneparameter semigroups see Chapter 3 below. Let it suffice to mention now that if X is a Banach space then they are the same.
Definition 1.2. Let T be a one-parameter semigroup over X . Then the generator o f T is the linear operator A in X defined by the following: 1
At = lim - ( T ( l ) t - t ) t-ot t
Chapter 1
4
where the domain of A is the linear subspace of x’s for which the limit on the right exists. We shall see that the generator of a one-parameter semigroup determines it (which fact justifies the term ‘generator’) and we shall use the conventional notation T ( t )= e t A in analogy with the finite dimensional case. The proof of the most elementary facts about one-parameter semigroups will be the first application of the central tool of this book, which is the averaging of representations with the help of an invariant measure. So it is useful to dwell on this a bit somewhat informally. In the case of a one-parameter semigroup T for any measurable ‘test-function’ ‘p we try to set
T(‘p)=
Lrn
‘p(t)T(t) dt
If ’p is locally Riemann integrable inside R+ (this is always the case in practice) then for a n y 3: E X and 0 < a < b we have even the Riemann integral Jab ’ p ( t ) . T ( tdt) ~ by the sequential completeness of X (see Appendix C and especially pp. 224-225 about Riemann integrals). If, in addition, ‘p.po[T(.)x]E L’(R+) for some E and for all p from a family of seminorms defining X then we get the improper Riemann integral T ( ’ ~ )=E ‘p(t)T(t);c dt for this E (the condition is sufficient but not necessary). The set of x ’ s for which this is true forms a linear subspace and T(’p) is a linear mapping from this subspace into X . The above can also be considered by writing T(’p)=
1
?Tdp m
where p is some measure on Rt which is absolutely continuous with respect t o the Lebesgue measure and m is its Radon-Nikodym derivative. This may have the advantage of getting a proper Rieniann integral on Rt . We stick to Riemann integrals in this chapter because of their more elementary nature but note that if a weak integral actually lies in the space then its properties are almost as good (see Appendix C ) . Now the invariance of the Lebesgue measure under translations comes into the picture by observing that T(s)T(’p)= T(’p)T(s)= T ( $ ) with $ ( t ) = ’p(t- s ) and $ ( t ) = 0 for 2 < s . Thus one can move differentiations from T to ‘p if the latter is nice. Since T(’p,) -+ I pointwise whenever ‘pn tends to Dirac’s delta in a rather weak sense, we can regularize the vectors of X with respect t o T . We shall expand these ideas for locally compact (semi)groups and especially for Lie groups in Chapter 3 below. Now we turn to the formal treatment of the case of one-parameter semigroups.
Definition 1.3. Let T be a one-parameter semigroup over the sequentially complete locally convex space X and let ‘p : R+ H C be some measurable function. If for some E E X the weak integral ‘p(t)T(t);c dt exists and, moreover,
The Hille-Yosida Theory
5
belongs to X then we denote i t by T(p)z. Clearly, the set of such x’s forms a linear subspace and T(p) is a linear operator on it. Most often in practice p(t)T(t)zdt can also be interpreted as an improper Itiemann integral and its belonging to X follows from this.
Lemma 1.4. Let p vanish for large t and be Riemann integrable on the corresponding finite interval. Then T(p) is a continuous everywhere defined linear operator and
where cps(2) = p(2 - s ) for t functions then
2s
and cps(t) = 0 for t
< s . If c p , $ are two such
PROOF:The continuity of T implies that { T ( t ); 2 E Zi} is a n equicontinuous family of linear operators for any compact K . In particular, Ii‘ can be a n interval such that (o vanishes outside it. Then for a n y z E X the function f = c p . T ( . ) z is bounded (in the sense of locally convex spaces); moreover, for any continuous seminorm p there is a continuous seminorm q such that p ( f ( t ) ) 5 q ( x ) for all 1 . On the other hand, f is continuous almost everywhere. Hence T ( p ) is everywhere defined and continuous (see Appendix C for the properties of vector valued Itiemann integrals). Then (1-1) follows immediately (cf. Proposition C . l l ) . Thus T(cp)T(qh)z= (o(s) qhs(2)T(1)3:ddl) d s and to prove (1-2) it is enough to show the equality of the weak integrals. This follows at once from Fubini’s theorem.
(s:
Theorem 1.5. Let T be a one-parameter semigroup over t h e sequentially complete locally convex space 3 and A be its generator. Then A is a densely defined closed operator, and if z is in the domain of A then T ( . ) x is differentiable in the strong sense everywhere, not just a t 0. Its derivative equals T ( . ) A x = A T ( . ) . . The generator A determines the semigroup; more precisely, if f ( 2 ) is a function satisfying f ( 0 ) = 3: and f’ = A o f then f ( t ) = T(2)t , so T is nothing else than the unique continuous extension of the solution of this differential equation with initial value varying in the domain of A . Let B be a continuous (everywhere defined) operator. Then B commutes with T if and only if it commutes with A ; more precisely, the relation
BT(2) = T ( t ) B for all t E R+ is equivalent to t h e relation B A c A B
.
Chapter 1
6
PROOF:If At = y then, evidently, the derivative from the right of the function T (.)tequals T (.)y . This implies that AT( .)z exists and equals the same, and using the joint continuity of the action we have the same for the left derivative.
1 St = T T ( ~ ~. ~Since , ~ ) S) ( t ) z is a Riemann integral and T is continuous, we obtain limt,o+Stz = t for any t (also using that X is locally Now let
convex; as a matter of fact, we only consider integrals in such). We assert that for any pair (t,y) E X x X the relation At = y is equivalent to the relation sty
(1-3)
=
T ( t ) x- 3:
Vt>O.
If Ax = y then T ( . ) y is the continuous derivative of the function T ( . ) z . Thus JhbT(s)yds = T ( b ) x- T ( a ) z and (1-3) follows. If (1-3) holds then its right hand side must tend to the same vector as the left hand side, and we know the latter tends to y when 2 + O+ . This means exactly that Ax exists and equals y. Since St and T ( t )are continuous operators, it follows from (1-3) that A is a closed operator. We prove the denseness of its domain by showing that (1-4)
AStx =
T(t)t- x t
Vx E X and Vt
>0 .
This will be enough, for S f z + 3: when t -t O+ . Now T ( t )- I 1 T ( t ) x- 2 and (1-4) is stx = T ( s ) l :ds - T ( s ) zd s ) = S, z tz t proved. Turn t o the statement about unicity. Let x E D(A) and f : R+ ++ X a differentiable function satisfying f ( 0 ) = x and f’ = A o f . Fix a t > 0 and let g(s) = T(t - s)f(s) for 0 5 s 5 t . Then o u r statement amounts t o g ( 0 ) = g ( t ) . The continuity of the action and the assumption that A is defined on the range of f imply that we can calculate g’ by the Leibniz Rule (cf. Proposition C.17), and thus we obtain g‘ = 0 . We prove the last statement by the same method. Of course, one direction is evident, so we assume B A c A B and show that B commutes with T then. Fixing t E D ( A ) and t > 0 we write this time g(s) = T ( t - s ) B T ( s ) x for s E [O,t] . Then we can use the Leibniz Rule as above (because B is continuous) and obtain g’ = 0 , T ( t ) B z = B T ( t ) x . Use the continuity of B and the denseness of D ( A ) to complete the proof.
(Lt+’
Ji
Our next objective is Theorem 1.8 below, which clarifies in some sense which operators can be exponentiated. We call this theorem in this book the ‘Hille-Yosida Theorem’ even though the classical result discovered independently by E. Hille and K. Yosida concerned with the special case of contraction semigroups over Banach spaces, cf. Corollary 1 . 9 . The crucial idea is to apply Laplace transform to the one-parameter semigroup. This heuristically means that writing R(X) = T(px) with px(t) = e - X t we should get R(X) = ( X I - A)-’ . This is unfortunately not so simple for a general one-parameter semigroup, see [Kom] for a possible approach. We restrict our attention t o the so called ‘contraction type’ case.
7
The Hille-Yosida Theory
Notation 1.6. We say that a one-parameter semigroup T is of contraction type if there is a real number C with the property that the family of operators { ewCtT(2); 1 E R+ } is equicontinuous. In this case we denote by WT the infimum of such C's (it is a real number or -m). If X is a Banach space then, clearly,
I-00
1
We say that T is weakly exponential if there is a C such that the sets
E(C,t)= { e-ctT(2)t; t E R + } are bounded for all x . It follows from the corresponding versions of the Banach-Steinhaus Theorem that if X is Bake or even if just barrelled then any weakly exponential one-parameter semigroup is of contractlion type (for barrelled spaces see, e.g., Theorem III/4.2 of [Scha] ). Finally, a vector t is called exponential for T if there is a C such that E(C,t) is a bounded set. We mention that if X is a Banach space then any one-parameter semigroup must be of contraction type (cf. Corollary 1.9 below). The following lemma consists of two related statements. In the proof of Theorem 1.8 below we need just the version (b), while the version (a) clarifies further i,he relation between semigroups and the differential equation f' = A f .
Lemma 1.7.(a) Let A be a closed linear operator in the sequentially complete locally convex space X and f : R+ H X be a continuous function which is continuously differentiable inside Rt , satisfies f ' = A o f there, and assume that there is a real number C such that the set { e - C t f ( 2 ) ; t E R+ } is bounded. For ReX > C consider the Laplace transform g(X) = e - A t f ( 2 ) dt (it certainly exists as an improper Riemann integral). Then g is holomorphic, the range of g ( k ) is contained in V ( A k t ' ) for any k = 0 , 1 , 2 , .. . ; namely
(b) Let T be a weakly exponential one-parameter semigroup over t h e sequentially complete locally convex space X and C be a suitable constant for the weak exponential property. Write pX(t) = e - x t , then for ReX > C the operators T ( c p A ) are everywhere defined, closed and T(pA) = ( X I - A)-' where A is the generator o f T . For any t E X the function T(cpx)t, considered in the variable X for ReX > C , is holomorphic and (-&)k7'(p~)r = (-l)kk!(AI
x=
1"
e-"(
-t)kT(l)x di!
Chapter 1
8
PROOF:The estimate on f ( t ) or T ( t ) z alone enough to differentiate under the integral sign with respect to X any times in the complex sense (we can use, e.g., Proposition (3.13 to justify this). Now we show the connection with A in the version (a). Let 0 < u < b < 00 , then (1-5)
Jo
b
e-xt(-t)kf'(t)dt = [e-xt(-t)kf(t)]: -
J
b
e-"(Xt - k ) ( - t ) " : - ' f ( t ) d t
a
(even weak integrals can be integrated by parts). Now we replace f' by A o f and then A can be interchanged with the Riemann integration by its closedness (see Proposition C.11). Then the estimate on f and the closedness of A enable us t o tend with [a,b] to Rt and infer that A g ( k ) ( X = ) Ag(')(X) kg("-')(X) for k 2 1 and Ag(X) = -f(O) Xg(A) , thus completing the proof of (a). Turning to (b) write h ( t ) = T ( t ) T ( p x ) z . Since T(2) is continuous, it can be interchanged with the limit in the improper integral and we have h ( t ) = Jm e - X ( 3 - t ) T ( s ) -dz s . Using the estimate on e-'T(s)z and also the continuity of the integrand we obtain h'(2) = -T(t)-z Xh(t) and, in particular, AT(cpx)z = --z XT(px)t . This was true for any I . Now assume that I E D ( A ) , then f ( t ) = T(t)-z satisfies the assumptions of (a) by Theorem 1.5 but we also have f ' ( t ) = T(2)A-z in this case. Hence from (1-5) we now infer T ( p x ) A z = - I + XT(px)z and so we have proved that T(cpx)= ( X I - A)-' whenever ReX > C . Since A is closed, the operators X I - A are also closed, hence their inverses are closed, too. Therefore we have T(cpx)u(t)dt = T(cpx)J; u ( t )dt whenever the function a is such that both integrals can be interpreted as improper Riemann integrals. Let a(2) = e ( l ) T ( t ) z with a continuous scalar function p satisfying le(2)l 5 Me-'' with some constants M > 0 and c > C . Then the above conditions are satisfied (use tha fact that T(cpx)commutes with any T ( t )) and the left hand side also equals T ( y ) z with y = cpx * e (this '*' is the same as in (1-2)).
+
+
+
+
JF
Writing & ( 2 ) = e-"
tk
= e k + l and hence it follows by induction k! on k that T ( ~ A ) ~=+T '( & ). Thus the proof is complete. - we have px * e k
Theorem 1.8 (Hille-Yosida Theorem). Let A be a densely defined linear operator in the sequentially complete locally convex space X . If A is the generator of a one-parameter semigroup T of contraction type with WT then for ReX > WT the resolvent ( X I - A)--' exists as an everywhere defined continuous operator and for any C > WT (and even for C = WT if it is a minimum) we have (1-6)
V p 3q :
(ReX - C ) k. p ( ( X I - A ) - k z ) 5 q(z)
for k = 0, 1 , 2 , . . . and Re X > C where p , q are continuous seminorms and their relation can be the same as in the description of the equicontinuity of e - C ' T ( 2 ) . Now suppose that A has resolvents satisfying (1-6) with some C a t least for a sequence A, ofreal numbers tending to +00 . Then A generates a oneparameter semigroup T of contraction type such that e - C I T ( t ) is equicontinuous with the
9
The Hille- Yosida Theory
same p , q relations as in (1-6). This one-parameter semigroup can be calculated by Yosida's formula
T ( t ) z = n-cu lim
(1-7)
2;
k
[ A (Z- k A , ' ]
z
k=O
as well as by Ilille's formula:
T ( t ) z = lim
(1-8)
n-oa
whenever kn/Xn
+
t
(Z
-
:,
-A
)-kn
.
PROOF: If e-C'T(t) are equicontinuous with the relations p(T(2)z) 5 e c ' q ( z ) then we have p ( T ( ~ ) z5) q ( z ) .
;1
lQ(t)lec' d2 whenever the sides of this inequality exist. Substituting the result of Lemma 1.7.(b) we obtain (1-6) (the fact l&(2)lec' dt = (ReX - C)-'"-' that, with the notation of the previous proof, can be checked by integrating by parts or also by observing that the integrand here is the k l'st convolution power of the positive function 9 ~ ~ x - c ) . Now assume that A is a densely defined operator satisfying the assumptions with suitable C and A, . Write R, = (A, - C ) . (X,Z - A)-' , B = A - CZ and, in discussions with fixed n , p = A, - C . Then we have (pZ - B)R,z = pz for any z and R , ( p - B z ) = p x for z E ? ) ( A ) . Hence obvious calculations yield
+
(1-9) Following the approach of K . Yosida we shall show that B, is the generator of some equicontinuous one-parameter semigroup V, and then prove that V(t)x = limn-oa V, ( t ) z again yields an equicontinuous one-parameter semigroup whose generator is B . First we show that limn-+oaR,z = z for all 3:. Because of the equicontinuity of the Rn's it is enough to prove this for a dense set of z's (see Proposition C.17). So 1 R,Bx we may assume z E ? ) ( A ) and then from (1-9) we obtain R,x-x = A, which tends to 0 for A, 4 00 and the R,'s are equicontinuous. Applying this t o x = By weget
c
(1-10)
lim B,y = By
n-+m
for any y E D ( A )
We know that the set { Rnkz ; k,n E N } is bounded for any 2 ;namely, the supremum of a continuous seminorm p is estimated by q ( z ) with the corresponding seminorm q from (1-6). Hence by the sequential completeness of the space we can see that for any t E R the sum Vn(t)z :=
xr=ot k B,'
3:
exists in X and equals
Chapter 1
10
CEO k! Rnk x (what we use here is the ‘complete associativity’of absolute convergent series; one may also consider the sums as Riemann integrals over N and use Fubini’s theorem). The same method also yields Vn(t+s) = Vn(t)Vn(s). Thus
e-Pt
(1-11) (for p
> 0 ) and
p (V,(t)e - e - p t
we have corresponding formulas for
Cr=o
R n k z ) for any N . This implies that for all z Vn(t)z
is differentiable at 0 and its derivative equals p(R,t - z) = B,I . We have already checked that V, turns addition into multiplication, so it follows from (1-11) that it is a continuous representation of R . Since V, is locally equicontinuous and Vn(.)z is differentiable for any z , the derivative of Vn(.)f(.) can be calculated by the Leibniz Rule whenever f is a differentiable function (see Proposition C.17). On the other hand, (1-9) and the continuity of the operator B, imply that it commutes with any V m ( t ) .Thus, also using the continuity of the integrand below, for any t , z and m we have
V n ( t ) z- Vm(t)z=
I’
rt
[ Vn(s)Vm(t- s ) z ]
ds
=
rt
N o w (1-11) implies equicontinuity with varying n’s for 1 2 0 . Hence we obtain by (1-10) that the sequence Vn(t)z is uniformly Cauchy for 2 E [O,Q] if I is a fixed vector from D ( A ) and Q is any fixed positive number. Then we use the equicontinuity again and infer that the same is true with any x . Since X is sequentially complete, we can set V ( t ) z := limn-,oo Vn(t)z for all t 2 0 and z E X and the limit is locally uniform in the variable t . Hence V ( . ) z is continuous for each Vn(.)z is such. We get the equicontinuity of V ( . )from (1-11) and then it is easy to check that V is a one-parameter semigroup (cf. Proposition C.17). We also obtain (1-7)
by observing that A(Z-?A)-’
= $(B+CZ)R, = $ ($Bn
+Cl)
and so the sum under the limit in (1-7) equals erCtVn(rZt)with r =
(use (1-9)), An -
A,-C‘ If y E D ( A ) then V,(.)By tends to V ( . ) B y locally uniformly while Vn(.)(Bn- B ) y tends to 0 in the same way by (1-10) and (1-11). So we can infer, using (1-3), that
and using (1-3) again we can see that B is a part of the generator G of V . But we know from (1-11) that V is of contraction type, so if R e p is large enough then
11
The Hille-Yosida Theory
p I - G is injective by Lemma 1.7 while pI - B is known to be surjective whenever p = An - C . Thus G = B = A - C I and A generates the one-parameter semigroup ~ ( t =) eC'V(t) .
It, remains to check Hille's formula.
Rn
= ec' . Thus it is enough
and we know from calculus that limn+m to show that with a , =
V(t
Rnkn2
A, - c
~
+ a,)z
-
-+
-+
V ( 2 ) z . Since
Xn ~
A,
-c
0 . First suppose that
V ( t ) z- a,BV(t
+ ay,,)z=
2
-+
1 , we have l i Q + m k n a n = t
E D ( A ) . Then
1'-
( V ( S )- V ( t
+ a , ) ) B z ds
and we apply R, to this equat,ion. Then substitute t = O,a,, 2 a n , .. . , apply powers of R, and add the equations to get
By the equicontinuity of the powers of the R,,'s and by the continuity of V we obtain now (1-8) for such z's (on a compact V ( . ) B z must also be uniformly continuous). But this is enough by the denseness of D ( A ) (cf. Proposition C.17). Corollary 1.9. Let X be a Bariach space. Then a densely defined operator A is the generator of a one-parameter semigroup if and only if there are real numbers C and h! such that for Rex > C the resolverits (XI - A)-' exist and satisfy
(1-13)
11 ( X I - A)-' 11 5 A4 . (Re X - C ) - k
for k = 0, 1, 2 , 3 , . . .
If we know (1-1 3) j u s t for a sequence of real A's tending to
+m then A is necessarily a generator. A is the generator of a semigroup consisting of contractions (i.e., of operators with norm 5 1 ) if and only if we can write C = 0 and M = 1 in (1-13); again it is possible to consider j u s t a real sequence. Of course, in this case (1-13) with k = 1 implies it for any k .
PROOF: Since
IIT(t + s)II 5 llT(2)ll . IlT(s)ll , we have
IIT(t)ll L.
(supssl IIT(s)ll)"It1 and so any one-parameter semigroup is of contraction type in this case. Writing p = 11. 1 1 and q = M . 11. 1 1 in (1-6) we obtain immediately our statements.
Corollary 1.10. Let A be a densely defined operator in the sequentially complete locally convex space X . It is the 'generator' (i.e., derivative a t 0) of a
Chapter 1
12
continuous almost equicontinuous representation of R over X (we call T ‘almast equicontinuous’if both TIR+ and T(-.)IR+ are oneparameter semigroups of contraction type) if and only if there is a C such that the family
(1-14)
(I Re XI - C ) k. ( X I - A ) - k , I Re XI > C
is equicontinuous (meaning also that these operators are everywhere defined). l t is enough to consider two real sequences of A’s, one tending to +m and the other to -m . The representation will be equicontinuous if and only if we may write C = 0 in (1-14) and again it is enough to consider two real sequences. We also have the now obvious refinements for Banach spaces (cf. Corollary 1.9).
PROOF:The only thing not proved already is the fact that if T and W are one-parameter semigroups with generators A and - A , respectively, then setting T ( - t ) := W ( t ) we obtain a representation of R . For this it is enough t o check that T ( t ) W ( t )= I for all t . By continuity we may restrict o u r attention t o T ( t ) W ( t ) x with E E D ( A ) and then $[T(t)W(t)x] will be identically 0 (cf. Proposition C. 17 ). On may naturally ask whether e l A could be calculated by the familiar power series at least for a dense set of E ’ S , i.e., whether we can interchange the linlit with the sum in Yosida’s formula (and so get rid of the limit) if E is “nice”. Unfortunately, this is not so in general (see Exercise 1 . 5 ) but the case is even a bit better for oneparameter groups with a dense set of exponential vectors by a classical result of I. M . Gelfand (Proposition 1.13 below) which, by the way, is older than the HilleYosida Theorem. We shall discuss the necessarily weaker extension of it to Lie groups in Chapter 4 . T h e other direction (the denseness of the set of analytic vectors together with some estimates implies that A is a generator) is contained in Theorem 1.14 below; in this form it was discovered by the author (cf. [Ma3]) independently of J . Rusinek who obtained essentially the same result a couple of years earlier (see [Rusl]). The corresponding extensions to Lie groups will be discussed i n Chapter 5 below.
Notation 1.11. Let A be a linear operator in a locally convex space Then for any positive number s we set (1-15)
d(s,A) =
{
E
EX;
($ A k . )
is bounded if 121
X.
<s
k€N
(this is a linear subspace) and call its elements the s-analytic vectors of A . Of course, for s1 < s2 we have d ( s 1 , A ) 3 d ( s 2 , A ) and it is easy to check that these linear subspaces are invariant under A . We write A ( A ) = U , d ( s , A ) , f ( A ) = n , d ( s , A ) and call the elements of these sets the analytic and entire vectors of A , respectively.
The Ziille-Yosida Theory
13
Of course, most operators have no other analytic vectors than 0. On the other hand, if A is an everywhere defined bounded operator on a Banach space then each vector is entire for it. Lemma 1.12. Let T be a continuous representation of R over t h e sequentially complete locally convex space x and A be itsgenerator. Assume that we have a vector x and a non-negative number C such that the set { e - c L 2 T ( t );zt E R } is bounded. For c > C set xc =
fiJ_", e-'"T(t)x
dt
x, = x and for any c > C we as an improper Riemann integral. Then lim,,t, can find some positive bound M such that { ( M & ) - ' A k z c ; k E N } is bounded.
PROOF:The assumptions imply the existence of the improper Riemann integral.
Observe that for any continuous seminorm p we have p(xc - x ) 5 00 - x)dt (because e-c12 dt = ). NOW write
Esrwe - ' 1 2 p ( T ( l ) x
s-,
fi
h ( t ) = e - c L 2 p ( T ( t ) z- x) , this is a bounded continuous function and h ( 0 ) = 0 . So lime,.+, 2, = 3: follows from the elementary facts that E J y , e ( c - c ) t 2 dt =
6
and for any Ii
>0
we have
cby,
fi/,,,,
e ( c - c ) t z dt = 0
.
I _
We have to show the existence of A k x c and then estimate it. Suppose that p is a C' function on R such that both u = p(t)T(t)zdt and 21 = p ' ( t ) T ( t ) xdt exist as improper Riemann integrals. Writing 21, = J: ,' p ( t ) ~ ( t ) x d t and v, = J_",p ' ( t ) ~ ( t ) x d twe o\>tain ~ u =, -v, p(n)T(n)x - p(-n)T(-n)x and hence if p is such that p(t)T(t)x ---* 0 when 121 -+ 00 then we obtain Au = -v because A is a closed operator. Let p be a polynomial times e - c t 2 with c > C , then p' is again a polynomial times e - c t 2 and the above conditions are certainly satisfied. Thus
s?,
s-",
+
(k)
A k x C= (-l)kEJ?w [E-~"] T ( t ) z d t and hence if
is finite then it can serve as M
. So we must estimate these integrals. The function
eta/' [ e - 1 2 / 2 ] ( k is ) a polynomial, denote it by P k ( t ) . Then (1-16)
Chapter 1
14
On the other hand, we have the recursion
PO = 1 ,
Pk+l(t)
= Pk'(t) - t ' pk(t) .
Therefore P k can be expressed as the sum of 2k terms, each of which is a result of j differentiations and k - j multiplications by -t applied on PO, where j = 0 , 1 , . . . , L . The terms for which 2 j > k are zero, the other terms can be estimated by k j 1 t 1 k - 2 j . Now if r > 0 and p 2 0 then it is easy to check that maxt lllPe-pi' = u > 0 we have
(&)"'
(this maximum is attained a t It1 =
e).
Thus for any
and hence IPk(ut)l 5 n i k .kk12 . e r t 2 where m depends on u and r but independent k . Substitute u = , choose 1' E ( 0 , c - C) and apply (1-16) t o complete the proof.
of
Proposition 1.13. Let T be a one-parameter group, i.e., a continuous representation of R , over the sequentially complete locally convex space X such that the set of vectors exponent,ial for both halves T(.)IR+and T(-.)IR+ is dense (see Notation 1.6; a stronger requirement would be that both halves be weakly exponential). Then, denoting the generator of T by A , the set
is dense in X . Consequently, & ( A ) is dense for it contains S .
PROOF: Let 3: be exponential for both halves, then the condition of Lemma 1.12 holds with any positive C . Theorem 1.14. Let A be a linear operator in the sequentially complete locally convex space X such that there is a positive SO with wich d(s0,A ) is dense and the following holds. We have a real number C and a real sequence A, tending to +cm such that each X , I - A is injective and (1-6) holds whenever it h a s sense (so we do not assume anything about the domains of the operators ( X , I - A ) - k ). Then A is closable and its closure is the generator of a one-parameter semigroup T ; for any 3: E d ( s , A ) and t E [O,s) we have (1-17)
If we have two sequences as in Corollary 1.10 then the closure of A is the generator of a group and (1-17) holds for t E (-s,s) .
15
The Hille-Yosida Theory
If X is such that any one-parameter semigroup over any closed subspace of it is weakly exponential then we can weaken the assumption about the denseness of d ( s 0 , A) by requiring only the denseness of d ( A ) . Remark. It is an open question whether the extra assumption on X is superfluous, i.e., the denseness of analytic vectors might be enough in general.
PROOF:We can assume that C = 0 because otherwise we can replace A by A - C I . This does not alter the analytic vectors, commutes with the taking of closures, the semigroups differ just by a factor ect and then (1-17) for them will be equivalent] as it is not hard to check. Denote by X, the closure of d(s,A ) . For t E [O,s) let V,(t) : d ( s , A ) H X, be the 'formal e l A ' , i.e., O0
tk
Akt .
V,(t)z = k=O
The existence of the sum is proved by choosing q E ( t , s ) and observing that
(:)
k
{ $ A k z ; k E N } is bounded and is ail absolutely summable series; the fact that the s u m belongs to X, follows from the invariance of d(s,A ) under A . Now let t E (0, s/2) and k, be a sequence of non-negative integers such that kn/Xn + t from below. Writing a,, = and yn = $ A k z we assert that
xiZo
(I- ~r,A)~"y,
(1-18)
+
z
To prove this fix a q such that 22 < q < s and consider the following functions on jtk
N x N : f ( j , k ) = (-l)k
(i)
(j:')
0
otherwise
8 t k
and the vector valued function F ( j , k ) = A j + k z . Then (j k)! ( I - C Y , A ) ~ " Y= , C j ! k f n ( j , k ) F ( j , k ) . On the other hand, Ifn[ 5 If1 and fn f pointwise. The function f is absolutely suminable by 2t < q and the function F is bounded, thus we have fn(j, k ) F ( j ,k) Cj,kf ( j ,k ) F ( j ,k) and the latter sum can be calculated by grouping the terms where j k is constant, thereby obtaining (1-18). This means that we have a sequence 2, tending to z such that p(y,) 5 q(zn) if p , q are from (1-6). Hence p(V,(t)z) 5 q ( z ) , i.e., the family { V S ( t )0; < t < s / 2 } is equicontinuous. Then the same holds if V,(t) is replaced by its closure T,(t) which is a continuous operator defined on X, for such 2's (and, of course, here we can allow t = 0 , too, for V,(O) is the identical operator on d(s,A ) ) . It is easy to check that if t E d ( s l A ) then [T,(.)z]'(O) = Az ,
-
+
xj,k
(1-19)
T,(t1)T,(tz)t= K(t1 +t2)z for
+
t1,tz
< s/2
Chapter 1
16
and V,(.)x is continuous. Then by the equicontinuity of the T ( t ) ' s we obtain that T ( t ) z is a continuous function on [O,s/2) x X, (cf. Proposition C.17). It also satisfies T,(t1)TS(t2)= T,(t1 + t 2 ) whenever t 1 + t 2 < s / 2 . Now for any t E R+ set T S ( 4 = T,(t/n)" with any n for which t / n < s/2 . It follows from the above that this definition is correct and, moreover, T, is a one-parameter semigroup over X, . It is also clear that T, 3 T, if r < s , Now if d ( s 0 , A ) is dense for some S O then denoting by T the corresponding T,, we can see that its generator G contains the operator B = AIA(A) and also that (1-17) holds (remember (1-19)). If we only know the denseness of d ( A ) then denote by V ( t ) the union of the operators T , ( t ) . These have a common dense domain, satisfy (1-17) and V ( t l ) V ( t z )= V( t1 1 2 ) but their equicontinuity or even their continuity is not evident a t all. The problem is connected with the question of how much greater the generator of T, is than the corresponding restriction of A . We show presently that if G, is the generator of T, and x E Y := flP=,D(Gf) then we can find a net x, E d(s,A ) such that limn A k x n = G f x for all k . For each seminorm p from a family defining X consider the seminorms x + p(Gtx) when k E N and x E Y , thereby endowing Y with a finer topology. We shall say that y, -+ y in Y if the convergence holds in this topology, while y,, --+ y means convergence in the original (coarser) topology. Since X is sequentially complete and G, is closed, we have that Y is also sequentially complete. Denote by 2 the closure of d(s,A ) in Y . We want to prove 2 = Y . Observe that D := G,I, is a continuous operator in the finer topology, and it follows from Theorem 1.5 that T , ( t ) D E z= D k T S ( t ) xfor x E Y , therefore
+
(1-20)
W ( . ):= T,(.)ly is a one-parameter semigroup over Y
and its generator equals D . Now we show that 2 is invariant under W ( 1 ) . It is enough to prove this for small t for the others are powers of these operators. So let 1 E [O,s) . By the continuity of W ( t )in Y it is enough to check that V,(t)x E 2 ti . ti if I E d(s,A ) . Writing yn = C;=, 3,413: we have Dky, = Cy=,3 A j A ' x and
D k x = A k z E d ( s , A ) , thus limn-m D'y, = V , (t )Dk z for any L . This amounts to limn+M yn = V S ( t ) xin Y and hence V,(t)z E 2 because yn E d ( s , A ) for all n.
Now let y E Y be arbitrary and xi be a net from d ( s , A ) tending t o y in the original topology. Let pm be a sequence from Cy((0, +m)) (so the supports are separated from 0, too) such that Pm 2 0 , Jpm = 1 and suppp, C (0, l/m] . Since the difference quotient of a compactly supported smooth function tends uniformly to its derivative, it follows from Lemma 1.4 and (1-20) that Dk pm(t)W(t)zdt = (-l)k & ) ( t ) W ( t ) x d t for any z E Y . M Writing Ym = p m ( t ) W ( t ) y d t and ym,i = pm(t)W(t)z, dt we can see that limi Dkym,,= Dkym because T, (.)xi + T, (.)y locally uniformly. Thus limiy,,, = ym in Y . We can view ym,i as a Riemann integral in Y and 2 is
so
SF
The Hille-Yosida Theory
17
invariant under W ( . ), thus we obtain ym,i E Z and hence ym E Z . Then using that ym is a Riemann integral in Y , too, the properties of the sequence (om and (1-20) imply that ym + y in Y . We can see that d ( s , A ) is dense in Y , hence we can write D instead of A in (1-6) because D is continuous in Y . Now assume that T, is weakly exponential. Then Letnma 1.7 shows that R, = (XnZ - G8)-' is everywhere defined in X, for large n . We also have R,G, C GSR, for such n's. Then R,G; C GfR,, for all k and hence R, leaves Y invariant. Therefore R,I, = (X,Z - D)-' is a continuous operator by (1-6). Now X, is the closure of d(s,A ) which is included in Y , and R, is closed, thus R, is the continuous closure of (X,Z - D)-' defined everywhere on 2,.Then (1-6) extends to R, and G, satisfies the assumptions of Theorem 1.8 with C = 0 ; so (1-8) implies that the local equicontinuity of the operators T,(t) is independent of t as well as s . Hence the operators V ( t ) are equicontinuous and their closures provide an equicontinuous one-parameter semigroup T . It remains t o prove that the generator G is the closure of A (for any X). We saw that B c G ( B w a s the restriction of A to its analytic vectors) and also that whenever x E fl?=.=,D(Gf) with some s then there exists a net xi such that limi B k x i = G k x for all k . Apply this with k = 0 , l and with x = (X,Z - G)-'y for any y E Y to infer that (&,I - B)-' is dense in the corresponding restriction of (X,Z - G)-l . Now the continuous resolvent (X,Z - G)-' exists because T is an equicontinuous one-parameter semigroup, and the restriction above is defined on the union of Y's which contains the dense set d ( A ) . Thus we have proved that G is the closure of B . Then the fact that both (X,Z - A)-' and ( X , I - G)-' are continuous implies that B C A C G . The statement about one-parameter groups now follows from Corollary 1.lo. As the Hille-Yosida Theorem shows, generators of one-parameter semigroups of contraction type are the operators whose spectrum is included in a left half-plane such that the distance from the boundary of this half-plane controls the resolvents in the right half in a certain way. Now in practice one can encounter operators with smaller spectra, e.g., the spectrum of the Laplacean operator over L2(R") is the non-positive half-line. Such operators generate nicer semigroups. This motivates the following.
Definition 1.15. Let -7r/2 5 a < 0 < /3 5 a/2 and G denote the union of the open sector { t 6 C \ { 0) ; LZ < arg t < ,f?} with the point { 0) . We consider G as a semigroup with the addition of complex numbers as its operation. A holomorphic semigroup is a representation T ofsuch a G over a sequentially complete locally convex space X such that the function T(.)x is holomorphic (i.e., analytic over C ,cf. Appendix D ) on the open sector for any x ;for any o < a1 < p1 < /3 the restriction ofT to the smaller closed sector {te" ; t 2 0 , 3 E [a',PI] } is continuous; and T is 'of contraction type' in the sense that for a n y a < a1 < p1 < P we can find a real number C with the property that { e-CIzIT(r); arg z E [al,PI] } is equicontinuous.
18
Chapter 1
We mention t h a t if X is a Banach space then t h e last condition is superfluous (cf. the proof of Corollary 1.9).
Proposition 1.16. If we replace the requirement in Definition 1.15 t h a t T ( . ) r be holomorphic for all 3: by the apparently weaker condition that the scalar functions ( u , T ( . ) t ) be holomorphic on the open sector when z belongs to some dense subset of X a n d u belongs to some w* dense subset of X' then we obtain an equivalent definition.
PROOF:Let r be a positively oriented circle in the open sector. Assuming the weaker condition we have
for any z inside r if u and t belong to the corresponding dense sets. Now by t h e dC exists for any z E fi continuity of T(.)t the Riemann integral and provides a holomorphic vector valued function in t h e variable z inside . T h i s function equals T(z)t if t belongs to the dense set above because their difference then yields 0 when tested with a w* dense family of continuous linear functionals. Now fix z and z and tend with a net z, to t such t h a t we know t h e equality for each 2, . Since T ( r )is continuous and the family { T ( ( ' ;) ( E is equicontinuous, we obtain equality for z, too.
& sr
r
r}
Lemma 1.17. Let A be a densely defined linear operator in t h e sequentially complete locally convex space X a n d U c C be an open region such that for a n y X E U the operator X I - A is injective a n d surjective; write R(X) = ( X I - A)-' . Assume further that the family {R(X); A E V } is locally equicontinuous. Then t h e function ( X , t ) -+ R(X)z is (jointly) continuous, holomorphic with respect to X and
(1-21)
(&)k
R(X)t = (-l)kb! R(X)k+lt
PROOF:Immediate calculations yield the 'resolvent identity' (1-22) whenever X # 1.1 in U . T h e local equicontinuity implies t h a t if I and X is fixed and p varies on a compact in U then t h e set of the points R(p)R(X)t remains bounded, which implies by (1-22) t h a t R(X)t is a continuous function of X on U for any fixed z ; and then i t is also jointly continuous in (X,z) by the equicontinuity (cf. Proposition C.17). Then use (1-22) again to infer t h a t this function is differentiable with respect to X and its derivative equals - R ( X ) 2 z . T h e n t h e higher
19
The Hille-Yofida Theory
derivatives can be calculated by the Leibniz Rule because of the equicontinuity (cf. Proposition C.17) and we obtain (1-21) (for Banach spaces all this can be seen more directly, see Exercise 1.6). Lemma 1.18. Let the assumptions of the previous lemma be satisfied and suppose that U contains a closed region Z of the form Z= { c reio ; r E R+ , I9 E [y, 61) where c > 0 , -a < y < -a/2 and a/2 < 6 < a ; and assume that { XR( X) ;X E Z ) is equicontinuous. For any I E X and t E C \ (0) satisfying a/2 - 6 < arg t < -y - a/2 denote by T ( z ) z the Riernann integral of the function R(X)I along the boundary of Z with respect to the finite complex measure &e"dX, or equivalently
+
r
ecz
T ( z ) z=
g
1
02
exp(i6
+ rsei6)R(c+ s e i 6 ) t - exp(iy + zseiY)R(c+ s e i 7 ) z ds
Then T ( . ) z is holornorphic, its derivative equals AT(.).; if I E D ( A ) then l i m z + ~ T ( t ) t = 2: ; if J is a compact subset of (a/2 - 6 , -y - a/2) then { e - c l z l T ( r ); arg z E J } is equicontinuous.
PROOF:We can differentiate under the integral sign because the difference quotient of e z x tends to its derivative in the L' norm at any t from our sector and R(X)z is bounded. From ( X I - A ) R ( X ) t = t we obtain XR(X)z = 2: A R ( X ) z ; substitute the right hand side into the derivative. Now X I - A is closed because it is the inverse of an everywhere defined continuous operator and hence A is closed, too. Since a closed operator is interchangeable with Riemann integration if both integrals exist (see Proposition C.11), the formula [ T ( . ) t ]=' AT(.). follows from the fact that the numerical integral e z x dX vanishes. If z = te" with 2 > 0 and 9 E J then -3a/2 E 5 I9 y 5 -a/2 - E and a/2 + E 5 d + 6 5 3a/2 - E with a small positive E which depends on J , hence with K = sin& > 0 we have (for b 2 0 )
+
sr
+
+
Re(tX) 5 bt - ritr
if lies on the translation rb of f by b - c (so b is the vertex of rb) and r = Ih - bl . The operators XR(X) are equicontinuous on r , 1/X is bounded there and ect
s r e x p ( c t - ~ t r ) d r= - , thus the difficulty arises when t + 0 . So far we K2 have not used the assumption that A has anything to do with the region right of f. We know by Lemma 1.17 that R ( . ) t is holomorphic on U and it is bounded on 2 , therefore we can change the path of integration from r t o fb if b 2 c . b with a positive constant e (independent of b ) , It is clear that 1x1 2 eb on r and then the fact that the operators XR(X) are equicontinuous on Z implies that { h ( l t J ) - ' T ( z ); a r g z E J ) is equicontinuous whenever h is a function of the form
Chapter 1
20
ebt where b 2 c may vary with 2 . Clearly, h = - which is constant if we write tcbt b = 1/t (but we can do that only for t 5 1/c ). 0 . It follows at once from complex Finally, examine the limit when z 2ai(T(z)z- z) = calculus that X - l e " dX = 1 , and hence e"'[R(X)x - X-'z] dX . We have R(X)z- X - l z = A-'AR(X)z and if z E D ( A ) then it equals X - 2 ( X R ( X ) A z ) . Since e z x X - 2 tends to X-' in the L' norm when XR(X)AzdX . z 0 in the sector, we obtain lim,,o 2xi(T(z)z- z) = In this last integral we can change the path and then the boundedness of the set { X R ( X ) A z ;X E Z } implies that this integral vanishes. Thus limz+o T ( z ) z= z for 2 E D ( A ) .
sr
s,,
-+
sr
-+
Theorem 1.19. Let T be a holomorphic semigroup with limits a , @for the sector and for any 21 E (alp) write To(t)= T(eiot) and denote the generator of this one-parameter semigroup by A8 . Then we have A0 = e"A0 ; therefore A := A0 is called the generator of T . A densely defined operator A is the generator of a holomorphic semigroup on a sector with limits a , /3 if and only if the following holds. For any small E > 0 we can find a real number C such that the 'U-domain'
U, = { z E C ; 321 E [ a + & , @ - &: ] Re(zeio) > C } (cf. Notation E.21) satisfies the condition t h a t the operators
(1-23)
T , ( X ) ~ ( X I - A ) -,~ X E U, , k € N
are equicon tinuous and everywhere defined, where r, ( A ) =
max
Re(e" A ) - C
oE[otc,P-~l
(if C 2 0 then r E ( X ) also equals the distance of X from the complement of the U-domain U, ). If we write k = 1 instead of k E N in (1-23) then the equivalence remains valid.
PROOF: Let T be a holomorphic semigroup with limits a , @ for the angle. Fix a 21 E (alp) and let C be a non-negative number such that { e - C l z l T ( z ) ; arg z E J } is equicontinuous where J is the segment with endpoints 0 and 21. Choose a positive number s such that d = Re(eiffs)- C > 0 . Then any continuous seminorm of the integral of the function f(z) = e - " T ( z ) z over the circular arc around 0 with radius r between the angles 0 and 21 is estimated by ICre-dr where the constant K depends on 2 and on the seminorm. Hence if these are fixed then this estimate tends to 0 when r +co or r -, O+ . with But f is holomorphic on the open sector, so we obtain T0((ps)= CTo(((pcS) C = eio , where cp'(t) = e-" as in Lemma 1 . 7 . Then this lemma implies that ( s l - A)-' = C . (CsZ - A o ) - l and hence sl - A = sl - ( ' - ' A s and A# = CA . -+
The Hille- Yosida Theory
21
Now when estimating ( X I - A)-’ we can change the angle of X because X I - A = e-’” (e’”XZ - Ad) and we have uniform estimates for the one-parameter semigroups T d when d varies in a closed subinterval of (a,P). Thus one direction is proved. Assume now that A satisfies the weaker variant of conditions, i.e., just with k = 1 in (1-23). The crucial difference from the case of ordinary semigroups (discussed in Theorem 1.8) is that the path of integration in the inverse Laplace transform can now be deformed into an angle to the left. So assume the equicontinuity of rc(X)(XI - A ) - ’ on some U = U, with vertex C and choose c and e such
>
ovc
+ +
e > 0 , 7 = -p - x/2 & e < -n/2 and COS(Q E ) A C O S (-~E ) ’ 6 = -a x/2 - E - Q > x / 2 . First check that the assumptions of Lemma 1.18 are satisfied. This is done if we show that IXl/rc(X) is bounded on Z . Any X E 2 is of the form X = c reiW with r E R+ and w E [y,6] , thus if 19 is the point of [a E , P - E ] which is closest to -w then Id W I 5 x / 2 - e , and consequently r,(X) >_ ccosd rsin e - C . Since r/lXI + 1 when r + 00 , we have liminf,,, r,(X)/lXl 2 s i n e in Z , while l/r, is bounded on 2 . that c
+
+
+
+
+
+
Thus we get a holomorphic function T ( . ) z for any t on the open sector between the angles x/2 - 6 = a + & @ and -y - x/2 = 0- & - Q from Lemma 1.18 which is hopefully the restriction of the desired semigroup. We first prove that this T is a holomorphic semigroup (setting, of course, T(0)= I ). Since D ( A ) is dense in X , the results of Lemma 1.18 imply a t once the continuity and equicontinuity assumptions of Definition 1.15. If z , w are two points in the open sector and t E X then set f ( s ) = T ( z ( 1 - s ) w ) T ( s w ) x for s E [0,1] . This is a continuous function whose derivative in ( 0 , l ) can be calculated by the Leibniz Rule (cf. Prop* sition C.17) and equals wT(z+(1-s)w)AT(sw)t-wAT(z+(1-s)w)T(sw)t by Lemma 1.18. Now A is closed (because ( X I - A ) - ’ is continuous for some A’s) and interchangeable with any R(X), while T ( ( )can be interpreted as a Riemann integral, thus T ( C ) A c A T ( ( ) for any C and the derivative above identically vanishes. Therefore f(0) = f ( 1 ) , i.e., T turns addition into multiplication.
+
+
Now check that the generator H of T equals A . Lemma 1.18 shows that H T ( t ) t = A T ( t ) t if t > 0 and t is arbitrary. We have seen that T ( 1 ) A c AT(1) and we know that T(t)H c HT(t) . Tending with t to 0 and using the fact both H and A are closed we obtain A = H . Since the generator of a holomorphic semigroup determines it (by the equality Ad = ei“A and by Theorem 1.5), a different choice of E , c and e yields the same T on the intersection of their domains (the same can also be proved by changing paths in the construcion described in Lemma 1.18). Thus the proof is completed by tending with E and e to 0.
We close this chapter by writing down the Cauchy formulas for a holomorphic semigroup. Observe that [T(.)t](’) = ACT(.)t on the open sector for any z and for all k (this can be shown by induction on k because A T ( z ) z = $ [ T ( z l ) t ] l , = , =
+
22
Chapter 1
T ( z - 20) -$[T(zo + 1 ) 1 ] l ~ = ~ = T ( z - z o ) A T ( z o ) e). Thus we have k! (dz)k T ( z ) I= A k T ( z ) x= 2 ~
(1-24)
I(< ir
-~)-~-~T(<)rd(
r
where e and k are arbitrary, z is in the open sector and is any closed rectifiable curve in the open sector whose encircling number around z equals 1.
EXERCISES Let A be the generator of some one-parameter semigroup T and be a continuous function such that for any functional u E D ( A ’ ) the complex valued function (u,f( .)) is differentiable and its derivative equals (A’u, f(.)) . Prove that in this case f ( 1 ) = T ( t ) f ( O )for any 1 . (Hint: use (1-4) and the fact that S t A c ASt ; show that % S t f ( s ) = IT14f;irf(s) .) 1.1.
f : [O,a]
H
X
1.2. Let X be the dual of a Banach space considered with the weak* topology. Show that X is sequentially complete and any one-parameter semigroup (or, indeed, any separately continuous representation of R+ ) over fi is weakly exponential. (Hint: use the Banach-Steinhaus Theorem thrice.) 1,2.(a) Prove that this X satisfies the extra condition in Theorem 1.14 by checking that w* closed subspaces are the duals of suitable factor spaces.
and
I
1.3. Let A be an operator in a sequentially complete locally convex space E d ( s , A ) . Show that
c m
f(Z1X)
=
k=O
X
zk
-A k z k!
defines a holomorpliic function on the open disc { z E C ; IzI < s } for any such I and f’(., I ) = f(.,A I ) there; furthermore, the points (f,f‘) lie in the closure of the graph of A for these z’s and 2’s. 1.4. Let X = L2(R) and A be the operator defined by Af ( 2 ) = t . f ( t ) (its domain consists of those f’s for which both f and t f are square integrable). Show that A is closed, Z ( A ) is dense and A is not a generator. More generally, show that any normal operator over any Hilbert space has a dense set of entire vectors.
1.5. Let X = CO((0, + m ) ) and T be the one-parameter semigroup of translations to the right, i.e., (1-25)
23
The Hille-Yosida Theory Show that the generator of T has no analytic vectors other than 0.
1.6. Let X be a Banach space, A a linear operator in it and X E C such that ( X I - A)-' is an everywhere defined bounded linear operator. Show that (pZ - A ) - ' is the same when Ip - XI < liminf,,, [ [ ( X I - A)-nll-l'n and
c(X 00
(pZ
- A)-' =
- p)'(XI
- A)-k-'
.
k=O
Note that the liminf above is actually a limit and also a supremum as it is easy t o check.
1.7. Assume that we know (1-13) for all real X > K with some Ii' 2 C but do not know that these bounded operators are everywhere defined. Prove that if X I - A is surjective for just one X > Ii' then for any; consequently, A is a generator then. 1.8. Let -7r/2 < a < 0 < p < 7r/2 and A be an operator in a sequentially complete locally convex space X such that the operators (1x1 -C)k(XI - A ) - k are everywhere defined and equicontinuous when k varies in N and X runs through two sequences tending to co on the two half-lines emanating from 0 a t the angles and -a ( C is some real constant). Show that A generates a holomorphic semigroup on the sector between the angles Q and 0 . This representation extends to the closed sector between these angles continuously.
-a
1.9. Let e be a non-increasing positive function on R+ , p = p.X where X is the Lebesgue measure on R+ and X = L2(R+,p ) . Let T ( t )be the translation by t to the right, i.e., it is defined by (1-25). Check that T is a one-parameter semigroup of contractions. Show that if e decreases rapidly enough then WT = --oo (e.g., e(t) = e - t ' will do). Note we can also consider a e which is just positive on a finite interval and then T ( t )= 0 for large 2's. The above example shows that WT = --oo is possible with never vanishing semigroups too.
t,+l/t,
1.10. Show that if a positive sequence t , 2 p > I for all n ) then
' 1
+ $00
I
"rapidly enough" (e.g., if
r
for all r > 0 . Using this result prove thzt if A is an unbounded normal operator in a Hilbert space then d ( s , A ) # d ( t , A ) for 0 < s < t .
This Page Intentionally Left Blank
25 2. Convolution and Regularization
Our aim in this chapter is to develop the concept of convolution of dist-ibutions on a general Lie group G . The usefulness of this concept lies in the fact that convolution from the right (left) is interchangeable with right (left) invariant differentiation. This makes it possible to use 'regularizations' and 'fundamental solutions'. The unfortunate thing is that the convolution of any two distributions can not reasonably be defined. Heuristically, the problematical case is when the two distributions are "not small enough a t infinity in nearly opposite direction". Conceptually, convolution is the way we compute how the sum of independent random variables is distributed (but readers unfamiliar with probability theory should not be frightened by this); namely,
where f and g are the densities of the two independent random variables, f * g is the density of t,heir sum, and we also have analogous formulas for random variables having no density-function and for more than two variables. First we define the version of convolution which is closest to this heuristic concept (of course, in the Lie group G the multiplication of the group replaces the addition of real numbers). In the course of 2.1-2.3 below C can also be any locally compact group.
Definition 2.1. Let p1, . . . , p k be finite complex Radon measures on G . Then their convolution (in this order) is the following measure u: (2-1)
(vjf) =
Jf(.l
" ' ~ k ) d ~ l ( ~ l ) . . . d ~ k ( ~for k )all
f E C,(G) .
Here the integration is meant with respect to the complex measure which arises by multiplying the product of the absolute values of the measures with the corresponding functions of absolute value 1 . So it can be considered (by Fubini's theorem) as a successive integration. Hence v is a bounded linear functional on (Cc(G),ll.llC0) with
and therefore it can be considered as a finite complex Radon measure on G . We mean the above definition in this sense and use the notation u
=
kp2*"'*pk
.
We assert this convolution is associative. If the integral is considered successively then pi * p 2 *. . ' * p k + i = ( p i * p a * . . ' * p k ) * p k + l , while the formula p1 *p2*p3 =
26
Chapter 2
*
*
p1 ( p 2 p3) follows from (the several variable version of) Fubini’s theorem. T h e commutativity of this convolution is equivalent to the commutativity of the group. We mention that one can define the convolution without invoking Fubini’s theorem but using instead that the successive integrations yield bounded continuous functions of the remaining variables. This also holds iff is just bounded continuous. For such f (2-1)extends because the integration on the right can be restricted t o A1 x . . . x Ak where each A, is a countable union of compacts, and then A l . . . Ak is also such in G . Hence associativity can be proved. It would be possible to consider non-finite measures but then some complicated assumptions should be imposed on them to ensure the correctness of Definition 2.1. Before passing t o more general definitions, write down the “functionversion” of Definition 2.1 .
Proposition 2.2. I f p is a left Haar measure on G and p1, p2 are absolutely continuous finite measures (with respect to p ) with p, = fjp then p1 * p 2 = gp with
(2-3)
where we integrate with respect to p , and A is the modular function of G (the integrals may not exist for a negligible set of z ’s).
PROOF:Replacing f1, f2 by Baire measurable functions does not affect the integrals here. Then for ‘p E C,(G) the function fi(y)f2(x)cp(yz) is integrable with respect to p x p and vanishes outside a countable union of compacts. Thus a n application of Fubini’s theorem and the left invariance of p yields the second line in (2-3). T h e equality of the four formulas follows from the assumption that p is a left IIaar measure. Remark 2.3. The equality of the four formulas in (2-3) is clearly valid for any pair f l , f2 of measurable functions (meaning that the integrals exist for the same x’s). The resulting function g is then defined just on a small subset perhaps; nonetheless it could be considered as the convolution of such a pair. This “convolution operation” has the property (2-4 )
((f * 9) * h ) ( x ) = (f * (9* h ) ) ( z )
whenever f ( + t ) g ( t - ’ s ) h ( s - ’ ) as a function of ( t , s) is integrable and vanishes outside a countable union of compacts.
Convolution and Regularization
27
Note that the identification of (locally integrable) functions with measures depends on the choice of the Haar measure, and hence the concept of convolution of functions (and consequently of generalized functions, i.e., distributions, too) must depend on it. From now on in this chapter we fix a left Haar measure on G and denote it by p . Formula (2-3) suggests a possible convolution of a distribution and a compactly supported smooth function. We shall formulate it in Definition 2.5 below. The difficulty with generalizing Definition 2.1 to distributions essentially is that f ( z y ) is not in general compactly supported when f is. Therefore we must tend to it with compactly supported functions in a suitable sense, and the question is what is the "most suitable". First we give a family of 'convolutions' in Definition 2.6 below, and then try to analyze the relations between the various convolutions.
Notation 2.4. I f f is any complex function on G then let f(z) = f(2-l)
(2-5)
and
f'(t)
,
ft(z)= f ( t - ' ) A ( t - ' )
=f+(z) .
If f l g are such that f g is integrable then clearly
Thus we can extend the involutions of (2-5) to distributions by setting (2-5a)
( G , p . p ) = ( u l cp+ Y)
7
(u+ , c p . P ) = (u,G . 4
?
(u', 9 . P ) = ( u , F C l ) .
It is also clear that if h = f * g in the sense of (2-3) then g*f = h , g t * f t and g' * f' = h' .
= ht
Definition 2.5. Let u E D'(G) and cp E Cp(G) . Then we define u*cp and cp * u to be functions, namely
We can write this out as
Of course, the above definition is in accordance with (2-3) when u is a locally integrable function. If t runs in a compact then cp(y-'z) and cp(ty-')A(y-') can be replaced by compactly supported smooth functions on G x G and hence we
Chapter 2
28
infer by Theorem E.9 that u * ‘p and ’p * u are smooth. There are versions of Definition 2.5 when ‘p is taken from some kind of Schwartz-space and u from the dual of that space. Now we shall formulate a very general notion for the convolution. Formally we should write (. * v 7 ’p . P ) = (uz x v y , ‘ p ( Z Y ) . ( P x P I ) but cp(zy) is not compactly supported in general, so the above has no sense for general u , v . But if we multiply ‘p(zy) by a compactly supported smooth function which “almost, equals 1” then we obtain an ‘approximate convolution’. Observe that if E E C r ( G x G ) then the mapping
is continuous from D(C)t o ’D(GxC) . These considerations motivate the following.
Definition 2.6. Let u , v E ’D’(G) and N be a non-empty set of nets from C r ( G x G ) . If (tn)E N then
defines a net of distributions. If these nets all converge (weakly) and, moreover, to the same distribution then we say that the convolution of u and v exists with respect to N ,and denote this limit by u *M v . Of course, this definition really yields a “convolution” only if N is a set of nets “converging to 1 in some sense”. If N is too large (the convergence is too weak) then too few convolutions will exist, but this convolution would have strong properties. If we consider some small N , we obtain a convolution with broader domain but weaker properties. A relatively large N is the following:
(2-8)
No := {
(en)
;
SUP n I, J
I
< 00 and
en
-+
1 locally uniformly }
.
Another candidate: (2-9)
Ni
:= {
(en)
;
tn
+
1 in C”(C x G) )
where C”(G x G) is endowed with the usual topology, i.e., locally uniform convergence of all derivatives. It is easy to check that the convolution of Definition 2.1 always exists with respect to NOand is the same. An application of Fubini’s theorem shows that if f 1 , f i are locally integrable functions with the properties that one of them vanishes outside some a-finite set and I f 1 I * If21 (in the sense of (2-3) ) is locally integrable then f l * f 2 exists and the same with respect to No as according to (2-3).
29
Convolution and Regularization
On the other hand, if u , v are distributions and one of them is compactly supported then for any y? E CF(G)
can be interpreted as
with any E which equals 1 on a large enough compact (the latter depends on 9). Now it is not hard to see that the convolution of such u and u exists with respect to Nl and can be computed by (2-10). Using Theorem E.9 we can now infer that Definition 2.5 is also a special case of the ‘convolution with respect to Nl’. So it is natural to seek for a “good” N inside NOnNl . We should obtain further motivation from the following lemma which is of central importance in this chapter . Lemma 2.7. Assume n/ 3 M are non-empty sets of nets such that for each left invariant vector field V we have
Then u *N v = w implies u *M Vv = Vw for any left invariant vector field V. The analogous statement, when right invariant vector fields are used in the first variable, is also true.
PROOF:If V is a left invariant vector field then
for any smooth (, cp. On the other hand, if u *cn v + w and u Qn = (n VY& then the difference tends to 0, and hence
+
*In
v
+
w with
Since Vt = -V+constant (see (F-21))l we obtain the result for the left invariant case. The right invariant case is quite similar, but then V+ = -V . Corollary 2.8. I f N satisfies the assumptions of Lemma 2.7 with M = N for both left and right invariant vector fields then for any left invariant differential operator L and right invariant differential operator R (2-11)
u *N 2) = w implies RU *N LV = RLw = LRw
.
30
Chapter 2
Note that Nl satisfies this condition. Of course, one should check that N1 is not empty, but we know from Proposition E.3 that for any compact K in a manifold there are smooth compactly supported functions which equal 1 on K . We are led to think that an optimal N should be a largest possible part of to which Corollary 2.8 is applicable. Therefore set
NOnN1 (2-12)
N2
:= { (En) ; RzLy€n(z,y)
+
RzLy(1) V R , L 1
where R ( L ) varies in the set of right (left) invariant differential operators and the ‘+’ means locally uniform convergence under a bound which may depend on R, L and on the net (t,).It is not quite trivial that N2 is not empty, but it follows from the following lemma which is important in its own right.
Lemma 2.9. There is a net en in C Y ( G ) with the following properties: Vli’ compact in G 3no : pnlK = 1 for n 2 no ;
05en51;
l l L ~ , l l ~< +m for any left invariant differential operator L . There is another net with analogous properties writing ‘right’ instead of ‘left’ in the third line. These nets can be sequences if G consists of an a t most denumerable number of connected components.
SUP,
PROOF:Let cp E CY(G) be such that cp 2 0 and S p d p = 1 . If Ii‘ is and cp * 1~ (defined by 2.5 a compact in G then consider the functions 1~ or equivalently by (2-3)). When K runs over a suitable net (sequence) then we obtain the desired Q, (from the first construction the left invariant case and from the second one the right invariant case). To check this we use Corollary 2.8 with Ni .
*+
Corollary 2.10.
N2 is not empty.
PROOF: Let Q,, 29, respectively. Then the net belongs to
N2.
be nets satisfying Lemma 2.9 with ‘left’ and ‘right’, t n , m ( z , Y) = d m ( z ) . en(Y)
From now on by ‘convolution’ we mean ‘convolution with respect to N2’ if not explicitly stated otherwise, and we shall denote it simply by ‘k’. Of course, all the other convolutions we studied, i.e., Definitions 2.1,2.5 and the convolutions with respect to NO and n/l are included in the domain of this convolution with the same result, and Corollary 2.8 applies to this convolution. Unfortunately, it is not associative in general. But it is associative for triples two of which are “small enough near infinity”. We obtain by immediate calculation that
(2-13)
(I(. *.$.) *tl WI , cp) ([. *.$ (. *tl W ) l l c p )
= (ur x = (% x
vy vy
x wz x wz
1
€(XI
Y ) V ( z Y l Z)cp(ZYZ) . ( P x P x P I ) z)cp(zyz) . ( P x P x P ) )
, €(z,Y Z ) V ( Y I
1
.
Con vol u tion and Regularization
31
Now if a , a in V ' ( G ) (i.e., weakly) then a , x 6 -+ a x 6 and 6 x a, + 6 x a (see the remarks after Theorem E.9), so if ( u * v ) * w and u * ( u * w ) exist then they can be computed by taking limits in i$first and in q after that for the first line in (2-13) and in the opposite way for the second one. Hence to check associativity for some fixed triple one can try to find nets (t,),(77,) such that the corresponding right hand sides of (2-13) tend jointly in n , m to the same number (different nets may be used for different rp's). We leave to the interested reader t o toy with this, but we mention that if a t least two of u , u , w are compactly supported then it is trivial to check associativity this way. 11 is frequently useful to "mollify" distributions, i.e., to replace them with smooth or even compactly supported smooth functions which are close enough to them. We can obtain such functions by taking the convolution of the distribution in question with a compactly supported smooth function which "almost equals Dirac's delta" and then perhaps multiply the result by en from Lemma 2 . 9 . This is especially fruitful for those groups where we can construct very nice mollifying factors. 13ut now we turn to the question of what can be said in general. We say that ( c p , ) is an approximation of Dirac's delta if it is a net in C,"(G) with the following properties: -+
where the last condition means that any neighborhood of 1 , the unit element of G, contains these supports eventually, i.e., for large enough n. Of course, it is possible to construct such nets and even sequences for any G.
Theorem 2.11 (Regularization Theorem). Let (rp,) be an approxirnation of Dirac's delta, u E V ' ( G ) and u, = rp, * u , tr, = u * rp, . Then u, and u, tend to u in the following senses: weakly for any u, in C"(G) if u E C"(G) , in C,"(G) if u E C,"(G), in Co(G) if u E Co(G) , in L P ( G , p ) if u E L p ( G , p ) and 1 5 p < 00 , in Lyoc(G,p) if u E Lfb,(G,p) and 1 5 p < 00 . PROOF: Let a , = S c p , ( z ) d p ( t ) and 6, = Srp,,(~)A(z-~)dpL(z) . It follows from (rp,)'s being an approximation of Dirac's delta that a,, 6, -+ 1 . If u is a locally integrable function and E C,"(G) then we know that u * and 11, * u can be computed by (2-3). Thus we have then
+
$J
32
Chapter 2
Assume first that u E Cc(G) and supp u = K neighborhood U of 1 in G such that
)(.I
- .(.)I
<E
if
2 - l ~E
. Then for any
u
or zz-l E
E
u
>0
we can find a
.
Hence it follows from (2-15) that un and v, tend to u uniformly in this case. On the other hand, their supports will be in a fixed compact neighborhood of A' for large n. Of course, u, and v, are smooth functions for any distribution u (see Definition 2.5). Now if L is a left invariant differential operator then Lu, = (P, * Lu and hence we obtain the statement for u E CF(G) for u, , and similarly for vn using right invariant differential operators. This implies the statement for u E C"(G) because if we consider u, and v, just locally then u can be replaced by any other distribution which is the same locally. Similarly can we pass from LP to Lyoc . To verify the statement for the Go- and LP-cases, observe that C F ( G ) is dense in all these spaces (see Appendix C) and its topology is stronger than their respective (linear) topologies. Therefore it is enough to prove that the linear operators u H u, and u H v, are equicontinuous on these spaces a t least for n 2 no with some no (see Proposition C.17). But this is a special case of a version of the Hausdorff-Young Inequality with p = r or q = r . We shall prove the Hausdorff-Young Inequality below as Theorem 2.12. It remains to prove the statement about general u . Observe that the nets @n and ( p i (at least after some no) also satisfy (2-14) and hence **Pn+$
,
P:*$+$
in C F ( G ) for any $J E G F ( G ) . Therefore we can apply u (more precisely, the functional ( u , . p ) ) to these convergences, and obtain our statement from the formulas (2-16) ( ( P * u , $.,u) = (%((P+ *$).,u) and ("*'PI $.,u) = ( u , (II.9I.p) . These are consequences of the associative property which is applicable because two of the three factors are compactly supported. Namely, these numbers equal ($+ *cp*u)(l) and (u*cp*$)(l), respectively (use Definition 2.5 and observe that a distribution which is a continuous function can be evaluated a t 1).
Theorem 2.12 (Hausdorff-Young Inequality). Let 1 5 p , q 5 +oo such that 2 1 (where = 0 ) and $ = + - 1 . Assume that f E LP(G,,u), g E L Q ( G , p )and i f p > 1 then also assume that g E Lq(G,p) . Then f * g exists in the sense of (2-3) almost everywhere, it is ,u-measurable and
+
(2-17)
&
Ilf*sll?. L Ilfllp . (IIYll,)~ . (Ilsll,)
l-!
where we omit the third factor i f p = 1 (and so q = r ). I f , on the other hand, q < r = $00 then the second factor can be omitted and the assumption g E Lq(G,,u) be dropped.
33
Convolution and Regularization
If r = +CQ then (2-3) yields a continuous function. If, in addition, p and q are finite then this function vanishes a t infinity. If p, q > 1 , r < +m and none o f f , g is 0 then we have equality in (2-1 7) if and only if there are: elements s,t E G , a compact open subgroup H of G and a continuous homomorphism c : H t* T such that f ( s r t ) = constant. c ( z ) and g(t-'z) = constant .c(z) for almost every z E H and f , g vanish (almost everywhere) outside sHt, t-'H , respectively. In particular, if G has no such subgroups then there are no such f , g . These statements are valid for any locally compact group G, not just for Lie groups.
PROOF:First we prove the special case when p = q = r = 1 . Switching to Baire measurable f , g we have by Fubini's theorem
so we also see that for non-negative f , g equality holds. Note that the pmeasurability of f *g follows from Fubini's theorem in this case. If r = +m , i.e., if P1 + 1Q = 1 then we obtain from Holder's inequality that
-
In this case a t least one of p and q is finite. Since Cc(G) is dense in LP(G) for finite p, and l f n - flip 0 , Ilgn - 81IQ -, 0 imply Ifn * gn - f * 91 0 uniformly, the statements about continuity and vanishing at infinity follow from the fact that the convolution of a compactly supported continuous function and a locally integrable function is continuous. To check this last assertion, we write
-
Here one of the expressions [.. I tends to 0 uniformly in y when z 2 (depending on which of f , g belongs to C c ( G ) )and a t the same time it is 0 outside some compact. Now if r < +cx, then ;+ > 1 and therefore p and q are finite, too. Let u , v be non-negative measurable functions on an arbitrary measure space, a, b E (0,1] such that 1 < a + b < 2 and c = a + b - 1 . Then applying Holder's inequality first t o the pair of functions ucve , u l - *v '-a with exponents and then to the pair
a, &
Chapter 2
34 with exponents
z,
( (
5
Juavb)'
(2-19)
.
Juv)
we obtain the following:
(J.,,)
(Iu)*
.
(J,,E~E)' % = (/.)
. (Jv)
5 .
(Iu)'-' (Iv) '-' .
where if Q = 1 or b = 1 then the objects with exponent 0 must be omitted. Recall that we have equality in Holder's inequality
J
f1f2
I llflllp, . llf211p2
(fly f 2
2 0 measurable)
fi2
exactly in the following cases: if the number on the left equals +m; if f f ' and differ only by a constant factor; or if one of the exponents, say pl, equals 1 and then f 2 ( i ) = Ilf211, almost everywhere on the set where f1 > 0 . Hence we infer that if a , b < 1 and u , v , u v are integrable then the only cases when
are these: u and v are constant times the characteristic function (almost everywhere) of the same set of finite measure; or uv = 0 almost everywhere. Now apply (2-19) with u(y) = If(zy)IP , v(y) = Ig(y-')IQ and Q = 1 b = 1Q. P' (then c = ) to get
1
Choose a sequence fn E C,(C) such that fn + f in L P ( G , p ) (recall that p is finite now) and apply (2-20) to see that a subsequence of fn * g converges to f * g almost everywhere, for - f l P * 1919 tends to 0 in L ' ( G , p ) by (2-18) and Proposition C.4 can be used. But fn * g is continuous and hence f * g is p-measurable. Applying (2-18) to lflP and 1g1q we obtain the Hausdorff-Young Inequality. We also see that equality holds if and only if we have equality in (2-20) for almost every z . If p , q > 1 and z is such that we have equality in (2-20) and I f 1 * 191(z)E (O,+m) then we must have I f 1 = c l l A , 191 = c21~3-1 almost everywhere with some constants c l , c2 and the symmetric difference A A z B must be negligible. If none of f , g is 0 but equality holds in (2-17) then f * g # 0 . So we can infer (in the case p , q > 1 and r < +m ) the following: If1 = C l l A , 191 = C Z 1 B - l almost everywhere, A , B are Baire measurable sets, 0 < p ( A ) < +CC and for almost every z : ( p ( Afl zB) = 0 or p ( A A z B ) = 0 ) and here the second case must occur for a non-negligible set of z's. Let s be a fixed element of this set, then p(AAsB)= 0 .
Ifn
35
Convolution and Regularization
Set H = { z E G ; p ( B A x B ) = 0) . Since p is left invariant, H is asubgroup. If 3: 4 H then s B A s x B is not negligible and hence A A s x B is not negligible (for " A M s B " ) and therefore p ( A n s z B ) = 0 for almast every such x . Thus we obtain that the function
vanishes almost everywhere outside H . On the other hand, H is exactly the set where h = 1 . Hence h2 - h = 0 almost everywhere but h is continuous and vanishing a t infinity because = p * @ with p = p ( B ) - ' I 2 l g E L 2 ( G , p ) . Therefore It2(.) = h ( t ) for all x and H is open and compact. Consider the (Baire) set {(t,y) E H x B ; z y E B } . It has measure p ( H ) p ( B ) because of the definition of H and since H is compact. Hence if C = {y E G ; j J ( H y \ B ) = 0 ) then
(2-21)
p(B\C)=O
.
If 2 E H then H z y = H y , thus HC = C . Fix a t E C . Then we assert C = H t . Assume xt E C . Then p(Hzt \ B)
+p(zHt \ zB) = 0
and hence
p ( B n t B ) 2 p ( H t t nt H t ) > 0 because H t t n t H t is a neighborhood of zt ( H being an open subgroup). Therefore h ( z ) > 0 and z E H . Thus C = H t , and comparing this with (2-21) and the definition of C we obtain that B A H t is negligible, i.e., B can be replaced by H i and A by s H t . We now have 1 ~ * 1 ~ -=' p ( H 1 ) l s and ~ If*g(z)l = Ifl*lgl (z)= c l c 2 p ( H t ) for almost every t E sH . But f , g E L 2 ( G , p ), so f * g is continuous and the above holds for all 1: E s H . It is enough to show the statement of the theorem if f , g are Baire measurable and If1 = l s ~ ,l Igl = lt-1H . Then the function co := f * g / p ( H t ) maps s H continuously into T . Let d = co(s) and c ( z ) = c o ( s t ) / d for z E H . Note that c(1) = 1 . Write u(t) = f(stt) , v(z) = g(t-'t) for t E H . Since If(xy)l = 1 = Ig(y-')l for ( z , y ) E sH x H t , we have for any z E s H : f(zy)g(y-') = co(2) for almost every y E N t , and writing z = s< , y = q-lt we obtain
(2-22)
V< E 11 :
u(
for almost every q E H
.
Substituting ( = 1 we obtain
(2-23)
u ( z ) = d/v(t-')
for almost all
t
EH
.
The function u((q-l)v(q) - d . c ( < ) is Baire measurable on H x H , so we find ~0 E H such that (2-22) holds for almost all ( with q = 90 . Writing ( = zvo into this and using (2-23) we obtain v ( q o ) / v ( z - ' ) = c(zq0) for almost all z E H
.
36
Chapter 2
This means that v ( q ) = Q / c ( ~ - ~ with ~ o )some constant Q E T almost everywhere on H . Writing this and (2-23) into (2-22) we get that for almost all Q E H the set o f t ’ s for which c(tq-’qo)/c(q-’qo) # c ( ( ) is negligible in H . But these are continuous functions, hence c : H H T is a continuous homomorphism, v = --“c 4%1 and u = fc(90)c. T h e proof is complete. In the remainder of this chapter we are trying to show briefly by examples how useful regularization is. First, as a simplest example, observe that if V is a left invariant vector field and u is a distribution such that u , V u E L P ( G , p ) for some 1 5 p < +a then cp, * u + u and V(cpn * u ) -+ V u in L P ( G , p ) at the same time by Theorem 2.11 and Corollary 2 . 8 . Analogous statement can be obtained for right invariant vector fields. Therefore the distributions in the ‘Sobolev spaces’ can b e mollified in a rather controllable manner. Lemma 2.9 is also useful in this respect. A central problem about distributions is how t o determine whether they are functions. We prove two propositions related to this.
Proposition 2.13. Let 1 < p 5 +cm and u is such a distribution that we can find an approximation cp, of Dirac’s delta with the following property: VA’ compact in G
3C :
sup ll(cpn * u ) . lKllp n
5C
Then u E LY,,,(G, p ) . The same is true if we have u *cpn instead of cp, * u in the formula above.
PROOF:Let U be an open subset in G whose closure is compact. Let B be the closed ball in L P ( U , p ) with center 0 and radius C, where C corresponds to the . Since p > 1 , we have L P ( U , p ) = LQ(Cr,p)*with = 1 and B compact is w*-compact (see Appendix C). Hence if u, = (cp, * u ) l u (or ( u *cpn)lu ) then there is a subnet u,, converging to some v E B in the w*-topology. This implies by Theorem 2.11 that ( u , $ . p ) = (v,$) for any $ E C p ( U ) , i.e., uIu = v . This is enough because G is paracompact (cf. Proposition A.3).
u
+
Proposition 2.14. Let f E L:,,(G) and a E g , the Lie algebra of G. Assume that there is some h E Ly,,,(G) with 1 < p 5 +cm such t h a t
and this ‘lim s u p ’ i s locally uniform, i.e., the supremum on ( I t I < 6) tends uniformly to its limit when 2 varies in a compact and 6 0 . Let A be the left invariant vector field corresponding to Q , i.e., for which A$(1) = ( a , $) . Then A f E Lyoc(G) and lAfl _< h (almost everywhere). The analogous statement for right invariant vector fields is also true. -+
PROOF:Let (cpn) be a sequence of non-negative functions approximating Dirac’s delta. Writing fn = cpn * f we know that Afn = cp, * ( A f ) . On the other
37
Convolution and Regularization hand, since
fn
is smooth, we have
Now if x is fixed then yx varies in a compact, and our assumption implies
We know from Theorem 2.11 that p,, * h + h in LP,,(G, p ) for any finite q which is not greater than p . We now infer by Proposition 2.13 that A f E L;,,(G,p) for some 1 < q < +oo . This implies (using Theorem 2.11 again) that Afn + A f in L;,,(G,p ) , and therefore a subsequence will converge pointwise almost everywhere on a fixed compact (see Proposition C.4). Comparing this with (2-24) we obtain lAfl 5 h . The version about right invariant vector fields is proved in the same way by considering f* cpn .
EXERCISES 2.1. Show that ifN=No,Nl orN2 then (.*NU)-= i r * ~ U , ( U * N V ) + = v+ *N u+ and (u*N v ) * = v* *N u* (meaning also that the existence of one side
implies the existence of the other one). 2.2. Show that the convolution is not associative if G = R" . (Hint: Consider u = 1 , v = a j 6 where 6 is Dirac's delta and seek such w that 1 d j w # 0 . Of course, for such a w the convolution 1 w can not exist.)
*
*
2.3. Let the function u be bounded and uniformly continuous from the right, - u(y)l is small when xy-' is near to 1 . Then show that pn * u + u i.e., I.() uniformly if (pn) is an approximation of Dirac's delta.
+f
2.4. Assume that f* E L P ( G , p ) and g E L q ( G , p ) where 1 = 1 . Show that f * g is a continuous function.
<
p , q and
Suppose f is an everywhere positive integrable function on G. Let 1 < q < +oo and g E L Q ( G , p ). Show that llf*g11, = llflll . llgllp if and only if g is constant. 2.5.
2.6. Check that dL(a)6 = dR(a)+6 for any a in the Lie algebra of G (here 6 is Dirac's delta (6, cp . p ) = p( 1) ; see Appendix F for the other notations). Then show that if u , v are distributions on G and one of them is compactly supported then u * ( d L ( b ) v )= (dR(b)+u)*v for any b in the complexified enveloping algebra.
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39 3. Smooth Vectors We saw in the previous chapter that functions and distributions can be regularized with the help of an approximation of Dirac’s delta. This can be considered as a special case (with substituting the right and the left regular representations, cf. Notation F . 2 , on various spaces) of the following. If T is some representation of a unital semigroup G possessing an invariant measure p and p is a measurable function on G then try to set T(p) = p(l)T(l)d p ( t ) in some sense; if p is nice and T is continuous then we hope that T(p)z “behaves better” than z when we examine the representation, while we also hope that we can tend with very nice 9 ’ s t o Dirac’s delta in such a way that T(p)z -+ I for all I , thereby obtaining a dense set of “well-behaved” vectors. The generalization of I . M . Gelfand’s method (see Proposition 1.13) to Lie groups (as proposed by E. Nelson) is the prime example of this and we shall devote the entire Chapter 4 to it; i n the present chapter concentrale on smooth vectors. We also use this method t o prove the continuity of a representation from weaker assumptions and also to deal with non-continuous representations, especially with dual representations. I n Chapter 1 we did not develop systematically the concept of ‘T(p)’;there we endeavored to reach theorems as soon as possible without burdening ourselves with lemmas of unnecessary generality. The author deems this better for a first experience. Now we try the other method in order to get the maximal value for our efforts and to diminish dull repetition later. Sometimes we form the statements in a more general setting than we ever use them; we do this firstly because the proofs are more transparent if we assume just about what is necessary for them and secondly to record a bit stronger versions for the sake of other applications the readers may find.
s,
Definition 3.1. Let G be a unital semigroup which is also the base set of a measure space (G,S,p) and T be a linear representation of G over the locally convex space X (a representation T is called ‘linear’ if T(1) is linear for all 2 ) . Suppose that I E X is such that the functions ( ~ , T ( . ) are I ) p-measurable when u varies in a w* dense subspace W of X’ . Let p be a p-measurable function. If it is such that p ’( u ,T ( .)I) is p-integrable for all u E W then we say that ‘TO( p, W ) I exists’ and for u E W set
i.e., To(cp,W ) z is the W-weak integral of cp.T(.)t.If this functional can be realized by an element in X then we denote it by T(p, W ) z (the unicity ofsuch an element is ensured by the w* denseness of W ) . If W = X’ then it is omitted from the notation, so To(p)z = To(p,x ’ ) ~ and similarly with T(p) . It is evident that To(p, W ) and T(p, W ) are linear operators defined on some linear subspaces. The reason for taking the trouble with W (instead of considering just the case W = x’ ) is that not all interesting representations are
40
Chapter 3
continuous and then the existence of T(cp)z might be problematical even for nice 9 ’ s . Furthermore, the existence of To(cp,W)z can be established for a certain W while for a smaller other W its actual analysis is perhaps easier. For some arguments w* denseness is not enough, therefore we introduce the following notation.
Notation 3.2. Let W c x’ be a linear subspace. We call a continuous seminorm p on X W-definable if there is a set H c W such that p ( z ) = supvEHI ( v ,x) I for all 2 . We say that W is sufficient if the set of W-definable continuous seminorms is numerous enough to determine the original topology on X and not a strictly coarser one. It follows from the Hahn-Banach Theorem that if T is any locally convex topology satisfying (X,T)’ = W then the sufficiency of W is equivalent to the requirement that we have a neighborhood base a t 0 for the original topology consisting of absolutely convex T-closed sets. In particular, x’ is always sufficient. We obtain a typical example of sufficient subspaces when X is the dual of some other locally convex space with the strong topology and W is this predual as identified with the corresponding set of functionals on X as usual. Since for any W the W-definable seminorms determine a topology which is a t least as fine as the W-weak topology (the latter is obtained from the seminorms lvl when v runs through W ) ,and the W-weak topology is Hausdorff if and only if W is w* dense (by the Hahn-Banach Theorem), a sufficient subspace is necessarily w* dense. If we replace X by its completion then the extensions of the W-definable seminorms are the same as the W-definable seminorms in this new setting. The readers may guess that Notation 3.2 was motivated by the obvious formula
which is valid for any W-definable seminorm p (cf. Appendix C, p. 224). Now collect the basic properties of these “smeared out representations”. T h e notations G , p , X,TIW of Definition 3.1 will be used and just the extra assumptions over those of Definition 3.1 will be mentioned. First we record the consequences of the assumption ‘ p is invariant’ which are too obvious to formulate as lemmas, and then discuss the slightly more complicated observations in Lemmas 3.3-7 . Assume that G has the unique left division property, which means that any equation sz = t has a t most one solution for z . Denoting these unique solutions by s - ’ t we set
(3-2)
f ( s - ’ t ) for t E sc 0 otherwise
for any s and any function f . Now suppose that (G, S,p ) is left invariant, i.e., if A E S then s A E S and p ( s A ) = p ( A ) for any s . Then it is clear that f + f3
41
Smooth Vectors
yields an isometric mapping of L'(G,S , p ) into itself for any s . Now let T, x and W be as in Definition 3.1. If T ( s ) # ( W )c W for some s then we obtain
whenever 'p is such that To(cp,W)x exists. Similarly, if G has the unique right division property and the measure is right invariant then the existence of To('p,W ) T ( s ) t implies T,('pos, W ) I = To('p,W ) T ( S ) Iwhere 'pa is the corresponding translation from the right.
I n Lemmas 3.3-7 below and also later we consider the set
for suitable complex valued p-measurable functions e . Reading through the chapter it beconres clear why. If nothing more is assumed of e then it will go without saying that p is a fixed p-measurable function (we use the letter e only in such context in the present chapter). We shall say that x is gbounded if x E X and B(e);c is a bounded set in X . Clearly, if G is also a locally compact space, p is Radon, eT(.)2:is continuous, I is gbourided and 11, is a p-integrable function then eT(.)Id(11,p) and it in the cornpledon of X we have the Riemann integral realizes To(4e)x.Moreover, if G U {GO} is metrizable then this Riemann integral can be approximated by sequences (not just nets) of Riemannian sums. In most applications T('p)tarises this way, though we shall encounter notable exceptions in the present chapter.
s,
In the following lemma we try to determine what is really needed for the heuristic statement T('p)T(x)= T('p* x) . We discuss after its proof how we can simplify it in most applications. We shall also talk about the measure and distribution versions there. Lemma 3.3. Let
I
be @-boundedand 11, be p-integrable, then To($Je, W)I
exists. Now suppose, in addition, that G h a s the unique left division property, p is left invariant, W is invariant under each T(s)# and p , 'p, x satisfy the following. ' p / p and x / e are integrable (we mean by this that 'p is the product of e and an
integrable function and similarly with x , so e might vanish on a large set), the function d ( s , 1 ) = ' p ( s ) x s ( t ) is ( p x p)-measurable, vanishes outside some u-finite set and l d ( s , t ) / p ( t ) l d p ( s ) is an integrable function of t . Finally, assume that T ( x ,W ) I exists. Then we have
s
42
Chapter 3
PROOF:The first statement is evident (remember that the assumptions of Definition 3.1 are in force, so ( u , T ( . ) z )is p-measurable for u E W ). T h e existence and integrability of w / e follows from Fubini’s theorem. Since we know (3-3) for all s now, what remains to prove is the equality
for any u E W (meaning also that we must show the existence of the left hand side). But this follows again from Fubini’s theorem. Now simplify the conditions for special cases. Perhaps the most annoying assumptions are the measurability and c-finiteness of 29. Let 7 be the family of ( p xp)-measurable parts of a-finite measurable sets (see Appendix B for the relevant definitions). Write R ( s , t ):= (s, s t ) (it is injective) and 29o(s,t) = cp(s)x(t) ; then we have to consider the sets d-’(B) = R(2901(B))for B open in C \ ( 0 ) . From the integrability of cp/e and x / e we infer that 2901(B) E 7 , so it is enough if R ( T ) E 7 for all T E 7 . Such semigroups might be called measurable. It is enough to require R ( T ) E 7 for T from a sub-a-ring whose completion is 7 because of the following. Fubini’s theorem is applicable to the function 1~ if T E 7 , and using the left invariance we can see that if T is negligible and R ( T ) E IT then R ( T ) is also negligible. Then it is not hard to check that if a subsemigroup GI in a measurable semigroup is also a locally p-measurable set (i.e., it intersects measurable sets in p-measurable ones) then it becomes a measurable semigroup with S(n)G, , p . We mention that measurable groups (at least if they are defined slightly narrower) are intimately related t o locally compact groups (see [Hal11). Unfortunately, the application of the above to locally compact groups (and their subsemigroups) is not quite automatic because the product of the Borel structures might be smaller than the Borel structure of the topological product. Lack of a-compactness may cause even worse problems, so we must be careful about “neglecting” (cf. Exercise 3.2). Taking the a-ring of Baire sets for S , we obtain that the a-ring generated by S( x ) S is the family of Baire sets in the locally compact space G x G , hence ( G , S , p ) with a left Haar measure p is a measurable group in the above sense. If we have the integrability of cp/e and x / e just in the sense of Radon measures (cf. Notation B . l ) then we find Baire measurable 91, x1 differing from c p , x on the negligible sets A , B . For any t the set A u tB-’ is negligible, hence w will not change. The condition about the invariance of W is most often disposed of by assuming W = X’ and T ( s ) is continuous for all s . If the function e satisfies le(a)l . l ~ ( b ) l5 A’ . le(ab)l with some positive constant K and we consider a locally compact group with Haar measure on the Baire sets then the integrability assumption for d ( s , t ) / e ( t ) follows from those for c p / ~and x / e by substituting (s, s t ) . If G is a Haar measurable subsemigroup then the same applies, because if any of s and s - ’ t lies outside G then 19 vanishes and so only elc counts. If G is an M1 topological group then it is known that the topology of G can be obtained from a
43
Smooth Vectors
left (also from a right) invariant metric. Denoting by 1.1 the distance from 1 with respect to such a metric it is clear that ~ ( t = ) C?-~I'I satisfies the above with I< = 1 whenever K 2 0 . On the other hand, if G is a connected Lie group, X is a Banach space and T is locally equicontinuous then e - K l t l llT(t)ll is bounded with a suitable K if 1.1 is a left or right invariant Riemannian distance (cf. Corollary 1.9 and (F-19)). Lemma 3.3 was tailored so as to handle semigroups as well as groups. We were also careful in assuming as little as possible of the continuity of the representation. The measure and distribution versions require a different approach. We do not use these versions in this book but they are important. Therefore we say a couple of words about them. If 'p is a measure then one tries to consider (To(cp,W ) z ,u ) = J, ( u , T(t)+)d'p(t) . If 'p is a compactly supported distribution and ( u , T ( . ) z )is smooth then we may consider the corresponding object accordingly. If 'p is not compactly supported but at least "small enough" at infinity then one perhaps obtains a T('p)similarly as one defines improper integrals. In any case if y = T ( x , W ) z exists and W is invariant under each T ( s ) # then (using, a bit unusually, the integral notation also for distributions) ( u ,T(S)Y) =
J ( T W u , W)+) dX(t) = J
(Ul
T ( s t ) z )d X ( t )
Thus tlie analog of Lemma 3.3 comes down to this question: under which conditions do we have
If, e.g., ( u , T ( . ) t )is a bounded continuous function and measures on a locally compact group then this holds.
'p,x
are finite Radon
In Lemmas 3.4 and 3.5 below 2: is p-bounded, $ or any integrable and use the notations 'p = $Q , cp, = $ i e . Lciiiiiia 3.4.
$i
is always p-
Assume that G is also a locally cornpact space, Y, is Radon,
X is quasi-complete and the function eT(.)z is weakly continuous. Then T('p)z exists.
PROOF:Let CI = { t E G ; e ( t ) T ( t ) z# 0 ) (it is an open set because (0) is a closed set for the weak topology), then we can assume that vanishes outside U for otherwise we can replace it by 11, . 1~ and 'pT(.)z remains the same. Since p is a Radon measure and I+!J is integrable, we can find an increasing sequence I<, of compacts in U such that - 1c,. l K , l l l + 0 . First we fix an n and show that T ( X Q ) exists Z for x = 4 . l ~ . ,Let L be the range of pT(.)+ over I(, , then L is compact in the weak topology. Multiplying it by the disc of radius llxlll (around 0 in C ) we obtain another w compact set L1 and then (by the quasi-completeness) the generalization of Krein's theorem (see Theorem C.10) implies that the closed convex hull P of L1 is w compact. Now if u E 3? is arbitrary then
Chapter 3
44
and hence To(xe)zE P (by applying the Hahn-Banach Theorem to the space 2 of all linear functionals on x' , Z is endowed with the w* topology). If the supremum of a seminorm p on the range of pT(.)z is M then we obtain p ( T ( p . l ~ , , ) z- T(p. 1~,)z) 5 M . Il$(lK,, - lK,)II1 ; so we have a Cauchy sequence here and it is easy to check that its limit realizes T ( ' p ) z . Suppose that W is sufficient, a topology is given on G , Lemma 3.5. eT(.)z is continuous a t 1, e(1) = 1 , $Ji is a bounded net from L ' (G, S , p ) such that T($J,e,W ) z exists for any i , $ J , 1 and J&u l$il -+ 0 for each U belonging to a neighborhood base a t 1 (meaning also that there exists a neighborhood base consisting of p-measurable sets). Then T($ie, W)z 2: .
s,
-+
PROOF: Let ai = s,$J, and neighborhood base above then we have P(T($ie, ~
)-z aiz)
b, =
ll$illl
. If U is
I supp( e ( i ) T ( t ) z- 2 ) .
I$i(f)l
t€U
( P ( z ) + wt €pG( e ( t ) T ( l ) z ) )
.
a member of the
+(i)
+
J,,, I$i(i)l ~
l
for any W-definable continuous seminorm p . Fixing such a p the first term is small for small U (independent of i ) by the continuity of pT(.)z a t 1 and by the boundedness of the net bi . The second term is small if i is chosen large enough for a fixed U . Thus the lemma is proved.
Lemma 3.6. Assume that G is a Haar measurable subsemigroup of a locally compact group G1 and p comes from a left Haar measure on G I (also denoted by p). Denote the GI-interior of G by H . Let T (defined on G ) be locally equicontinuous on H (i.e., let any point of H have a neighborhood U such that the family (T(2); 2 E U } is equicontinuous). Let W be suficient and invariant under each T ( i ) #. Finally, suppose that y = T('p,W ) z exists for a certain 'p E C c ( H ) (we extend this 'p to be 0 outside H ). Then T(.)y is continuous. Furthermore, if C1 is a Lie group, 'p is also continuously differentiable and T ( w , W ) z exists for all w E C c ( H ) then T ( . ) y is continuously differentiable (strongly, cf. (C-5)) and can even be extended into a C1 function defined on a neighborhood of G . We have the following rule for computing the derivative. If V is any left invariant vector field on G1 then denoting the corresponding right invariant vector field by V, (cf. ( F - I ) ) we have V[T(.)y] = T(.)T(V,'p,W ) z or, more precisely, if v is the element of the Lie algebra corresponding to V then
whenever s is such that the filterbase for the limit exists, i.e., sexp(hv) E G at least for a sequence of non-zero h's tending to 0 .
)
45
Smooth Vectors
PROOF:Write I< = supp ‘p and fix an s E G , then { (a,b ) E GI x GI ; cab E H } is an open set containing {s} x K (for even GH is open in GI ), therefore we can find a neighborhood L of s in G1 such that L K c H by the usual compactness argument. Choose a compact L , then { T ( t ); 2 E LI< } is equicontinuous for L K is compact in H . The function ( p z can be defined by (3-2) for any z E G1 and if z E L then it vanishes outside LA’ and equals cp(t-’t) for t E L K . Since ( z , 2 ) c p ( t - ’ t ) is continuous on the compact L x I<, we obtain that (pz (ps uniformly when t s in G1 . Thus Ipz - (psI .p(T(.)x) tends uniformly to 0 and vanishes outside a compact if t s in L and p is a continuous seminorm. Our assumptions imply that T(cp,,W)z = T ( z ) y for t E G (because (3-3) can be used) and if p is W-definable then we obtain p ( T(cpz, W)x- T ( p s, W ) E ) 0 . -+
-+
-+
-+
-+
If vergence
‘p
is continuously differentiable then on compact parts of charts the con-
limu,o
-t ).
‘(‘
- ’(‘)
- ”(‘).
1.1
= 0 is uniform in t (we denoted the
images of everything through’a’chart by the same letters). On the other hand, the convergence li mu-,0
.-It
- 2 + a,@(o,t). 1.1
I,
= 0 is also locally uniform in 2 (with
I,
the same notational conventions excepting that the image of 1 is 0 as usual; Q, is the multiplication in the corresponding chart). Hence we obtain that the difference q s exp(hu)
- ‘Ps
quotients tend uniformly to (V,cp), i n LA’ and vanish outside of it h for small h , and this whole business can be done simultaneously for u’s in a compact subset of the Lie algebra. This implies the statement i n the same way as we proved the continuity (the fact that the derivatives along the translated one-parameter subgroups yield a derivative in the sense of (C-5) can easily be checked by using a translated logarithmic chart). I n the following lemma we show that much weakcr conditions on p are enough to ensure the continuity of T(.)T((p, W ) E . For differentiability the crucial problem is that T(.)xmay be “very large” near to co . Accordingly, i n most applications (o is smooth and decreases “very fast” close to 00. Another problem is the boundary of G (cf. Exercise 3 . 4 ) . It is interesting that minimal smoothness conditions are enough. Lemma 3.7. Let G, H , p , T , W be as in Lemma 3.6 but now assume the local equicontinuity on G , not j u s t on H . Also assume t h a t T(w,W ) x exists for all w E C , ( H ) . Let e vanish outside H , be continuous in it (so e is not necessarily continuous on G ) and let x be e-bounded. Then the existence of y = T ( $ e ,W ) x implies the continuity of T(.)y whenever $ is p-integrable. For the differentiability assume the following (together with the above). (i) G1 is a Lie group, its identity 1 belongs to the closure of H , V is a fixed left invariant vector field and V, is the corresponding right in variant vector field. (ii) p satisfies le(a)l 5 C . le(ab)l for a E H and b E J where J is some open subset in H such that 1 is in the closure of J , and C is some positive constant. (iii) V,.($e) = with an integrable $1 (in the sense of distributions on the manifold GI , not j u s t on H ; in particular, we suppose that p = $ e and V,cp are
46
Chapter 3
locally integrable on (GI , p ) ). (iv) z = T(11,le,W ) z exists. (v) The point s in G is such that { h E R ; sexp(hv) E G } is connected near 0 (i.e., either 0 is an interior point of it or it locally coincides with one of the closed half-lines).
PROOF:Since Q vanishes outside H , we can assume the same for 11, without affecting cp = $Q . Then we can find a sequence X , E C c ( H ) tending to 11, in L ' ( H , p ) . Writing w, = xne we obtain continuous functions with support in H for e is continuous on H . Thus we can consider y, = T(w,, W ) z and using W-definable seminorms as above we can see that y, tends to y . Since T(.)y, is continuous by Lemma 3.6 and T is now locally equicontinuous on G , we obtain the continuity of T(.)y . Now turn to the proof of the differentiability. Denote by Z the set of the pairs ( a , b ) in Xx X for which both T ( . ) Qand T(.)bare continuous and V [ T ( . ) a = ] T ( . ) b in the sense required in the statement (at every such s ) . Observe that the last condition (in the presence of the others) is equivalent to the following. ~ (exp(.rv))b s d7-
T(sexp(h2v))a - T(sexp(h1v))a = Jh I
whenever it has sense, i.e., if h l , hz, s are such that sexp(rv) E C for 7- E [ h l ,hz] (the integral is a Riemann integral, though its being a weak integral is enough for us). Then, using the local equicontinuity of T , we can see that 2 is closed. Thus the task is t o approximate (y,z) from Z . Extend e to be 0 outside G . The assumption on e implies that
whenever f and f e are locally integrable on ( G 1 , p ) and a E C p ( J ) (extended t o be 0 outside J ). Here we used the convolution (2-3) which is in accordance with the more general convolutions. Let U = { t ; e ( t ) # 0 ) and Y be the normed subspace of L1(G1,p) consisting of those f's for which f e is locally integrable on ( G 1 , p ) . Then set R,f (2) = [(fe) * a ]( t ) / ~ ( t )for 1 E U and R , f ( t ) = 0 for 2 4 U whenever f E Y and a E C p ( J ). Then (3-5) shows that R, is a bounded linear operator from Y into Y with llRall 5 C . llalll . Let a, be an approximation of Dirac's delta in the sense of (2-14) (such one exists because 1 is in the closure f in Y if of J ), then the operators Ran are uniformly bounded and Ranf f E C c (U) (use Theorem 2.11 and the fact lei varies between positive bounds on compact subsets of U ). Hence we infer that Ramf + f whenever f E Y vanishes outside U (for C c ( U ) is dense in L 1 ( U l p ) ) . We may assume that 11, and t+bl vanish outside U (for otherwise we can multiply them by 1~ without altering 'p and Vrcp). So the above can be applied. On the other hand, (Vrcp)* a = Vr(cp*a) by Corollary 2 . 8 . Observe that if A ' = suppcr
-
47
Smooth Vectors then cp sr a vanishes outside H K . Thus we have a sequence is smooth on G1 and vanishes outside U ,
fn
such that each
fn
and for each n there is a compact K , c H such that f, vanishes outside H K , . Choose a net g , tending to the function 1 “in the right invariant fashion” by Lemma 2 . 9 . Then the function ( g i - 1)j,/p and also the function
is small in L’ for large i, n . Thus we can approximate the pair ($, $1) in Y x Y by pairs of the form Vr(gifn)) . The usual argument with W-definable
($
1
e
seminorrns will complete the proof if we can show that ( T ( g i f,, W ) x ,T ( V r ( g ,f,), W ) z ) E Z . But this follows from Lemma 3.6 by checking that s u p p g i f , c H . If M = s u p p g ; , fn vanishes outside K H and 2 is in the closure of M n ( I C H ) then by the compactness of M and K we find a net ( r n j , k j ) converging to ( t , k ) in M x I\‘ such that k r ’ r n j E H for all j . But k E ’I\ c H and hence there is an open neighborhood S of 1 in G1 such that Sk c H , SkkJT1rnj c H for any j . Now t belongs to these sets whenever t r n f ’ k j k - ’ E S , which certainly holds for large j .
Theorem 3.8. Let G be a Haar measurable unital subsemigroup of a locally compact group GI such that any neighborhood of 1 in G has nonempty interior in G1 . Let T be a locally equicontinuous linear representation of G over a locally convex space X . Assume that for a weakly dense subset of x’s the function T (.)zis continuous in the weak topology on a neighborhood of 1 (the neighborhood may depend on x). Then T must be a continuous representation.
PROOF:We may assume that X is complete, because otherwise we can take its completion and extend T to it by the local equicontinuity. Assume that z satisfies the weak continuity on some neighborhood U of 1. Let V c U be a neighborhood of 1 in G , then its G1-interior is not empty, therefore we can choose a non-negative cp E C,(GI) with integral 1 such that K = suppcp lies in this G1-interior. The condition V c U implies by Lemma 3.4 (choosing e = cp and $ = 1~ ) that T(cp)x exists, and so we can see from Lemma 3.6 that T(.)T(cp)x is a continuous function. T(cp)x can also be considered in the weak topology (as a matter of fact, there it is a Riemann integral) and hence z is in the weak closure of the set of such T(cp)z’s (this follows from Lemma 3.5 with e = 1~ where L is some suitable neighborhood of 1 but also follows from the definition of Riemann integrals). So we have proved that the set of those y’s for which T(.)y is continuous is weakly dense. But this
48
Chapter 3
set is a linear subspace and therefore it is dense in the original topology (by the Hahn-Banach Theorem) and then the local equicontinuity implies the statement (cf. Proposition C.17). There is a number of versions of this theorem with the following common pattern: a t the price of assuming something more about G or X we can considerably weaken the conditions about continuity. The easiest observation is that if G is such that for any s E G and any neighborhood V of 1 in G we can find an n such that { 2 1 . . . t , ; t i E V } is a neighborhood of s then it is enough to know local equicontinuity a t 1 because it implies the same everywhere. But if G is a connected group then this condition is always satisfied, and also holds, e.g., for G = (R+)' . If G = G1 then the weak continuity of T ( . ) z a t 1 implies it everywhere for T ( z ) c - T ( s ) z = T ( s ) ( T ( s - ' t ) z - z) and the continuous operator T ( s ) is continuous in the weak topology, too. Since weak boundedness is the same as boundedness (see, e.g., Corollary 2 to Theorem IV/3.2 of [Scha]), we have the following. If the operators T ( t ) are continuous, T ( . ) z is weakly continuous for any z and X is barrelled then the local equicontinuity follows and consequently T is a continuous representation. Of course, in the group case it is enough to know the weak continuity a t 1. If X is nice enough then continuity can be weakened into measurability. We mention two typical results whose proof can be found, e.g., in MOO^]. If G is a locally compact group and X is separable (i.e., has a denumerable dense subset) then it is enough to know weak measurability (i.e., the measurability of each (u,T(.)3:)) for a weakly dense subset of z's (in the presence of the local equicontinuity) and, moreover, if we know this for just one 3: then T ( . ) z must be continuous (in the originial topology) for that z . If G is a locally compact group and X is a separable Banach space then the weak measurability of T ( . ) z for all z and the continuity of each T ( s ) imply the local equicontinuity and so in this case something much less than separate continuity implies that T is a continuous representation. Proposition 3.9. Let T be a continuous representation of a locally compact unital semigroup (we do not assume any connection between the topological and algebraic structure) over a locally convex space X . Denote by E the semigroup obtained from G by reversing the order of multiplication (so E = G if and only if G is commutative). Then T* is a separately continuous representation of E over (x',w*) and a locally equicontinuous (but generally not continuous) representation over (x',p) (we denote by ,B the strong topology, see Notation C.9). Setting
ZJ = { u E x' ; T * ( . ) uis P-continuous } we obtain a &closed linear subspace. If the operation in G is partially continuous then ZJ is invariant under T' and T' is a continuous representation of E over (ZJ, p) . Fix a Radon measure p on G and let a(@)(see (3-4)) be equicontinuous.
If 11, E L 1 ( G , p ) then T'(11,p)v (with respect to the w* topology) exists for a11 v and the resulting linear operator is (P,P)-continuous. In particular, if 'p E C , ( G )
49
Smooth Vectors
then T*(p) (constructed in the w* topology) is a continuous linear operator over
(x'lP). PROOF: T h e statements that T* is a linear representation of E and it is separately continuous in the W* topology are evident from the definitions. If A is an equicontinuous set of linear operators from X into X with p , q relations for the seminorms and H is a bounded subset of X then p ( S z ) 5 supH q for any S E A and z E H , therefore HI = {Sz ; S E A, z E H } is also bounded. Then pIf(S*v) = supzEH I (v,Sz) I 5 p ~ , ( v ), i.e., the family {s' ; S E A } is equicontinuous with respect to 0. Thus the local P-equicontinuity of T' follows and hence T*(.)vn T * ( . ) v locally uniformly when v, + v in p . So we have the P-closedness of ZJ while its being a linear subspace follows from the linearity of p . The T*-invariance clearly follows even ifjust the operations z z s i n E (i.e., the operations z -+ s z in G ) are continuous for any s . Then the local equicontinuity implies the continuity of the action (cf. Proposition C.17). Write A = a(@) then A*v = { v o S ; S E A } is an equicontinuous set of linear functionals for any II E x'. On the other hand, we saw that A* is a (P,P)-equicontinuous set of operators. T h e first fact implies that T,($Q)IIis continuous (as a linear functional on W = X),so T ' ( $ p ) v exists. T h e second fact then implies the ( P , P)-continuity because p ~ ( u5) supzEIf I ( f (.), z) I whenever the weak integral u o f f (with respect to the w* topology) belongs to x' . -+
-
Observe that the dual of a locally convex space does not change as a linear space when the space itself is replaced by its completion but the w* topology becomes finer. Thus the statements of Proposition 3.9 can slightly be sharpened. Note that any space with its weak topology is the dual of another one, hence (x',w*) does not have any completeness property in general. Therefore the existence of the integrals above follows not from the separate continuity alone but from the continuity of T .
Notation 3.10. Assume that C in Proposition 3.9 is actually a locally compact group. Then t + t-' is an isomorphisrn from C onto E ; we call the corresponding continuous representation of G over (9, the contragredient of T and denote it by T , i.e., T ( t ) v = T * ( t - ' ) v = [ T * ( t ) ] - ' v for v E 9 . We mention that if X is a Hilbert space and T is a unitary representation then T "coincides" with T ; more precisely, ? ( t ) = J T ( t ) J - ' where J is the canonical conjugate linear isometry from X onto x' (i.e., (Jy)z = (z,y) if (., .) denotes the scalar product in the Hilbert space).
a)
Theorem 3.11. Let T be a continuous representation of a locally compact group G over a locally convex space X and fix a right Haar measure p on G (then p is a left Haar measure on the reversed group E ) . Then T*(p,X) (we write X in the expression to emphasize that we also use other topologies on X' than the weakf topology) is a (P,P)-continuous linear operator from A? into 9 (defined in Proposition 3.9) whenever a continuous p and a p-integrable $ can be found such
Chapter 3
50 that
B(Q)is equicontinuous and
'p
= Il,e . In particular, the set
{ T * ( ' p , X ) v ;'p € CC(C),
'u
E x'
1
can be composed and the linear subspace generated by this set is invariant under T and dense in x' with respect to T , where r is the strongest locally convex topology on x' whose dual is the completion of X . If G is a Lie group then consider the subspace of the above generated by the T*('p,X ) v ' s with 'p E C y ( C ) and denote it by G*(T)('dual Girding subspace'). It is also ?-invariant and r-dense in x'. For any v E G*(T) the function ?(.)v is /?-smooth (in the sense of (C-5)) and if v = T*('p,X ) u with 'p E CF(G) then for any left invariant differential operator D we have D[?(.)v] = T(.)T*(D'p, X ) u .
PROOF:The existence and continuity of T*('p,2) was proved in Proposition 3.9. Then Proposition 3.9 and Lemma 3.7 together imply that T*('p,X)v E 9 for any v E X' . If 'p E Cc(G) or 'p E C,"(G) then the same holds for any ps (constructed for the reversed group E ) , so the ?-invariances follow from (3-3). The representation T can be extended continuously to the completion of X and this extension still yields the same T' . Hence we can tend with suitable T * ( ' p , X ) v ' st o v in the 3-weak topology by Lemma 3.5 which is applicable by Proposition 3 . 9 . Then the generated subspaces are also r-dense by the HahnBanach Theorem. Writing f ( t ) = ? ( t ) T * ( ' p , X ) z l with some u E x' and p E C F ( G ) we have by Lemma 3.6 that in the reversed group E (3-6)
( d J , -at) = T(1) T*(d,!,(a)'p, X ) u
for any a from the Lie algebra of E (see (F-1) for the notation d L ) . But in charts N we have analytic functions h,j such that ( - a t ) , = CjZ1 hij(t)aj in the chart ( N is the dimension of G ) , and these functions form invertible matrices a t each t . Taking the transposed inverse of these matrices we obtain another matrix valued analytic function x,j with which (3-6) can be written in the form ( j ' ) i = N x;j f J where fj(t) = ?(t)T'(dL(ej)p, X ) u (the ej's denote the basis of RN used in the chart). Thus the /?-smoothness follows from Lemma 3.6 by induction on the order of differentiations. Since the left hand side of (3-6), as a function of 2 , is nothing else than d L ( a ) f , we also get our formula for vector fields because the right invariant vector fields on E are exactly the left invariant vector fields on G. Obvious calculations show that the set of differential operators D with which D[?(.)T*('p,X > u ] = T(.)T*(Dp, X ) u holds (for a fixed u ) is a complex algebra containing the constants. Since the smallest such algebra containing each left invariant vector field is the set of left invariant differential operators, the theorem is proved.
cj=l
We remark that we can switch from right to left Haar measure by substituting and this is controlled by the modular function of the group (see (F-12)); in fact, if T is anything (not necessarily a representation) to be smeared out and 2
+ 1-'
51
Smooth Vectors
we write Tt(cp,W) and Tr(cp,W) for the corresponding objects constructed with a left Haar measure 1.1 and with the right Haar measure d p ( t - ’ ) , respectively, then we have
for any cp and W . Thus it is useful to calculate the differential operator [ l / A ] D [ A ] for left invariant D’s. If D is a vector field then D [ f ] = [ f ] D [of]for any smooth function f and applying (F-21) we obtain the formula [l/A]dR(a)[A] = d R ( a ) ( a , A ) = -dR(a)+ . Consequently,
+
+
(3-8)
[l/A]dR(b)[A] = dR(6’)’
for any 6 E LL, (now and in the sequel we use Ihe symbol LL, as an abbreviation for the cornplexified enveloping algebra C @R U = LL@ i . U of the Lie algebra g). In the remainder of this chapter G is a fixed Lie group, its Lie algebra is denoted by g and p is a fixed left Haar measure on G. In integrals with respect to this p just, the variable will be indicated (similarly to the common practice with the Lebesgue measure). We leave to the interested reader the possible generalizations to local groups and to certain (maybe local) subsemigroups. This remark also pertains to Theorem 3.11 above. The letter T will denote a continuous linear representation of G over a sequentially complete locally convex space X unless explicitly stated otherwise. Definition 3.12. The tangent dT of T is defined to be its derivative a t 1 along the one-pararneter subgroups; more precisely, d T is a mapping from the Lie algebra g of G into the set ofgenerators (cf. Chapter I ) defined by
d T ( a ) z = lim h-0
T(exp(ha))z - 2: h
We note that in the special case when X is finite dimensional d T is a continuous linear operator from g into B ( X ) and coincides with the tangent of morphisms between Lie groups introduced in Appendix F (identifying the Lie algebra of a matrix Lie group with the corresponding Lie algebra of matrices as usual). In general, the domains of the dT(a)’sare not the whole space and differ from each other, thus just a “degenerated version” of the tangent’s being a Lie morphism could be hoped for. This version is in fact true (see Theorem 3.23 below). A classical theorem of Lie theory asserts that any morphism of Lie algebras from g into B ( C ” ) can be integrated (some say ‘exponentiated’), i.e., a morphism can be found from the simply connected covering group of G into the matrix group whose tangent equals the morphism of Lie algebras in question (cf. Appendix F ) . For infinite dimensional
Chapter 3
52
spaces the situation is considerably more complicated, we shall devote Chapter 5 to its discussion. Since exp(h Ad(t)a) = i(exp(ha)) (this is a special case of (F-4)), we obtain the formula dT(Ad(t)a) = T(t)dT(a)T(t)-'
(3-9)
Proposition 3.13. Proposition 3.9).
PROOF:Evidently, if
d?(a) = -dT(a)*
o is the limit of
n (gx g) (with the notation of
T'(exp ha)u - u in the w* topology h
= dT(a)*u . Now assume that u , v E fi' and v = dT(a)*u . dT(a)T(exp h u ) t = &T(exp h a ) z for a dense set of 3:'s and hence
then
o
We have
d f ( h , z ) = (T*(e xpha )v,x) = - ( u , T ( e x p h a ) t ) dh
so t
which is a continuous function of h . Taking integrals we have f(h,t)dh = ( u , T ( e x p t a) t - t ) for a dense set of 2's. Since T is continuous we have the convergence f ( . , t , ) --+ f ( . , t )locally uniformly if t, 3: . Thus the above holds for any t ; tending with t to 0 and using the fact that f ( . , t ) is continuous T*(expta)u - u in the W*topology. we obtain o = liml,o ---+
t
If, in addition, o E 9 then the corresponding integral means (which exist in the X-weak topology) converge strongly to o because X is sufficient for the strong topology. Thus the proposition is proved. We can see from the proof above that dT(a)* itself can be considered as a generator for T' (though T' is not a continuous representation in general). Observe that the proposition above implies that the domain of the dual of any (densely defined, but that will follow from Proposition 3.14 ) algebraic combination of d T ( a ) ' s contains E*(T) (by Theorem 3.11). Since G'(T) is w* dense, we can see that any such combination is closable.
The 'Girding subspace' is the linear subspace defined by
(the existence of T(cp)z as a Riemann integral follows from the sequential completeness of X for any cp E Cc(G) and even for more general cp's). Lemma 3.5 implies the convergence T(cp,)z + t if cp, is a n approximation of Dirac's delta
53
Smooth Vectors
in the sense of (2-14). Thus the Girding subspace is dense. The properties of the Haar measure and Lemma 3.6 immediately imply the following. Proposition 3.14. The GBrding subspace is invariant under each T ( s ) and d T ( a ) ; in fact, the following formulas hold. Using the involutions f ( t ) = f ( t - ' ) and f t ( t ) = f(t)A(t)-' introduced in (2-5), the notations at = d R ( a ) , a, = & ( a ) (see ( F - I ) ) and writing A = d T ( a ) we have for any cp E CF(G)
((;.+ means first
"
and after it
-t
, i.e., P + ( t ) = cp(t)A(t)-' ).
Note that the last equality can also be written i n the form T(t9)A= T ( a f 1 9 ) (with 29 = p/A ) by (3-8) and by the facts that A is densely defined and the right hand side is a continuous operator. Taking duals we obtain A'T'(I9) C T * ( a t I 9 ) (we use our left Haar measure ,u here, so the notation differs from that of Theorem 3.11). But the left hand side is everywhere defined by Proposition 3.13 and Theorem 3.1 1 . Thus, using also the corresponding other equality from Proposition 3 . 1 4 , we can infer the following important pair of formulas:
B T ( p ) = T(dL(b)cp) ,
(3-11)
cp E C,"(G) , b E tl,
whenever
B*T*(cp)= T'(dR(6)'cp)
and B is any version of " d T ( b ) " , i.e., B =
c:='=, c ; d T ( a ; , l ). . . d T ( a , , k , ) where E C , E g (we may have k; = 0 for b= some i's and an empty product is interpreted as identity) and c:='=, ... . We mention that the second formula for vector fields could also be proved immediately from Theorem 3.11 also the remark after it) by c;
ci
ai,l
ai,j
ai,k,
(see
considering
(T*(cp)u, T ( t ) - ' e ) ( , = ,for
I
E D ( d T ( a ) ).
Proposition 3.15. Let I E X , u E X' , f ( 2 ) = ( u , T ( t ) x ) and B be a version of d T ( b ) as above. Write bt = d R ( b ) and b, = d L ( b ) . If u then we have b: f = ( v , 7 ' ( . ) 1 ) for all I (in the sense of v = distributions). Analogously, if y = ( B ' Is.(T)) * I then 6tf = (21, T(.)y) for all
(
U .
Note that the assumption y = ZI is stronger than the above.
PROOF:Our assertions can be reformulated as follows: ( u , T ( b , c p ) x ) = (v, T(cp)z) and ( T ' ( b f c p ) u , z ) = (T'(cp)u,y) for all cp E CF(G) . So these assertions follow from (3-1 1 ) . Definition 3.16. A vector 3: E X is called C k if the function T ( . ) I is C k from G into X where k = 1 , 2 , 3 , .. . , m , w (Cw is read 'analytic', while 'smooth'
54
Chapter 3
is synonymous with C"). The set of C k vectors form a linear subspace, we denote it by C k ( T ) . Since T ( s t ) x = T ( s ) T ( t ) x , we obtain the T-invariance of each Ck(T) by fixing t and running s , while fixing s and running t on a neighborhood of t o we obtain that the C k property a t t o implies the same a t sto for each s (because t --* s-lt is an analytic function), i.e., a vector is Ck whenever it is Ck a t one point. Using a logarithmic chart we can see that if I is C' then a --* d T ( a ) z is linear (this also follows in the same way even if we just assume that T ( . ) z is differentiable at 1 but that assumption is, in fact, not weaker by Proposition 3.18 below). The Girding subspace is contained in C"(T) (e.g., by Proposition 3.18 below). Consequently, C"(T) is dense in X . It is known that if X is a FrCchet space then G(T) actually coincides with C"(T) and, moreover, if one writes C"(T) instead of X in (3-10) then the resulting (apparently smaller) set still equals C"(T) in this case (see Theorem 3.3 of [D-M] ; it also follows from the results of [D-M] and from our Lemma 3.3 that these two different "Glrding subspaces" are, in fact, the same for any sequentially complete X ;similarly, we have G*(T)= G(f') ).
Remark 3.17. An interesting consequence of the relation between smoothness and weak smoothness (see Theorem C.18) is that x € C"(T) is equivalent to the apparently much milder requirement that ( u , T ( . ) x )be smooth for a boundedness-showing set of functionals u . This fact may greatly facilitate deciding the smoothness of a vector in practice. We mention an important application in Theorem 3.28 below (it is essentially due to N . S. Poulsen, see [Po,]).
Proposition 3.18. Let a l , .. . , a ~be a basis of g and Aj = d T ( a j ) . Then for any finite k we have C k ( T )= n { D ( A j , . . . A j , ) ; 1 5 j i 5 N V i } . On the other hand, C"(T) equals the tjoint analytic space' of the A, 's, i.e., x E C"(T) if and only if a positive number s can be found such that the set
F A j , , , - . . A j k , k; ~k = 1 , 2 , . .. , j is arbitrary is bounded in 3E
PROOF:Let Vj = d R ( a j ) denote the left invariant vector field corresponding to aj (cf. (F-1)). Write f ( t , z) = T(i!)x for any 2: ( f ( . , x ) is continuous for any x then) and set
for
2
(3-12)
E D ( A j ) . It is clear that
6 [T(.)I]= f j (., x )
Smooth Vectors
55
whenever x E C' (7') (use translated logarithnuc charts; differential operators of order 5 k are defined on C' functions through charts in the sense of (C-5)). Hence one direction of the statement for finite k is evident. For the other direction it is enough to prove the C' property around 1 . We proceed by induction on k ; the induction step also follows from the following more precise formulation of the starting step. We assert that for any x E njN,lD(Aj) the function f ( . , z ) is differentiable and its derivative in a chart 'p around 1 can be written in the form
with some analytic functions x i j . The proof of the differentiability causes the difficulty; when this is done we easily check that the matrix valued function x is N the inverse of h where d t ' p ( t ~ ; )= Cjzlh;j(l)ej (ej is the canonical basis of R N ) . We prove the differentiability just a t 1 (cf. the remark after Definition 3.16) with the help of a prjduct chart 'p with respect to the basis ~ 1 , . .. , Q N . With this chart we write u ( z ) = f('p-'(z),z) (for a fixed z E njN=,D(Aj) ) and let zJ = ( z l , . . . , zj, 0 , . . . ,0) for t E RN . Then we have N
u ( z ) - u ( 0 ) = z ~ ( ' p - l ( z j - ' ) ) (T(exp(zjaj))x - Z) j=1
and thus the continuity of T implies that u is differentiable a t 1 in the sense of (C-5) and, in fact, u'(0)z = Cj"=,t j A j x . If z belongs to the joint analytic space of A l , . . . , A N with some s then and, using the notation (D-6) about we infer by the above that E E C"(T) non-commutative multi-indexes and applying (3-12), we have the boundedness of
{
~&o[T(l).]
; 1E K
, cr is arbitrary
for any compact K by the local equicontiriuity of T. Here la[!can be replaced by a ! a t the expense of changing s into s / N . The characterization of analyticity with the help of suitable vector fields (cf. Remark D . 4 ) completes the proof.
Definition 3.19. Let A, be as in Proposition 3.18 and p be a continuous seminorm on X . Then for finite k the Ck serninorm induced by p is defined on C k ( T ) by setting
(3-13)
pk(x) = max p(Aoz) loll'
(this also depends on the chosen basis of the Lie algebra but from another basis we obtain a seminorm varying between positive bounds times this one for dT(.)x is linear for C' x's). We always consider C'(T) (for finite k ) as endowed with
56
Chapter 3
the locally convex topology defined by all pk 's; while C"(T) as endowed with the projective topology, i.e., defined by all pk 's with all finite k's. Collect several easy observations about these spaces. We can see from (311) that AQT(cp) = T ( d L ( ~ ) ~ c for p ) any multi-index Q and any cp E C Y ( G ) . Consequently, T(cp) is continuous from X into Cm(T). It follows from Proposition 3.18 that Aj maps C"(T) into itself continuously. Furthermore, if bt is the left invariant differential operator corresponding t o the element b = C,"==, cp . u a p of L1, then (3-12) implies that m
bl[T(.)t]= T ( . ) C c ,. A a p t
(3-14)
p=l
for any 3: E C m ( T ) (we use the non-commutative multi-index notation of (D-6)). In particular, the corresponding continuous linear operator C,"==, cp . A"pl,,(,) in the space C"(T) depends only on b and we adopt the notation dT(b) for it. Since the involution f + f induces an isomorphism between left and right invariant differential operators, we obtain from (3-14) its twin formula
(3- 14a)
b r [ f ( . , t ) ]= f ( . , d T ( b ) t ) with / ( t , z ) = T ( t ) - ' t
E C"(T) with br = d L ( b ) = C,"==, cp . d L ( ~ ) .~ p It follows from (3-14) that dT is a morpliism of complex unital associative algebras and consequently dTI, is a morphism of Lie algebras. An important corollary of Proposition 3.18 is that C k ( T ) is sequentially complete for k = 1 , 2 , . . . , 00 (because the generators d T ( u j ) are closed operators). Similarly, these spaces are complete if X is complete. We infer from (3-9) and from the linearity of dT(.)y for C' y's that
for
t
(3-15)
for any 2: E C k ( T )with some analytic functions y a p if la1 = k . Thus we can see that the restriction of T to C k ( T ) is a continuous representation with respect to the topology introduced in Definition 3.19. The same follows for k = 00 . Then (1-3) implies that the tangent of this representation can be obtained by taking ~ T ( un ) ( ~ ~ ( 5x"c) ~ ( T ). ) The following proposition has rather old relatives. It can be found in [Po.] in almost the same form as below (more precisely, stated just for Banach spaces but with essentially the same proof). Proposition 3.20.
Let Z be a 5"-invariant subspace in Ck(T)(where
k = 1 , 2 , .. . , 00 ) which is dense in the topology of X. Then Z is also dense with respect to the finer topology of C k ( T ) .
57
Smooth Vectors
Note that this proposition can also be applied to T o F where F is some morphism from another Lie group into G. In particular, the restriction of any generator d T ( a ) to G(T) is dense in that generator.
PROOF:Let Y be the closure of 2 in C k ( T ) .Since T is continuous over C k ( T ) ,Y is also T-invariant. If 'p E CT(G) then the Riemann approximations of the integral T('p)y lie in Y whenever y E Y and so T('p)y itself belongs to Y (for C k ( T ) is sequentially complete and T is continuous over it). Now let t E C k ( T ) and choose a net 2 , E Y converging to 3: in the topology of X . Then T('p)tn---.* T('p)z even in the Cm topology by the corresponding continuity of T('p).Therefore T('p)t E Y , i.e., the Girding subspace of the representation T over Ck(T) is contained in Y ; the proposition is proved. The main feature used by the proof above is the fact that T is also continuous over Ck('Z')and this fact, in turn, relies on the important property (3-9). T h e latter states that even though T does not commute with dT in general, we know how t o intertwine them. This idea is crucial for the theory of integrating morphisms of Lie algebras (see Chapter 5 below). Now we show another tool t o deal with this problem of non-commuting. We begin with a simple lemma.
Lemma 3.21. For a E g denote the differential operator d L ( a )- d R ( a ) + by [a].Then for a n y C' function cp [alp can also be calculated by the following formula: [a]'p(t) = dlh(0,t) where h ( z , t ) = cp(exp(-za)texp(za)) . A(exp(za)) (the variable z varies in R).
PROOF:Writing f(t1, z 2 , t ) = cp( exp(-zla)t exp(z2a)) . A(exp(z2a)) we have &f(O,O, .) = dL(a)cp , & f ( O , O , .) = dR(a)cp + (a,A) . 'p and d l h ( z , t ) = (8, + a,)! (2, z , t ) . Therefore (F-21) completes the proof. Proposition 3.22. We can find a sequence cpn from CF(G) such that the following holds. Whenever a1 ,..., a t E g , A . = d T ( a j ) and t E n{ D ( A j , . . . A j , ) ; 1 5 j l < . . . < j, 5 k } we have limn-m A1 . . . AkT('p,)t = A1 . . .A k z (for any k including k = 0 ).
,
PROOF: It follows from Proposition 3.14 and Lemma 3.21 that for any 'p E CF(G) we have BT('p)x= C, T ( [ a ] " ' ~ A ) a z where the sum runs through the strictly increasing multi-indexes a (including 0), a' is the complement of a (i.e., the strictly increasing multi-index taking the values not taken by a ) and B = Aa0 with a0 = (1,. . . ,k) = 0' . Separating T('p)Bz from the sum we have a' # 0 for the remaining terms, and so it is enough to find a special approximation of Dirac's delta (in the sense of (2-14)) whicli also satisfies the following: limn+m T([bl]. . . [bp]'pn)x = 0 for all t E fi with any p = 1 , 2 , . . . and any bjEg.
Chapter 3
58
Observe that J [ b ] ' p = 0 for any b E g and 'p E C r ( C ) because this integral equals J'p. (dL(b)+ - d R ( b ) ) l and dL(b)+ - d R ( b ) is a vector field. Hence by the continuity of T (. )z at 1 it is enough to find an approximation of Dirac's delta such that the sequence [ b l ] .. . [bp]'pn is bounded in L ' ( G ) for any fixed b l , . . . , bp . Let ( U , Up) be a logarithmic chart whose range is star-shaped with respect to 0. Write m = e 8 , p (i.e., m is the Radon-Nikodym derivative of the Haar measure with respect to the Lebesgue measure on d ( U ) , cf. Notation E.6). Denote the images of everything through the chart I9 by the same letters. Choose a nonnegative p1 E C r ( U ) such that J ' p l ( t ) d X ( t ) = l/rn(O) where X denotes the Lebesgue measure. Then we assert that 'pn(t) = n N ' p l ( n t ) (where N is the dimension of C ) defines a sequence with the required properties. Since I ' p = J ' p ( t ) m ( t )dX(t) for any 'p supported by U and m is a continuous positive function on the compact support of 91, we can see immediately that 'pn satisfies (2-14) and the other requirement will follow by checking the boundedness of the sequence [bl] . . . [bp]pn in L'(U, A) . The function h whose derivative is [b]'p can be written in the chart I9 as h ( z , t ) = p( e- zadbt ) . e-ztr(adb) because I9 is logarithmic (cf. (F-5) and ( F - 1 1 ) ) . Therefore [b]'p(t)= - ( ' p ' ( t ) , [b,t])-'p(t).tr(adb) in this chart. Hence we obtain by induction on p that [bl] . . . [6,]'p = ClpllP rp@p where the smooth functions r p are IPI-homogeneous, i.e., r p ( z t ) = zIplrp(t) for z > 0 . Thus we can see that [ b l ] .. . [bp]'pn(t)= nN ClbllP r p ( n t ) . ( @ p l ) ( n t ) and the required boundedness follows. Theorem 3.23. The tangent dT is a rnorphism of Lie algebras in the following sense. Let a , b E g and A = d T ( a ) , B = dT(b) . Then a A B = d T ( a a b) for any scalar a and A B - B A = d T ( [ a ,b ] ) .
+
+
PROOF: We know that these operators agree on C"(T). In particular, they agree on the GIrding subspace (this special case already follows from Proposition 3.14). Now choosing a sequence pn with the property described in Proposition 3.22 we can see that these restrictions are dense in the corresponding operators and the closedness of any generator completes the proof. Corollary 3.24. Let a l , . . . , a k be a Liegenerating subset of g (i.e., such that the smallest Liesubalgebra containing them is the whole 9 ) and A, = d T ( a j ) . Then n,D(A") = C " ( T ) . PROOF: A basis can be chosen from the iterated Lie products of ( 1 1 , . . . ,a k . Denoting by B 1 , . . . , BN the corresponding generators we obtain from the theorem that any contains a suitable combination of As's and then we apply Proposition 3.18.
Notation 3.25. I t is customary to use the notations L ( s ) f ( 1 ) = f ( s - ' t ) and R ( s ) f ( t ) = f ( t s ) for any function f on a group (cf. Notation F.2). Then L and R are representations over any set which is invariant under them and these
59
Smooth Vectors
representations are linear whenever the base set is a linear subspace of all complex valued functions. Since the set of negligible functions is invariant under both L and R , one can factorize by them in suitable cases. We call any such representation (of a locally compact group) left (right) regular but "the" left (right) regular representation is meant to be the one over L2(G) whenever the context does not explicitly distinguish another base set (here L 2 ( G ) is considered with o u r fixed left Haar measure also for R ) . Most often a left (right) regular representation is denoted just by L (by R ) without indicating the base set. This slight abuse of language is used to avoid such uncouth formulas as, e g ,
a ('IC(G))
(')
'
Over most naturally arising locally convex spaces the regular representations are continuous (bear in mind that if the local equicontinuity of a linear representation T can be established then it is enough to check the continuity of T ( . ) z for a dense set of z's). In particular, this is true for P ( G ) with finite p , for Co(G) and for C(G) (we mean C(G) as endowed with the topology of locally uniform convergence as usual). It is also not very hard to check that for a Lie group the regular representations are continuous over C"(G) and Cp(C). Moreover, they are smooth over these spaces (this can be shown immediately or also by checking just the fact that the notations d L ( a ) and d R ( a ) we used hitherto as defined by (F-1) are in accordance with Definition 3.12 over these spaces and then applying Proposition 3 . 1 8 ) . Observe that over C"(C) we have d L ( a ) = aL(a) for any a E U, (on the left we mean the differential operator which is also applicable to any distribution) and this ~ L ( Q is) continuous. Of course, similar statement holds for R . Proposition 3.26. Consider the regular representations over C ( C ). Then for any a E g the generators d L ( a ) and d R ( u ) are included in t h e differential
(
(
operators denoted by the same symbol. Both C" LlC(,-)) and C" Rlc(G)) equal C"(C) as locally convex spaces. Both sets of analytic vectors are included in the set of analytic functions. For any smooth f and any differential operator D we have
D".)fI(t)
(3-16)
(1) = Df(t)
where on the left we mean the differentiations with respect to the topology of
C"(G).
(
PROOF:If g = d Rlc(G))(a) f then for any p E CF(G) we have
and applying the properties of the Haar measure and (F-21) we can see that the right hand side above equals J f . d R ( a ) + p . The corresponding statement concerning L is proved in the same way.
60
Chapter 3
(3-16) is an obvious consequence of the fact that the substitution of 1 is a continuous linear functional on C”(G) and therefore commutes with any differential operator. It is also continuous over C(G), hence the C‘ spaces of the representations must be included in C’(G) for k = 1 , 2 , . . . , m , w . Recall that the locally uniform convergence of the difference quotients of a C’ function to its partial derivatives holds in the “curved version”, too; in particular, we can infer by induction on k that C’(G) is a part of the C’ spaces of the representations for any finite k and consequently also for k = 00 . It is easy to check now the coincidence of the topologies. Note that quite often we have other analytic functions than the analytic vectors of these representations because an analytic vector is “uniformly analytic” in a sense, e.g. if G = R then we can see from Proposition 3.18 that the analytic vectors are exactly those functions which can be extended holomorphically to some strip. In contrast to this, if G is compact then one can show the coincidence using Theorem D.3 and Proposition 3.18. Now we are going t o examine the “proper” regular representations, i.e., those over L2(G).At this point it is appropriate to introduce the ‘complex conjugate’ and the ‘adjoint’ of an element b E U, . T h e complex coiljugate of b is defined through d R or d L (cf. Definition E . l l ) or equivalently the conjugation is the conjugate linear involutive automorphism of L1, which does not move the elements of the Lie + algebra. T h e adjoint is the transposed conjugate h* = h+ = (sl) . This definition is motivated by the connection with representations over Hilbert spaces (especially with unitary representations). Namely, if X is a IIilbert space and we identify X* with X as usual then Proposition 3.13 can be reformulated with the Hilbert space dual as
where, of course, T(2) = T(2-l)’ in the Hilbert space sense (because the whole X now by Theorem 3.11), and consequently
(3-18)
equals
a?(b*) C aT(b)* for b E U,
(note that taking contragredients in the Hilbert space sense is involutive, so you need not remember on which side was the ‘ A ’ ) . Proposition 3.27. Consider the regular representations L and R (over L 2 ( G ) ) . L is a unitary representation, while R can be “corrected” by a scalar valued representation to become unitary; namely, & ( t ) = A(t)’/’R(t) defines a unitary representation. For any a 6 g the generators d L ( a ) a n d dR(a) equal the intersections of the corresponding differential operators with L2(G)x L 2 ( G ) . Furthermore, for any b 6 U, the corresponding intersections equal tlL(b*)* and ak(b*)*, respectively, where k(t)= R(2-l)’ = A ( t ) R ( t ).
61
Smooth Vectors
C"(L) equals the set of those smooth functions f for which Df E L2(G) for each right invariant differential operator D and the analogous statement holds for R . Both sets of analytic vectors are included in the set of analytic functions. Any function in C"(R) vanishes a t infinity. We mention that C"(L) is not a part of Co(G) in general (for an example see Exercise 3.6). Of course, this pathology can not occur for unimodular groups (use the reversed group). Notice the different advantages of L and R .
PROOF:T h e statements about unitarity are trivial. For 'p E C r ( G ) the difference quotients vanish outside some compact and so the argument in the proof of Proposition 3.26 yields that C" spaces of both L and R and for any v E U, we have
(3-19)
d L ( v ) ' p= a L ( v ) ' p and
'p
belongs to the
dR(v)'p = aR(v)'p
where on the left hand sides we mean the differential operators introduced in Appendix F . Since R equals an analytic function times R , any 'p E C r ( G ) is also a smooth vector for R and (3-18) implies that
a k ( b * ) p= aR(6)*(p .
(3-20)
If D is any differential operator, f E L2(G) and 'p E C r ( G ) then obviously (f, D p ) = (D+f ,Cp) where on the left we mean the scalar product in L2(G)while on the right we consider distributions. This implies that -
D+ n ( L z ( C )x L 2 ( C ) )= DG
(3-21)
where Do = Dlcy(G) . Applying this to D = dR(6) and using (3-19) we see that in (3-20) we can also write dR(b)+'pon the right. Having this version of (3-20) we can apply (3-21) to D = dR(b)+ and infer that the corresponding intersection is the dual of ak(b*)I . But C Y ( G ) is dense and R-invariant, thus it can be C,"(G)
omitted by Proposition 3.20. The analogous statement for L can be proved in the same way (by observing that L - = L ) or a part of the proof can be replaced by the observation dL(b*) = dL(b)+ which follows from (F-21). If Q E g then ~ T ( Qis)dense in ~ T ( Qfor ) any continuous representation T by Proposition 3.20 and we obtain the desired equality by comparing the above with (3-17). The characterization of the spaces of smooth vectors now follows from Proposition 3.18 and Sobolev's lemma (together with the elementary facts that if the first partial (distribution) derivatives of a continuous function are continuous then the function itself is C' and consequently if the partial derivatives are continuous up to the order k then the function is Ck). Now let f E C"(R) and apply the version (E-8) of Sobolev's lemma to it around 1 with Aj = d R ( ~ j where ) ~ 1 , . .. , a ~ is a basis of g . These differential operators are left invariant and hence we obtain
62
Chapter 3
where K is a compact neighborhood of 1. Since f E C " ( R ) , the functions to be integrated over S K are integrable over G and so we can choose a compact P such that the integrals over G \ P are small and therefore I f 1 is small outside the compact set P K - l . Finally, prove the statement about analytic vectors. Let f E C"(L) or f E C"(R) . We again resort to (E-8). We obtain (also using Proposition 3.18) that if P is a compact set in G then some constants C1, Cz can be found such that m)! where Aj = d L ( a j ) or Aj = d R ( a j ) and maxp 1A"fl 5 C1 . Ckttm.1.1( (k m)! < m = [$ 11 . Here C2 is independent of P . Since for large k we have
+
+
+
k! (2k)m < 2k , we can see by Proposition 3.18 that f is also an analytic vector of the left (right) regular representation over C ( G ) and the proof is completed by Proposition 3.26.
In some theorems below we shall be concerned with 'elliptic' and 'strongly elliptic' elements of U, , so it is time to clarify what we mean by this. Let b E U, , then the characteristic forms of d L ( b ) and d R ( b ) (see Notation E.14) are right and left invariant, respectively, and at the point 1 the first equals ( - l ) k times the second if k is the order of b (it is d L ( b ) + whose characteristic form a t 1 coincides with that of d R ( b ) ) . Hence they are elliptic or strongly elliptic (see Definition E.15) a t the same time; namely, if a l , . . . , U N is a basis of g and b = CI _ c , . acT then they are elliptic if and only if the polynomial P(<) = CI,I=k c a t a has no root in RN \ (0) and they are strongly elliptic if and only if Re(ikP) is negative there (of course, the latter can only occur if k is even). So we can talk about the ellipticity or strong ellipticity of b . Evidently, the mapping which assigns the polynomial i k P to 6 is multiplication-preserving and on the set of elements with a fixed order it is linear. Furthermore, the i k P of b' equals the conjugate of that of b . Thus we have the following elementary facts. If b l , bz are elliptic then blb2 is elliptic; If b l is elliptic (or strongly elliptic) and 62 is of smaller order then bl b2 is also elliptic ( or strongly elliptic); if b l , bz are strongly elliptic and of the same order then bl bz is strongly elliptic; if b is elliptic then - ((b'b)") is strongly elliptic for all n .
+
+
Theorem 3.28. Let b be an elliptic element of order Ic > 0 in U, and let b = CIalsk c, .aa be an expression for i t where a l , . . . , U N is a basis of g . Denote by B the closure of CI _ c, . d T ( a ) , . Then C"(T) = n,w,lD(Bn); and if X is such that (X.,p) is sequentially complete (this is so, e.g., if X is barrelled or bornological) then C"(P) = nF!-lD((B*)n). Furthermore, the topology on C"(T) defined by the collection ofseminorms z 4 p(Bnz) where n = 0,1,2,. . . and p varies in a family defining X coincides with its own (as defined by Definition 3.19). The existence of B follows from Proposition 3.13 (see the remarks after it). This B apparently depends on the special expression, not just on b . In fact, Theorem 3.32 will imply that actually B = a T ( 6 ) = af'(b+)* for an elliptic b .
Smooth Vectors
63
PROOF: Let 2 E D(B) , then we know from Proposition 3.15 t h a t ( u , T ( . ) B z )= dR(b) ( u , T ( . ) x ) for any u E X' in the sense of distributions. Hence it follows by induction on n t h a t ( u , T ( . ) B " r )= dR(b") (u,T(.)2:)whenever 2 E D(B"). Since ( u , T ( . ) t ) is continuous for any t , we obtain by elliptic regularity t h a t for 2: E nr=lD(Bn) ( u , T ( . ) z ) is smooth (use Theorem E.16 or the more advanced Theorem E.19). Now Remark 3.17 finishes the proof of the first statement. T h e proof of the dual statement is completely similar; t h e only differencrs are the following. We should consider dL(b)+ = d L ( b t ) instead of d R ( b ) and in the end we must notice that X is boundedness-showing for (X*,a) (see Proposition C.20). T h e topology of C"(T) evidently contains the other one. If X is a Frkchet space t h m so are both these spaces (because B is closed) and thus their equality follows from the Closed Graph Theorem. Now apply this result to the right regular representation R of Go over the Frkchet space C(G0) where Go is t h e connected component of G containing 1 . Fixing a t~ E U, the seminorm p(f) = I d B ( v ) f (1)l is continuous on C"(R) = C"(Co) (in fact, we could consider a n y differential operator in the place of d R ( v ) here by Proposition 3 . 2 6 ) . Then by the equality of the two topologies we have a natural number n , a constant C and a compact A' such t h a t p(f) 5 C . maxK IdR(b)kfl for any f E C"(G0) (stronger results are known i n the theory of partial differential equations but this is enough for us and the proof we presented is pleasantly siniple). Apply this formula to the restrictions of the functions ( u , T ( . ) z )to G o , where u is arbitrary and 2: E C"(T) . Then we get 1 ( u , a ~ ( i i ) a : I) 5 C . C',"=, maxK [(ti, T ( . ) B ~ z. )If ~y is a continuous seminorm on X then we find a u such that ( u , aT('u)r)= y( a T ( v ) z ) and (u1 5 y . T h e n the expressions under the sum are estimated by maxK y( T ( . ) B k 2 : )which, in t u r n , can be estimated by some q l ( B k z )because of the local equicontinuity of T . Since the relation of y and y1 depends only on A' and that depends on 'u and b , the theorem is proved.
C;=,
It is clear from the proof above that if we consider any sequence b, of elliptic elements with orders tending to infinity and denote by B, the corresponding closure then we have C"(T) = nr=,D(B,) a n d , for the spaces having sequentially complete strong dual, C"(f') = nr=lD(Bz) . By essentially the same melhod it can be proved t h a t D(dT(u,)*) also equals the set of smooth vectors (Theorem 1 . 1 of T GOO^]); in particular, if a vector is smooth for each one-parameter subgroup then it is smooth.
ny=,flrZl
It follows from Proposition 3.13 t,liat a'f'(b+) c aT(b)* for any b E U, . Here the right hand side is a w* closed operator and an interesting (and highly nontrivial) question is: when will a?(bt) be w* dense in it? Note t h a t this condition is equivalent to the requirement t h a t d T ( b ) be dense in af'(b+)* (this dual is meant with respect to the w* topology). We can find useful results of this kind for unitary representations in [N-St] (it is especially instructing to learn their counter-examples, i.e., examples when d T ( b ) is not dense in a'f'(b+)* ). T h e "elliptic theorem" (Theorem 2.2 of [N-St]) a n d some
64
Chapter 3
earlier results were extended to continuous representations over Banach spaces in [Lan] . The ideas of the latter can also be applied to the most general continuous representations. We are going to discuss this here. Furthermore, we shall be able to prove a “commuting-with-elliptic theorem” (cf. Corollary 2.4 of [N-St] ) for Banach spaces (see Theorem 3.35 below). In the course of 3.29-32 b always denotes an element of U, , Z and 2’ are Cm-dense subspaces in C m ( T ) and G”(T) , respectively, and Bo = a T ( b ) l z , (the letter c in the subscript stands for ‘contragredient’). ReBo, = Xf’(b+)l Z* member that for invariant subspaces denseness in the original and in the strong topology, respectively, implies this Coo-denseness (see Proposition 3.20 ). Since the operators a T ( b ) and a?(b+) are continuous in the Cm-topology, the role of 2 and Z’ will not be substantial; in particular, they do not affect the operators B1 := B& and B1, := B,’ (the first one is meant with respect to the w* topology). We bother about these subspaces because they naturally arise sometimes. A typical case is when 2 = G(T) and 2’ = G’(T) (cf. Proposition 3.15). We shall denote by B a linear operator between Bo and B1 , i.e., a version of “dT(b)” in the widest reasonable sense. It is useful to notice that the power of a statement ‘ BO is dense in B1 for such and such b’s and T’s’ can be multiplied by the following simple trick. When we want to check such a statement for a certain b and T then, of course, it would b e enough to find an operator S c and a w* dense linear subspace W c Cm(T) such that S is dense in (S*lw)’. Now such S and W may be found in the shape of B and Z’ for some representation of another group over 2 .For a simple application of this method with Theorem 3.30 see Exercise 3.8 (then the other group is R ) . Instead of using other groups we can also try “nice enough” semigroups (see Lemma 3.34 below). Lemma 3.29. For I E 2 and u E 2’ write f ( t ) = (u,T(t)z). If u E V(B’) is such that an appproxirnation (P, of Dirac’s delta (cf. (2-14)) can be found satisfying
(3-22)
lim( ( d R ( b ) - d L ( b ) + ) f
, (on) = 0
for all I then (u,B’u) belongs to the w* closure of the operator Boc (the differential operator in (3-22) is applied to f in the sense of distributions). Analogously, if I E V ( B ) is such that (3-22) holds for all u then ( 2 ,B E ) belongs to the closure of
Bo.
PROOF:Since BO c B c B1 , we have B’u = Blcu or Bx = Blx , respectively. Thus we can apply Proposition 3.15 to infer that in the first case dL(b)+ f is continuous for all I and at 1 equals (B*u, I) while in the second case . On the other hand, the d R ( b ) f is continuous for all u and a t 1 equals (u,BI) ‘B’ considered in (3-11) also lies between Bo and B1, and hence (3-11) certainly holds if we write B1 instead of B and B1, instead of B’ in it. But these operators are considered in (3-11) just a t vectors belonging to the G t d i n g subspaces, so they
65
Smooth Vectors
can be replaced by dT(6) and aT(6+),respectively, and the latters are in the Cm-closure of Bo and Boc . Thus if (P, satisfies the assumptions then in the first case (T*(cp,)u,Z T * ( c p , ) u ) + (u, B * u ) in the w* topology and in the second case (T(cp,)z,KT(cp,)z)+ ( 2 ,B z ) in the w topology. This is enough by the Hahn-Banach Theorem applied on X x 2 . If 6 is such that dR(6) = dL(6)+ on a neighborhood of 1 then the assumptions above are always satisfied and hence Bo and Bo, are dense and w* dense in B and B* , respectively, for any continuous representation T . On the other hand, we have the following theorem.
Theorem 3.30. If d R ( 6 ) = dL(b)+ on an open neighborhood U of 1 then also on the connected component Go of G containing 1 . The latter holds if and only if 6 is central in U, (i.e., ab = ba for all Q E U, or equivalently for a subset of a's generating U, as a complex unital associative algebra; in particular, a Lie generating subset of g can be considered).
PROOF:Fix a b E U, and write D = d R ( 6 ) - d L ( 6 ) + . If T is some continuous linear representation and B = aT(6) then for any 2 E C w ( T ) we have by Proposition 3.15 that (3-23)
D[T(.)XI = T(.)B Z - BT( .)x
(since the sides of this equality exist, it is enough to test them with a w* dense set of functionals). Write T = R into this, where R is the right regular representation over the space Cm(G). Then d T = d R (on the right we mean the differential operators considered on their original domain, i.e., on C"(C)). If DI, = 0 then we have R ( t ) d R ( b ) R ( t - ' )= dR(6) for t E U . It is known that d R is injective, hence the last property is equivalent to Ad(t)6 = 6 by (3-9) where Ad(t) is extended from g to U, to become an automorphism (of complex unital algebras). On the other hand, if R(.)B = BR(.) on Go then comparing (3-23) with (3-16) we obtain that 03:= 0 for z E C m ( C ~, )i.e., DlG0 = 0 . Thus the theorem will follow from the facts that if 6 is stable under Ad(U) then 6 is central while if 6 is central then 6 is stable under Ad(G0). We just hint a t the proof of this standard result: check that Ad(exp za)6 = Ad(exp z a ) [ a ,61 (the differentiation is meant in the finite dimensional subspace of elements with order not greater than that of 6 ) .
5
If 6 is not central then we may still hope for (3-22). Use the method of the proof of Proposition 3.22 to show the following statement.
Lemma 3.31. Let k be a positive integer and 6 E U, be an element of order 5 6 . Iff is a Ck-'function on G and (P, is the sequence constructed in the proof of Proposition 3.22 then for this f , 6 and (P, (3-22) holds.
xi
PROOF: Let b = ci ci E C , ai,, E g and each ki
. a;,' . . . a;,k, be an expression of order k for 6 (i.e., 5 k ; some of the k ; ' s may equal 0). Since any left
Chapter 3
66
invariant vector field commutes with any right invariant vector field, we can infer by induction on p that * *
. vP)+ =
C d L (va‘) [-val,l].
. . [-val]
a
where the sum is added over the strictly increasing multi-indexes a,the complement of a is denoted by a’ as in the proof of Proposition 3.22 and v, E g . Therefore if h is any distribution and cp E CF(G) then we have
(3-24)
( h , (dR(b)+ - d L ( b ) ) p )= C i
C i
.
C*( d L (u?’)+h
D(i,a)~)
a
where C: means the sum over the strictly increasing multi-indexes except 8 (so we have 2k*- 1 terms in the i’th sub-sum) and each D ( i , a ) is of the form [vl]. . . [v,] with p 2 1 (in fact, D ( i , a ) = [ - u . t,al,l] . . . [ - ~ i , ~).~ Substitute ] h = f and cp = (P, , observe that each a’ in (3-24) is shorter than k and consequently each derivative o f f on the right is a continuous function. Therefore the whole expression tends to 0 by the properties of (P, we checked in the proof of Proposition 3.22. Observe that instead of requiring f to be Ck-lwhat is really needed is the condition that certain differential operators (the ones applied on h in (3-24) ) change f into distributions which are function-like on a neighborhood of 1 and continuous at 1. Comparing Lemmas 3.29 and 3.31 we can see that the condition B = is equivalent to the apparently weaker condition that the restriction of B to Y is dense in it where Y is the set of vectors t for which ( u ,T ( . ) z ) is C”-’ for all u ; a similar statement holds for the duals. The simplest application is when k = 1 (though it does not give anything new because Proposition 3.20 already implies the denseness of d T ( a ) in d T ( a ) as we remarked there). An application with k = 2 is shown in Exercise 3.9 below. Probably the most important application is the following theorem.
Theorem 3.32. If b is elliptic then Bo is dense in B1 .
PROOF: One should just gather Proposition 3.15, Lemmas 3.29 and 3.31 and the fact that any (distribution) solution of an “inhomogeneous” elliptic equation of order k and with continuous right hand side is a Ck-’function (we prove the latter statement in Appendix E , see Theorem E.19). For unitary representations the condition d T ( b ) = -dT(b)* (dual in the Hilbert space sense) implies that this operator generates a unitary representation of R . Similarly, if aT(b)= dT(b)* and this is a negative operator then it generates a holomorphic semigroup on the half-plane and it extends continuously to the closed half-plane and equals a unitary representation on the imaginary axis. These facts follow easily from spectral theory as well as other related results concerning normal
Smooth Vect ors
67
operators. The following theorem (taken from [Lan]) can be considered as the counterpart of the statement about negative operators and shows, in particular, that if h is elliptic and is a Banach space then aT(h*h)" is "good enough". Though we formulate the theorem for Banach spaces, the proof works more generally in the 'ultra-continuous' case (cf. the remarks after the proof).
n
Theorem 3.33. If SE is a Banach space and a E U, is strongly elliptic then A = &"(a) = d F ( a t ) * generates some holomorphic semigroup.
PROOF:We are going to check that the condition of Theorem 1.19 is satisfied by A (naturally, we just check the weaker version with k = 1 in (1-23) and it is enough to find just one U-domain because we do not bother about the limits of the domain of the holomorphic semigroup). So the task is to find a suitable Udomain U , prove that XI - A is injective and surjective for X E U and estimate I l ( X I - A ) - ' ] ] . Recall that a densely defined operator has a dense range if and only if its adjoint is injective (by the Hahn-Banach Theorem). If we can show that XI - A is bounded from below then (by its closedness) its surjectivity is equivalent to the condition that it has a dense range and so if we also show that X I - A* is bounded from below then the required surjectivity follows. We shall prove the estimates for X I - A and X I - A* by the same method but we must take care of some minor differences. Hopefully, it will not confuse the reader that the same letters may mean different things in the two versions. Let X E C . In the first case fix 3: E D ( A ) , write y = ( X I - A ) . and choose a functional u such that 111~11 = 1 and llzll = ( ~ ~ 3 .: ) In the second case fix u E D ( A * ) , write v = (XI - A * ) u and choose a vector 3: such that 11x11 = 1 and llull 5 2 (11, 3:) . In both cases write f ( t ) = ( u ,T ( t ) z ). Then f is continuous and we have by Proposition 3.15 that in the first case ( u , T ( . ) y )= Xf - d R ( a ) f and in the second case ( v , T ( . ) 3 : = ) Xf - d L ( a + ) f . I f f were smooth then we could apply Theorem E.22 to it (in fact, we included that theorem in the Appendix exactly for the sake of the present proof). We assert that in the special case when the manifold is a Lie group and the differential operatforD is left or right invariant the formula in Theorem E.22 can be extended to f's of which we only know that on a neighborhood of zo f and Df are function-like and continuous. If we have such an f then choose an approximation pn of Dirac's delta in the sense of (2-14) and set fn = pn * f or fn = f * pn if D is left or right invariant, respectively. Apply Corollary 2.8 and then Theorem 2.11 to infer that fn -+ f and Dfn + Df uniformly on a neighborhood of zo (when applying Theorem 2.11 we replace f and D f by some compactly supported continuous functions equalling them on a neighborhood, then their convolutions with pn will not change on a smaller neighborhood for large n ) . Since the domain R and the constant C in Theorem E.22 are independent o f f and the formula holds for each fn , it must hold for f , too. Now suppose that X belongs to the R's (in Theorem E.22) of both D = d R ( a ) and D = d L ( a + ) (with 3:o = 1 ). Let W be a compact neighborhood of 1 , A4 = suptEwllT(t)ll and C be good enough in Theorem E.22 for this W and
Chapter 3
68
for both D’s. Then we have in the first case
and in the second case
In other words, we have obtained a lower estimate for llyll/ llzll or for llzlll / )1(. if .(A) is large enough and this estimate is not worse than constant times .(A) if it is even larger. Thus if U is a U-domain lying far right in the intersection of the two R’s then the weakened (first power) version of (1-23) holds for U and the proof is complete. Now examine where the assumption ‘ X is a Banach space’ was used essentially. T h e norm can be replaced by seminorms p and one can choose a u such that p ( z ) = ( 2 1 , ~ ) and 1.1 5 p (applying different ti’s for different p’s). T h e difficulty arises when we want to estimate supw If1 by p ( z ) and this can be done if T is ‘ultra-equicontinuous’ on W in the sense that a constant M exists such that p ( T ( t ) z )_< M . p ( z ) for t E W and for all p from a defining family and for all 2 . If this holds then the other path of the proof is also passable but this is not important because the ultra-equicontinuity is stronger than the ‘contraction type’ property from which we shall prove the same conclusion in the next chapter (see Theorem 4.5 ; its proof will rely on the Banach space case we just proved). Serious further relaxation of the condition about T seems hopeless because we have counter-examples even if X is a separable Frkchet space and G = R (for one such example see Exercise 3.10; it is due to the author who, by the way, has not seen any previously published example of this kind and even the existence statement seems new). Note that it follows by an easy compactness argument that the domain of the holomorphic semigroup generated by A can a t least be stretched to include any ray in the right half-plane whose angle d satisfies the condition that e i f f u is strongly elliptic (the fact that the “union” of the holomorphic semigroups on the corresponding small sectors still has the semigroup property can be checked, e.g., by applying the unicity theorem of holomorphic functions). The same conclusion could also be inferred from the result in Exercise 1.8. In particular, if the characteristic form of u is real (and negative) then the domain is the whole open right half-plane plus the origin.
[N-St] inspired the author to try to obtain a common generalization of the two classical cases (central and elliptic) when aT(b) = a?(b+)’ is known to hold. Complete success has not been achieved but a remarkable result is contained in Theorem 3.35 below; first we prove a lemma. Lemma 3.34. Assume that a l , a2 are commuting elliptic elements oforder that both A1 = aT(u1) and A2 = aT(u2) generate one-parameter
> 0 such
69
Smooth Vectors
semigroups over X . Suppose that X is such that its strong dual is sequentially complete. Then for any b in the complex unital associative algebra generated by the pair a 1 , q we have 8T(b)= 8!f(b+)*.
PROOF:Denote by T, the one-parameter semigroup generated by A , . We knowfromTheorem 3.28 that C m ( T )= n;==,D(A?) for j = 1 , 2 and consequently C"(T) is invariant under T j (by Theorem 1.5). Furthermore, Theorem 3.28 also implies that the C" topologies of these representations (defined analogously as in the group case) coincide with the topology of C"(T) and consequently with each other. The restrictions of 7'1 and Tz to this common C" space are one-parameter semigroups with respect to this common C"-topology by Theorem 1.5 which also implies that they commute because the restrictions of the Aj's are continuous in this topology and commute. But the are continuous on X and therefore they To(s, t ) := T l ( s ) T z ( t ) is a continuous representation of commute; consequently ( R , ) ~ over X . Now, of course, we would like to apply Lemma 3.29 t o To but, unfortunately, TOis just the representation of a semigroup. T h e good news is that this semigroup is nice enough. The details are the following. More generally, consider a subsemigroup of a Lie group as in Lemma 3.6 and assume that for a certain a E g we have e x p r a E G for small non-negative t . If V is a continuous representation of this semigroup over X and A = d V ( a ) (by which we now mean the derivative from the right instead of the derivative in Definition 3.12) then it follows from Lemma 3.6 that AV(cp) = V(dL(a)cp) whenever cp E C r ( H ) (where II is the interior of the subsemigroup) because X is sequentially complete. We can similarly deal with V * (see Proposition 3.9 for the existence of V * ( c p , X ) ) .Notice that V * is a representation of the reversed semigroup and so in the formula obtained from Lemma 3.6 we have left Haar measure and right invariant vector field with respect to this, i.e., a right IIaar measure and a left invariant vector field for the original group. The local equicontinuity (required in Lemma 3.6 just inside) holds for the X) (with respect t o the strong topology and so we have BV*(cp,X) = V*(dL(a)cp, reversed semigroup) with the corresponding strong derivative B . Thus B is a part of the w* derivative which is evidently contained in A* (in fact, the w* derivative equals A * , cf. Proposition 3.13). Switching from right to left Haar measure by (3-7) and (3-8) and observing that the '&(a)' of the reversed group equals - d R ( a ) in the original group we obtain that (3-11) holds in this setting (i.e., writing a , A and V instead of b, B and T and for cp E C,"(H) ). If several elements of the Lie algebra satisfy the assumption on a above and bo is an algebraic combination of these then we obtain (3-11) for bo accordingly. Now restrict our attention to the case when 1 g ?r . Let Z be the " H Girding subspace", i.e., the linear subspace generated by the ranges of the operators V(cp) when cp runs over C T ( H ); and similarly let 2' be the subspace in x' generated by the ranges of the operators V*(cp). 2 is dense (by Lemma 3 . 5 ) be. Let Bo be the restriction of the version of dV(b0) cause V is continuous and 1 g used in (3-11) to 2 . Then, by (3-11), we have B:V*(cpn)u- V*(cpn)B;u 0 in the w* topology if (3-22) holds with cpn E C?(H) for a certain u 6 D(B{) and
-
Chapter 3
70
if bo is central then B:lz. is w* dense in B: . Returning to the original question, we set V = TO and let bo be a polynomial in two variables over C such that b = bo(a1,az) . Then we should just check ' c Cw(F). We saw that 2 c n:=lD(Ay) and that Z c Cw(T) and 2 Z* c n:=,D((A;)") and Theorem 3.28 completes the proof. for all
I . Thus
We remark that the lemma remains true if we omit the condition 'a2 is elliptic'. Then C w ( T ) is not necessarily 7'2-invariant but a t least T l ( t ) a T ( a 2 )c d T ( a 2 ) T l ( t ) follows for every t 2 0 as above. Since A2 = a T ( a 2 ) and T l ( t ) is continuous on X , we get TI(t)AZ c A2Tl(t) and applying Theorem 1.5 (in X this time) we obtain the commutation of the two one-parameter semigroups. The rest of the proof is unchanged.
Theorem 3.35. Let X be a Banach space and b E U, be such that a self-adjoint elliptic element (other than a scalar) can be found commuting with it. Then d T ( b ) = dT(b+)* .
PROOF:Let h be a self-adjoint elliptic element of positive order such that hb = bh . Then b also commutes with v, = - (h'") for all n and these are strongly elliptic elements with orders tending to infinity. Thus with a suitable n the order b of v, is greater than the order of b and in this case a1 = v, and a2 = v, form a commuting pair of strongly elliptic elements. Comparing Theorem 3.33 and Lemma 3.34 we obtain the result.
+
Remarks. Instead of requiring that X be a Banach space we could consider the following weaker conditions: the strong dual of X is sequentially complete and T is of contraction type, i.e., { e-CIzlT(z); z E Go } is equicontinuous for a suitable C (see Theorem 4.5 in the next chapter). If we do not know that b commutes with a self-adjoint elliptic element but it commutes with an elliptic element h and also with h' then it commutes with h*h , so this condition is equivalent to the one in the theorem. Several slightly stronger versions can be inferred from the same proof. E.g., it is enough if b commutes with a real elliptic element. The weakest assumption with which the proof works is to require a strongly elliptic element commuting with b and having greater order than b has. For unitary representations it is enough if b*b commutes with an elliptic element (because then the conclusion for b'b implies the same for b by Lemma 2 . 3 of [N-St] ).
EXERCISES 3.1. Check that if W is a dense subspace of a Banach space sufficient for ( Pp) ,, i.e., for the dual Banach space.
X
then it is
71
Smooth Vectors
3.2. Let G = R x Rd where Rd is the group R considered with the discrete topology. Show that the sum A B of two negligible sets may not be a-finite with respect to the Haar measure of G.
+
3.3. Let G be a semigroup which is also a locally compact space and p be a non-negative Radon measure on it. Consider a linear representation T of G over a quasi-complete locally convex space X . Show that if 2 is pbounded and Q . T ( . ) x is continuous then for any $ E L 1 ( G , p ) the weak integral T ( $ p ) z equals an improper Riemann integral of $Q . T ( . ) z ,i.e., a sequence I<, of compacts (depending on $ ) can be found such that yn = $ ( t ) e ( t ) T ( t ) t d t exists as a Riemann integral and T ( $ Q ) x = Iiw-" y,, . (Hint: apply Luzin's theorem.)
,s,
3.4. Prove that the formula for the derivative in Lemma 3.7 would not be valid if we meant the condition V,($Q) = $ 1 ~just on the interior H ; show a counter-example for C = R+ .
3.5. Let X be a Banach space and T be a separately continuous linear representation of a locally compact group over (r' , w*) . Show that T is the dual of a continuous representation of the reversed group over X . Readers better acquainted with topological vector spaces may prove the same conclusion if X is not necessarily a Banach space but satisfies the following conditions. It is sequentially complete, infrabarrelled and such that its strong dual is barrelled (the sequential completeness is used through Proposition C.20).
3.6. Let C be the matrix Lie group
(this is the unique two dimensional connected non-commutative Lie group). Check da db da db that in this chart - is a left Haar measure, - is a right Haar measure a2 a and the modular function equals l / a . Furthermore, if we denote the basis of the Lie algebra arising from this chart by e l , e2 then we have (in the chart) d L ( e 1 ) = -aa, - b& and d L ( e 2 ) = -& . Show that i f f is a smooth function on the positive half-line, g E C,"(R) , h ( a , b ) = f ( a ) g ( b ) and D is a right invariant differential operator of order k then D h ( a , b ) = C,"=, a j f ( J ) ( a ) g j ( b )with gj E C,"(R) . Find unbounded functions in the C" space of the left regular representation of this group G over L 2 ( G ) .
3.7. Prove the following analog of a theorem of R. Goodman (cf. the remarks after Theorem 3.28). Let T be a continuous linear representation of a Lie group over a sequentially complete locally convex space whose strong dual is sequentially complete, and a l l . . . , a N be a basis of the Lie algebra. If ZL E D ( ( a T ( a j ) " ) * ) for all j and n then u must be a smooth vector for T .
72
Chapter 3
3.8. Let T be a continuous linear representation of a Lie group over a reflexive Banach space. Show that aT(b) is dense in Bf(b+)' if b is a polynomial of some a E g . (Hint: C"(T) and C"(p) can serve as 2 and 2' for the corresponding representation of R .)
3.9. Let T be a continuous linear representation of a Lie group over a sequentially complete locally convex space and A , B be two generators, i.e., A = d T ( a ) and B = dT(b) with Q , b from the Lie algebra. Show that C ' ( T ) f O ( A B ) c V ( B A ). 3.10. Let T be the right regular representation of R over C"(R), i.e., T ( s ) f ( t ) = f ( s t ) on this space. It is easy to check that its generator is the differentiation Af = f' which is a continuous operator on C"(R). Show that the second differentiation A* does not generate any one-parameter semigroup over C"(R). Since this exercise is harder than usual, we give more hints. Assuming the contrary denote by V the one-parameter semigroup generated by A2 and denote by V2 the analogous one-parameter semigroup over L2(R) (the latter is known t o exist and, in fact, V2(t) = F - ' S ( t ) F where S ( t ) f ( z )= e-lZ2 . f(z) and F is the Fourier-Plancherel isometry of L2(R)). Show that V2(t)p = V(t)cp for cp E C r ( R ) . Then get a contradiction by constructing a sequence cp, in CF(R) such that cp, + 0 in C"(R) but V~(l)cp,, f* 0 there. To this purpose it is suitable to try P n ( t ) = M,, . cpo(2 a,) with arbitrary non-zero cpo E C r ( R ) and a, + +m ; and then choose the sequence M,, depending on these data so that SUP,, Il/z(l)Vn( be unbounded on a finite interval (observe that 1/2(l)p, is a n entire function).
+
+
73 4. Analytic Mollifying Throughout this chapter G is a fixed Lie group, Go is its connected component containing 1, g is the Lie algebra of G and U, is the complexified universal enveloping algebra of g . We fix a left Haar measure p on G and shall not indicate it in the integrals. To mollify analytically on a Lie group (analogously as we did in Proposition 1.13 on the line) one needs nice analytic kernels as the Gauss-kernel on R ” . The Gauss-kernel is “the best” fundamental solution of the classical heat equation and so it is natural to look for nice kernels among good solutions of highly symmetric “heat type” equations over G . It seems that the first application of this method for general Lie groups is due to E. Nelson (see “el]). There the Laplacean of R” is replaced by an operator of the form d L C .a2 , where ~ 1 , .. . ,Q N is a 1 1 ) basis of the Lie algebra. This generalized “Laplacean” and its relatives have become widespread in mathematics in the meantime. Of course, the best case is when the Lie group can be endowed with a Riemannian structure which is right and left invariant and the basis above is orthonormal. It is known that such a Riemannian structure exists on G if and only if the group Ad(C) is a bounded set in B ( g ) and if G is connected then this is equivalent to the condition that G be the direct product of a compact Lie group with some R” . T h e geometric considerations can also be applied on general Lie groups to obtain the most precise known results for nice enough “Laplaceans” (cf. [Vrs] ). The simplest proof of the denseness of analytic vectors is found in [Girl (and in a pleasantly expository style, too). In “el] a bit more was asserted of the corresponding heat kernel but the proof had a gap which, it seems, was only filled thirty years later in [Ma4]. I n the last mentioned article essentially no geometry is used and, in spite of this, fairly accurate estimates are obtained. Interestingly, in a very short time after “el] and [Girl R. P. Langlands succeeded in proving important results if the Laplacean is replaced by any invariant strongly elliptic differential operator on G (see [Lan] ; unfortunately, this is an unpublished Thesis). We shall proceed on the base of the ideas of [Lan] in the present chapter. We recommend [Rob] to readers interested in related developments.
(
If other thing is not explicitly stated, i n this chapter Q is a fixed strongly elliptic element of order k > 0 in U, , A = ~ T ( Qwhenever ) some representation T is fixed by the context, D = d L ( a ) and Dt = d R ( a ) . We denote by S the maximal open sector in C provided by the remark after Theorem 3.33, i.e.,
s = { te’” ; t > o , 29 E ( - T / z ,
7r/2) ,
2”.
is strongly elliptic }
We shall construct “the best fundamental solution” of the “generalized heat equation” $ u ( t , z ) = D z u ( t , z ) (it will also be a “fundamentalsolution” of g u ( t , z ) = (D$)=u ( t ,z) and continues holomorphically to S times a “complex neigborhood” of G ) .
Chapter 4
74
Lemma 4.1. Let T be the left regular representation over the Banach space Co(G) and V be the corresponding holomorphic semigroup on S U (0) generated by A . Denote by vi the finite complex Radon measure defined by ( v t , h ) = [ V ( t ) h ] (1) for h E Co(C) . Then V(t)f = vt * f for all 2 E S U (0) and f E Co(G) (of course, YO = 6 ). On S we have (4-1) where the differentiation with respect to 1 is meant in the complex sense and in the norm topology of Co(C)' .
PROOF:Fix an I E G , then the right translation R ( z ) commutes with T and hence R(z)dT(v) C dT(v)R(z) for any v E g by Theorem 1.5. Consequently, R(x)aT(b) c BT(b)R(z) for any b E U, and hence R ( z ) A c AR(I) . Using Theorem 1.5 in the other direction we can see that R ( z ) commutes with V . Then R ( z ) also commutes with any A k V ( t ) and these are everywhere defined bounded operators if t # 0 by the Cauchy formula (1-24). Thus V ( t ) f ( z )= [ V ( t ) R ( t ) f ] ( l ) = Sf(y-'z)dv,(y) . Then it follows that
llvtll = IlV(t)ll
(4-2)
where we mean the usual norm, i.e., the total variation of tile Radon measure (which equals the functional norm over Co(G)). Clearly, the integral above equals even vt *N,f (cf. Definition 2.6 and (2-8)) and so it is certainly a convolution in the sense we considered in Chapter 2 eventually. Writing ( v l k ) , h ) := [AkV(t)k] (1) we obtain similarly (for 1
# 0 ) that
A k V ( t ) f = v , ( ~ f) and
(4-3) The notation vl(k)does not contradict t o the usual short notation of derivative; we k show presently = vt on S with respect to the norm topology of Co(G)'. Indeed, it follows from (1-24) that a holomorphic semigroup over a Banach space is also holomorphic on the open sector with respect to the operator norm, while f ( 1 ) is a bounded linear functional on f ---* f is an isometry of Co(G) and f it. Observe that A c D because the topology of Co(C) is finer than the weak distribution topology, D is a continuous operator in D'(G) (consequently, its corresponding restriction is closed in Co(C) ) and d L ( v ) f equals the weak distribution derivative %L(exp tv)f),,o for any locally integrable function f and
(5)
for any v E g . This implies by Corollary 2.8 that vt(k)* f = ( D k v t )* f for all f E Co(G) . On the other hand, if f E C"(T) then this further equals v t * A k f = vt * Dk f because any holomorphic semigroup commutes with its generator. If f is also compactly supported then this equals vt * f by Exercise 2.6. Now for f E C,"(G) c C"(T) both these conditions are satisfied and evaluating
(
)
75
Analytic Mollifying k
at 1 we can see that v1(') = D k v t = ( o f ) vt as distributions (the same can be proved without invoking Exercise 2.6 by more calculation; namely, [vt* of]( 1) = (vt 1 (W)" ) = (Vt = (D, +vt , = [ P t v t )*fKl) 1. 9
ai)
i)
We note that (4-1) and Theorem 7.4.1 of [IIor] imply immediately that vt is a smooth measure for any t # 0 because any Radon measure on a n N-dimensional manifold is clearly included in B:;, for any s < - N / 2 (the notations are those of [Hor] ). By similar but more careful methods it could also be established that these vt's are analytic. We do not go into the details because we shall prove stronger properties in a slightly different way.
Lemma 4.2.
vt * v, = ut+, for all 1 , s E S .
PROOF:Since the convolution is associative on the set of finite Radon measures, we have (4* v,) * f = V(t)V(s)f = V t f s * f for any compactly supported continuous f 1 we obtain the statement.
Lemma 4.3. C,"( R x G ) set
Let
. Evaluating these elements of Co(G) at
c be a number of absolute value 1 in S and for
p E
(g
( A is the Lebesgue mea~iireon R). Then u is a distributioii and - CDc) u = 6 where 6 is Dirac's delta (6, p . ( A x p)) = cp(0, 1) . This distribution is, in fact, a locally integrable furiction which vanishes for t 5 0 and smooth on (R x G) \ ( ( 0 , l ) ) . Its restriction to { t } x G equals t h e Radon -Nikodym derivative d v ~ t for a// positive t (not j u s t for almost every t ). dP
PROOF:The facts that u is well-defined and is a distribution follow from (4-2) if we can prove that the inner integral yields a measurable function of 2 for any 'p. In fact, this function is smooth for positive t ' s l ~ yLemma 4.1 and by the fact that 1 + p(t, .) is a smooth function from R into the Banach space Co(C). We mention that one can prove by similar methods that u is actually a Radon measure but we do not need this because we shall prove the stronger statement of the lemma otherwise. h'(t) = The continuous derivative of the inner integral h ( t ) is (vet, d,p(t, .)) + (CDvCt,p(t, .)) by Lemma 4 . 1 and here the second term can be written as S,
-srh ' ( l ) d t =
(u,
(& - < D , ) + p ) . But h ( t ) = [V(Ct)p(t,.)'](1) and
2
-
p(t, .)" is continuous from R into Co(G), thus h is continuous from the right at 0 and h(0) = p(0,l) .
Chapter 4
76
Then the restriction of u to (R x G) \ {(0,1)} is a smooth function by Theorem E.17. Denoting this function by u ( t , z ) we have for any cp E CT(R\{O}) and $ E CF(G) the following:
h ( t ) = (vet,$) for t > 0 , h ( t ) = 0 for t < 0 and h l ( t ) = for t # 0 then h = hl as distributions on R \ (0) . But hl is continuous because u ( t , x) is continuous for 1 # 0 and h is continuous because h ( t ) = [V(
0 . Since this is true for any $, we have u ( t , .) = qt for t > 0 and u ( t , .) = 0 for t < 0 as distributions on G . Then Fubini’s theorem and (4-2) imply that u(2, x ) is also locally integrable around the point ( 0 , l ) and realizes the distribution u . Hence if
&$(z)u(t,z)dz
d4 From now on we use the notation p t = - , i.e., for any 1 E S we denote dP
V(t)f = pt * f where V is the holomorphic semigroup generated by 8 Llco(Gl)( a ) . We also write p ( t , r ) := pt(x) sometimes; and we set p ( 2 , x ) = 0 for 1 E -SU {0} , so the distribution u in the lemma above can be written as u ( t , z ) = p ( < t l x ). We mention that another usual approach starts not from Co(C) but from L 2 ( G ) and, using Theorem 3.28 and Proposition 3.27, yields a p t for which pi+ E L 2 ( G ) (cf. Exercise 4 . 1 ) . Then one can apply the same to a+ (cf. the remark after Notation 4 . 7 ) and then p t will be smooth by Lemma 4 . 2 , Theorem 3.28 and Proposition 3.27. Here the validity of Lemma 4.2 can be established several ways
by p t the unique smooth integrable function on G with which
(
but the most natural one would be to show this p t coincides with our one (defined by Lemma 4.1 ). Unfortunately, this is not so easy, Exercise 4.2 shows a possible path. Arguing along these lines the rather hard parabolic rcgularity theorems can be circumvented to a considerable extent for nice a’s (see [Nel] and especially [GBr] ). Since we want to prove eventually the estimate of Theorem 4 . 1 0 , these theorems are indispensable for us (though just for rather symmetric operators, cf. the remarks before Lemma 4 . 9 ) . Therefore we did not restrain from them even a t this early stage. We also mention that the correction of the gap in “el] is essentially moved by the fact that p t ( z ) becomes “negligible” if t + O+ and 2 is separated from 1 . Now to show this by the method described in the end of [Glr] one should first check that p is locally an L2 function there, and it is difficult to do that, for lirnt,o+ llpt112 = 00 . Of course, the results of the present chapter (especially Exercise 4 . 5 ) clarify this picture but they depend on parabolic regularity. The following lemma forms the main bulk of the proof of Theorem 4 . 5 . T h e idea is that a certain family of regular representations provides the “worst instances’’ of the representations considered in the theorem. T h e reason for separating this lemma from the theorem is that its results are also of independent interest.
77
Analytic Mollifying
Lemma 4.4. The function p vanishes outside the identity component Go. Let 121 denote a left or right invariant distance from 1 in Go and write xr(z)= erlzl for r 2 0 . Then X , . Fpt is integrable for any r 2 0 , t E S and right invariant differential operator F and also for any left invariant differential operator F . In S we have (4-4)
Fix a closed subsector S1 of S U (0) and also fix an r . Then we have constants M1 and Mz such that (4-5)
llxrPtlll I exp(M1
+ Mz111)
for t E S1 \ (0)
.
PROOF: Let P f = lco . f , then P is a bounded operator o n Co(G). Obviously, L ( z ) P L ( z ) - ' f = l z ~.fo and so P commutes with any L(exp v) ; hence P also commutes with V (cf. the analogous argument i n the proof of Lemma 4.1 ). This implies by Lemma 4.1 that Go supports ut . Consider the Banach space X = X, .Co(Co), i.e., the elements of X are the functions f on Go for which x;'f E Co(G0) and the norm o f f is the maximum of x;'ljl. Clearly, Co(G0) is contained in X and its topology is finer than the subspace topology (because r 2 0 ). On the other hand, x, . C,(Go) = C,(Co) as a set and therefore Co(G0) is dense in X , One can easily check that X' equals the set of those finite Radon measures u for which X, . u is still finite and the norm of u i n X' equals the total variation of Xr . u . Let T, be the left regular representation over X , it is a continuous linear representation and (4-6)
llTr(z)ll
I xr(z)
(because the x's are submultiplicative and x(xc-')= x(z) ). Denote by A , and V, the corresponding operator and holomorphic semigroup. Since TO= Tlco(co)is a part of T, and the topology of Co(G0) is finer than tliat of A', we obtain A0 C A, and so [Vo(.)f]' = A,Vo(.)f is valid with respect to the coarser topology of X whenever f E D ( A 0 ) and hence Vo(t)lD(Ao)c V,.(l) for any 2 (we have used Theorem 1.5 for both semigroups on the rays). VOequals the restriction of V to Co(G0) and D ( A 0 ) is dense in Co(Go),thus on S we have (4-7)
Vr(2)f = PC* f
for any f E Co(G0) . Furthermore, the fact A. c A , now yields for any t E S (4-8)
ArVr(t)f=(&t)*f
~ ~ E C O ( G. O )
Since f + f is an isometry of X and Dirac's delta is a functional of norm 1 on X , we can infer from the description of X' the following. If v is a distribution on
78
Chapter 4
Go such that a continuous operator B in X can be found satisfying u * f = B f for f E Cr(G0) then u is, in fact, a finite Radon measure satisfying (4-9)
llxr . ~ lI l IlBll .
Thus (4-5) follows from (4-7) because X is a Banach space. Similarly, (4-7) and (4-8) yield (4-4) (because V, is holomorphic on S in norm, cf. the proof of (4-1)). To prove the statement concerning Fpt for a right invariant F set B = aTr(b)Vr(t) if F = d L ( b ) . First show that B is an everywhere defined continuous operator. We know that aT,(b) is continuous in the locally convex space Coo(Tr)and the latter can be described by A , (see Theorem 3.28). Thus we can infer from (1-24) that V r ( t )is continuous from X into Coo(Tr) , and hence so is B . T h e fact that ( F p t ) * f = B f for f E CP(G0) follows from (4-7) by Corollary 2.8 (also observing that aT, c d L ; this is a consequence of X ’ s topology being finer than the weak distribution topology, cf. the proof of Lemma 4.1 ). We turn to the problem of a left invariant F and reduce it to the already proved right invariant case. Since xr is locally bounded and submultiplicative, we may assume that the invariant metric came from an invariant Riemannian structure (cf. (F-19)). Then we know that F can be written in the form f j F j such that each F, is right invariant and the functions f j satisfy inequalities of the type Ifj(x)l 5 M . eClzl on Go. We sketch the simple proof of this fact. Let k be the order of F and F l , . . . , F,, be a basis of the complex linear space of right invariant differential operators of order 5 L . Then they form the basis of the module of all differential operators of order 5 k over the ring of smooth functions and therefore we obtain a matrix valued smooth function h such that Ei = h,,Fj where E l , . . . , En is some basis of the linear space of left invariant differential operators of order 5 k . This space Z is invariant under the adjoint representation Ad of the group over the enveloping algebra and this Ad can explicitly be given on the left invariant operators as A d ( i ) E = R ( x ) E R ( x ) - ’ . Since the F j l s commute with the R(x)’s, one can easily check now that R(x)hij = C:=l Ad(x)k;hkj (where we denoted by Ad(x) , somewhat informally, the matrix representing Ad(x)l, with respect t o the basis E l , . . . , En ). In particular, h(x) = Ad(x)* .h(1) and the proof is completed by observing that []Ad(.)I1 is submultiplicative and locally bounded (the norm is some operator norm over 2 ) .
cj
cj”=l
We can also obtain some information about the dependence of the constants M I ,A 4 2 in (4-5) on the parameter r . Since the following theorem can be proved without this, we postpone it to 4 . 6 . Another proof of Theorem 4.5 may use Theorem 4.6 instead of Theorem 3.8 (cf. Exercise 4 . 4 ) .
Theorem 4.5. Let T be a continuous linear representation of G over a sequentially complete locally convex space X , fix a right or left invariant Riemannian structure on G and let 1x1 be the distance ofx from 1 if I E GO. Assume that T is such that the family { e-ClzlT(z) ; 1: E Go} is equicontinuous for some C . Then U(2) := T ( p l ) is an everywhere defined continuous operator for any 1 E S , and U is a holomorphic semigroup with generator A = aT(a) .
79
Analytic Mollifying
Note that the exponential estimate on T is satisfied for any continuous linear representation if X is a Banach space (by (F-19)).
PROOF:Writing e(z) = e-ci+I for z E Go (with the C of the condition) and e(z) = 0 otherwise, we have B(e) is equicontinuous (see (3-4) for the notation). Then any 3: is gbounded and T ( p t )is everywhere defined (because X is sequentially complete and Go U {m} is metrizable) and, moreover, we have by (4-5) that the family U ( t ) satisfies the equicontinuity assumption of Definition 1.15. We also have T ( p t ) T ( p , )= T(pt * p 3 ) because the conditions of Lemma 3.3 hold. Then Lemma 4.2 implies that U turns addition into composition of operators. It follows from (4-4) that U ( . ) e is holomorphic on S for any c and we also obtain the formula
(4-10)
-$J(t)z
= T(Dpt)c .
Now fix a direction ( (i.e.l a complex number of absolute value 1 in S ) . We want to show that t --+ U ( ( t ) is a one-parameter semigroup. Since the local equicontinuity has already been established, it is enough to check the weak continuity of t + U(Ct)z on Rt for any 3: because of l'heorein 3 . 8 . Of course, we saw that these functions are actually analytic inside R+ ( i n the original topology) but this is not enough to infer anything a t 0. So fix c E X and u E x' ; then, writing h ( g ) = (u,T ( g ) x ) , we have (u,U ( ( t ) 3 : )-= J p C t ( g ) h ( g ) d g . We assert that this integral equals V,((t)hl ( 1 ) with h l = hlG0 whenever r > C and V, is the holomorphic semigroup from the proof of Lemma 4.4 (the fact that x;'hl E Co(G0) follows from the compactness of the balls, cf. Appendix F, p . 2 7 8 ) . Indeed, the functional f + V,((i)f(l) is continuous and so it is a Radon measure by the description of X' in the proof of Lemma 4.4 ; and then it must equal pct by (4-7). Thus the desired weak continuity follows from the continuity of Vr(Ct). So U along any ray yields a one-parameter semigroup. This and the homomorphic property of U imply U is a continuous representation of the closed subsectors and with this we completed the proof of the assertion U is a holomorphic semigroup. Denote by B the generator of U ; we are going to prove B = A . We know from (4-10) and (1-24) that B U ( t ) = T ( D p l ) for any t E S . Let c E X , u E D ( A ' ) and consider the function
f(t) = (ul B U ( t ) c )=
J DPt(!l)/+l)
on S where we use the notation h(g) = (u,T ( g ) z ) again. Let pn be a sequence approximating the function 1 on Go in the "right invariant fashion" from Lemma 2.9 and set
fn ( t ) =
J D(pnpt )( 9 ) . h(g)dg = ( D+
11
1
p n pt
) .
We know from Proposition 3.15 that the distribution Dth is nothing else than the continuous function ( A ' u , T ( . ) c ) , so both e . h and e . Dth are bounded continuous functions. The boundedness of the latter implies by (4-5) that lim fn(t) = n-oo
J
Dth . p t = ( A ' u U ( t ) c )
80
Chapter 4
On the other hand, we assert that this limit equals f ( t ) . We have D(cppt) = cpDp, EjcpFjp, for any cp E CF(G) where the differential operators E,, F, are right invariant, depend only on D and the E,’s have no constant term, so Lemma 4.4 implies this assertion because p . h is a bounded function. Thus by the closedness of A we have A U ( t ) = B U ( t ) for all t E S . Since Dp, = D f p , , the method above (using this time the other versions from Lemma 2.9 and Proposition 3.15) yields U ( t ) A c B U ( t ) . We also have U ( t ) B c B U ( t ) because B is the generator of U and then using that both A and B are closed we obtain their equality by tending with t to 0 along a ray.
+ cj
Theorem 4.6. Considering a distance arising from an invariant Riemannian structure in Lemma 4 . 4 , it is possible to write M2 = max(c1, c2rk ) in (4-5) so that M1 and the positive constants c1,cz depend only on the element a, the subsector S1 and on the distance 1.1 we use. Here k is the order of a . This estimate is tolerably sharp (try to examine nice a’s for G = R ). For a non-Riemannian invariant metric one obtains an estimate by joining (F-19)with this theorem.
PROOF: Because of (4-7)and (4-9)it is enough to estimate the function IlV,(t)ll . Now we may forget about the exact origin of V, and should just use the estimate (4-6) of the continuous representation T, which yielded V, by Theorem 3.33. If we fix a subsector then we have a positive number y such that 111 221 2 y(ltll 1121) whenever t1 and t2 lie on the two boundaries of the sector. Therefore it is enough to prove the inequality for the degenerated special case when the sector is a ray and changing the notation we can assume that this ray is the real half-line. It is not hard to check that for any chart cp around 1 and satisfying cp(1) = 0 we have Ilcp(g)ll /Igl varies between positive bounds on any compact subset of D ( p ) , where the norm means the Euclidean norm on RN (this holds even for Riemannian structures which are not invariant). With considerably more effort one can also prove the existence of a chart cp satisfying IIcp(g)ll = Igl for all g E ’D(cp) (see, e.g., in [Hell]) but this is not important for the purposes of the present proof. What we shall need is the following consequence of (4-6):
+
+
(4-11) for small g’s with some positive constant c (where we fix some chart cp and, with the customary abuse of language, we denote the images of anything in this chart by the same letters except that the unit 1 of G goes t o 0 in RN ). We must examine carefully the proof of Theorem 3.33 in order to get a good estimate. Our starting point is the inequality (E-34)which depends on the differential operator and the neighborhood U considered there. Now we use this for the two differential operators D+ and Df and fix U so that (4-11)hold on it. Bear in mind that the two operators have their different U-domains going into (E 34). We shall utilize that in (E22) the test function $ can be chosen freely. Let $1 E CF(U) such that $l(O) = 1 , max1$11 = 1 and the set where $1 does not
81
Analytic Mollifying
vanish is star-shaped with respect to 0 (in fact, we could take $l(g) = $,o(llgll) with a non-increasing non-negative $0 equalling 1 around 0 ) . Then for s 2 1 set $J8(g) = $l(sg) . Thus we obtain a family E C F ( U ) satisfying 1ClS(O) = 1 = max($,I and laa$d)5 C(a)slaI for any multi-index (Y (the constant C ( a )equals maxlaa$ll but we may forget this). Furthermore, we have a positive constant w such that $J8(g)= 0 whenever llgll 2 w / s . Write these into (E-22) and apply (E34)to obtain the following (we change the notation r(X) t o d(X) in order to avoid confusion with our non-negative parameter r ; similarly, we denote the differential operator by E because the letter D is occupied and, i n fact, the two cases E = Dt and E = Dt must be considered) If(0)l 5
for any s 2 1 and any f E Ck(U) . Here the constant I'\ differs from the I( in (E-34) but, luckily, has nothing to do with our parameter r , T h e invariance of E enables us to use this formula (exactly as in the proof of Theorem 3.33) even if we just know that f and Ef are continuous on U . Because of (4-11) the number A4 = exp(cwr/s) can serve as M in the proof of Theorem 3.33 and for the 2,y, u,v there we obtain (with this M ) the following. 1 1 2 1 1
5 IiM . ( d ( X y
llull
5 2KM
'
( d(X)-'
v
llyll+ ( s k d ( X ) - ' IlVII
+ (skd(X)-'
sd(X)-"k
v
sd(X)-'lk
) .112:11) ) . llull) .
Thus if d ( X ) > (1 V 21\'M)k sk for both U-domains then the resolvent ( X I - A,-)-' exists and satisfies / ] ( X I - A,)-'ll 5 2I\'M/d(X) . At this point wernake the choice s = r V 1 which has the benign effect that M can be replaced by ecw and so we have a constant C such that d(X) > (Cr v C ) k implies Il(XI - A,)-' 5 C / d ( X ) . Emanate two half-lines from 0 in C to left-up and left-down a t angles from vertical not greater than any of the corresponding angles of the half-lines in the boundaries b the path consisting of llie translation of these of the two U-domains. Denote by r two half-lines by the positive number b . We have positive constants bo, bl such that d(X) 2 b l ( b - bo) for X E r b and therefore if 6 > bo b;'(Cr V C ) k then rb lies in the resolvent set of A , . The estimate we derived for the resolvents and the holomorphic property makes it possible to choose rather freely the path along which we compute the integral of f(X) = etX(XZ - A , - ) - ' z (for 1 > 0 ) and we saw in the proof of Theorem 1.19 that this inverse Laplace transform does yield V,(t) when the path is an angle to the left whose slopes are close to vertical and whose vertex lies sufficiently to the right. Therefore Vr(t) can also be computed for any t > 0 by this integral along rb for any b above. Collecting our estimates we see
11
+
with some positive constant
K
and for all b
> bo
+ 6;'(Cr V C ) k .
stipulate b 2 2bo then the estimate above is stronger than M
If we also
obt
o with ~ some
82
Chapter 4
constant A40 while the two conditions o n b are implied by b 2 max(c1, c2rk ) if c1 , c2 are suitable constants. Choosing b = l / t whenever this last inequality is satisfied by that and the smallest b otherwise we complete the proof. The theorem we just proved is so powerful that rather precise pointwise estimates can be obtained from it by the methods of [Ma41 (cf. Exercise 4.5). Moreover, the right as well as the left invariant derivatives can also be estimated and so we have a different approach to some results in [Rob]. I n this book we are mainly interested in the behavior of operators of the form d T ( b ) T ( p t ) for any b E U, when T is fixed and concentrate our efforts in reaching such results on an optimally short path (from which Theorem 4.6 was a digression). Notation 4.7. The element a+ has the same characteristic form as a has, so on S x C we have a function q which is the p of a + . Then we define p + ( t , z) := q ( - t , z) . Of course, this notation would be too liarsli an abuse of language unless it is in accordance with (2-5); but this condition does hold, i.e., p + ( t , t ) = p(-t,z-')A(z)-l for t E -5' and z E G . This fact is not important for us, so we just hint at its proof identify L2(G) with its dual this time through the bilinear form J f g (in order to avoid the unnecessary burden of conjugating back and forth) and check that o(t)= V ( - t ) tlirough this identification, where U and V are the holomorphic semigroups generated by 8 L ( a ) atid 8 L ( a + ) over L ' ( G ) . Lemma 4.8. Let ( be a direction in S , II c R x G be a n open subset, and u E V ' ( H ) be a solution on H of the equatioii - (LIZ) u = 0 . Then u is a smooth function and whenever ( s , y ) E H and 11, E C,"(H) are such t h a t 11, = 1 on a neighborhood of (s, y) then
(g
where, denoting by H1 the set where 11, difers from 0 and 1, Ho = H1 n { (2, z) ; t 5 s } and it can be replaced by any greater measurable set for the integrand vanishes outside Ho .
PROOF: We can see from Lemma 4 . 3 (applied to a + ) that the function h(2,x ; s ,y) = p + ( C ( t - s), zy-')A(y)-' is smooth on the set { ( t , z ; s , y ) E ( R x C ) ~( ;t , x ) # ( s l y ) } andforanyfixed(2,x)and V E C , " ( R X G ) the integral J h ( 2 , z; s, y)cp(s, y) ds d y exists and equals 7' * cp ( t , z ) with r(i, x) = p + ( C t , x) (then r is a locally integrable function on R x G ) . Now r * c p is smooth and E ( r *p) = ( E r )*'p for any right invariant differential operator E ; apply this to E = - & - ( D : (then E r = 6 byLemnia4.3for p + ( t , z ) = q ( - t , z ) ) toinfer that h satisfies the assumptions on the kernel I< i n Proposition E.13. It remains to check that the integrand vanishes outside H o . If t > s or if 11, = 1 around ( t , z ) then this is obvious. If $J = 0 around ( t , z ) then ( 1 , ~ )# ( s l y ) and hence (-$ - (0:) h ( t , z; s, y) = 0 by the right invariance of Dt and by Lemma 4 . 3 .
83
Analytic Mollifying
We mention that analogous formula holds for solutions of the equation = 0 . Namely, e ( t , z ) = ~ ( - 6 1 , ~ )satisfies the equation (-$ -Cot) e = 6 (this follows from the proof of Lemma 4 . 3 using (4-1)) and hence the convolution by e from the right provides a kernel with which the argument above works.
(i -Cot) u
The next lemma would be a special case of a more precise version of Theorem E.17.a (which version, in fact, follows from the discussion in Appendix E ) . We include a proof for the sake of the following niceties. It may be useful t o know in some applications what is the exact origin of the constants; the fact that the problem of analytic continuation of an arbitrary solution is reduced to the similar problem about p + might open up the way to a simpler proof than Eidelman's one, thanks to the high symmetry of p+ ; and finally, to prove just Theorem E.17.a instead of the sharper version may be easier another way. Lemma 4.9. Let I'\ and l i l be compact sets in G such t h a t 1 E I< c int(l\'l) and A'lI<-' c D(p) for some analytic chart (p, and let J and J1 be compact sets in R such that J c i i i t ( J 1 ) . Also fix a direction in S . Then we find positive constants e and w (depending on these data) such that whenever u satisfies the same equation as in the previous lernrna OIJ some open set containing J1 x l < l we have the following. Writing ut(.c) = u(L,p-l(x)) and P = { z E C N ; IzJ I 5 e V j } a smooth function ii(t,z ) exists around J x (p(Z\') P ) (smoothness is meant in the corresponding 2 N 1 real variables) such that it is holornorphic in z for any fixed t E J , G ( t , x) = u I ( x ) for ( 1 , x) E J x (p(li)+ R e p ) and
<
+
+
where, of course, X is the Lebesgue measure on R
PROOF:Choose a 4 E CY(R x G) which equals 1 on a neighborhood of J x K and its support is contained in Jl x l i l . Denote by 2 the interior of the set where 4 = 1 . Then M := ((51 x l i l ) \ Z ) ( J x A')-' (the division is meant in the Lie group R x G ) is compact and ( 0 , l ) 4 M . Furthermore, the projection of M into G is contained in l
x ( t , z) := p+(Ct,p-'(z)) is a solution of the corresponding parabolic equation around MO = (id x p)M by Lemma 4 . 3 , and so we iiifcr from Theorem E.17.a and from the compactness of Mo the existence of a compact neighborhood PO of 0 in C N and of a smooth function 2 defined around Mo PO such that g(2, .) is holomorphic for any fixed t and 2 = x on Mo Re PO . Since the division in G is analytic and the iiiodular function is also analytic, we can find a neighborhood PI of 0 i n C N such that the mapping t , y -+ zy-' (meant through the chart) continues holomorpliically onto (p(K1) P I ) x (p(l<) P I ) and d = l / A o p-' continues holomorphically onto p(K)+Pl . Furthermore, if PI is small enough then we have J x ( p ( l i ) + R e P I ) c (id x cp)Z and (id x p)( (51 x 1<1) \ Z ) ( J x (A' R e p , ) ) - ' C MO PO . Thus
+
+
+
+
+
+
84
Chapter 4
if ,Q is so small that P is contained in such a small PI then the corresponding integral expression from Lemma 4.8 yields the desired ii (differentiations under the integral sign can be performed because the integrand is smooth around the compact J1 x K1 x J x ((o(K) P ) ; the kernel itself is also smooth there, so the maximum of its absolute value over this compact can serve as w ).
+
An analogous argument can be applied to left invariant equations (cf. the remark after Lemma 4.8). One may guess that ,Q and w can be chosen uniformly when the direction C varies in a closed subsector. For this just p+ has t o have the corresponding property, i.e., the analyticity of x around Mo should be “uniform” for C from the subsector. This is indeed true and by similar reasoning we could even prove rather more general statements because Eidelman’s estimates (see Appendix E and [Eid] ) depend only on the various positive numbers describing the parabolic property, Holder continuity, etc. of the system. In our case this can be circumvented utilizing the convolution identity from Lemma 4.2 (and using Theorem D . 3 , cf. the proof of the next theorem). Theorem 4.10. Fix a closed subsector S1 of S U {0}, a left or right invariant Riemannian distance 1.1 from 1 and a basis w l , . . . , W N of g . For any 1 N-dimensional non-commutative niulti-index cr write E, = - d L ( w ” ) . Then for cr ! t E S1 \ {0} and r 2 0 we have
where k is the order of D . Attention: we assert that C is independent of r
PROOF:For brevity we shall write B ( n , t ,r ) = max
I4=n
/
IEapt(z)le‘lZIdx
Go
(it is finite by Lemma 4 . 4 , though the present proof will establish this fact independently). First just consider the case when t is real. Striking a t the heart of the matter, we begin the proof around the singularity ( 0 , l ) . Fix an analytic chart (o around 1 such that (o(1) = 0 , and apply Theorem E.18 to obtain a function v for which - D ) [ p ( t ,z) - v ( t , ( o ( z ) ) ] = 0 holds and its extension 6 satisfies the estimates and holomorphy described in Theorein E.18. Choosing a suitable compact neighborhood K of 1 and taking J = [0, d] with a small enough d we can apply Lemma 4.9 to the difference and see that it extends holomorphically around J x ((o(K) P ) . If co is the maximum of the absolute value of this extension on this compact then &laf[p(t,(o-’(t))- v ( t , t ) ] l 5 c o . Q-IPI on J x (o(K)for any multi-index /3 by Cauchy’s formula. Now estimate the derivatives of v ; for this we apply Cauchy’s formula on tori depending on t . Namely, let w1 be a positive number such that, writing PI = { z E C N ; Izjl 5 wld’Ik V j } , the domain of 6 contains
(6
+
85
Analytic Mollifying
+
+
J x (cp(K) P I ) and apply Cauchy's formula with the torus p(z) ( w l t ' / ' T ) N for 0 < t 5 d and z E K . Writing (E-14) into the inequality thus obtained we get the bound C l ' W 1-PI , t-(IPItN)/k
)(
-b
+
The expression under the exp can be estimated by - c 4 . ~ ~ t - l ~ '--l k p ( c5~ )with ~ ~ some positive constants c4, c5 (in fact, for any c4 < c2 one can find such c5 ). Now applying Theorem D.3 to the analytic vector fields d L ( q ) , . . . , d L ( w N ) on cp(1t') we have
= wad'/' 1 V (w1d'lk)-' V e - l ) if w o is the constant arising from Theorem 0 . 3 ) . Denote by B K the maximum of the corresponding truncated integrals. Using the dP fact that - is bounded on I<, writing c7 = maxZEK1x1 and substituting t = dX t - ' l ' ( p ( z ) we can see for 0
(4-12)
(
and
E
E A' with some positive constants w2, CO, c4 and
B , y ( n , t , r )5 ( w z t - l / k ) n' c 8 . e e l r
for 0
c6
(here
w2
Now turn to the domain W = (R+ x GO) \ ((O,d] x I<) . T h e function = p ( t s , y t ) is a solution of ( 8 , - D ) u = 0 on H , = R x G \ { ( - t , z - l ) } by Lemma 4.3 and by the right invariaiice of D . Now if ( 1 , z) E W then [ - d , + c o ) x Z<-l c H , , and for any right invariant differential operator E we have E,p(t,z) = Eyu(O,l) . We are going to apply Cauchy's formula t o the hotomorphic extension of u(0,p-'(.)) and then invoke Theorem D.3 as above. Choose a compact neighborhood Q of 1 such that Q c D(p) n I<-' and write { 0 } , { l } , [ - d , d ] and Q in the place of J , K ,J1 and Z i 1 i n Lemma 4 . 9 . Then Cauchy's formula and Theorem D.3 yield .(sly)
+
(4-13) for ( t , z ) E W (here the upper limit d could be replaced by 0 because of the proof of Lemma 4.9 but this is not important for us). Thus we see from Fubiiii's theorem that
w: . w
.lxd(/
GoxGo
)
1Q(Y)IP(s,y")IePlZldEddy ds
estimates B ( n , t , r )or B ( n , t , r ) - B ~ ( n , t , r for ) t > d or 0 < 15 d , respectively. Applying left translation in the variable z and using the properties of the distance
86
Chapter 4
we obtain the estimate SQ dy S ., lps(x))erIzIdx for the inner integral. But we know from (4-5) it is finite and comparing this with (4-12) we arrive a t the desired estimate of B ( n , t , r ) for all t > 0 . It is amazingly easy to pass from this to any t E S1\ (0) . Since IzI 5 lxyl+ I y - l I forany z , E~GO, w e h a v e Il(f*g)xrIII 5 Ilfxrlll.llgxrll1 w h e n e v e r f , g a r e measurable functions on Go such that the right hand side is finite, where x r ( z ) = erlzl . But Ept+s = ( E p t )* p s for any right invariant differential operator E and any t , s E S by Lemma 4.2 and Corollary 2 . 8 . Now if 211 5 212 are the angles bordering S1 then we choose y1 < 211 , y2 > 212 and consider the above with t , >~0 and s = eiYiT , j = 1 , 2 . Then each point in S1 \ (0) can be obtained as s t and if s t E S 1 then t V Is1 5 Is t l 5 ~t with some positive constant K . Then applying (4-5) we obtain the estimate for each 2 E S1 \ (0) (with a C worse by the factor nilk than the C we obtained for t > 0 ).
+
+
+
The estimate we proved is quite appropriate for our purposes but we note that one can obtain more precise estimates simply by more care. E.g., we could use Theorem 4.6 instead of (4-5) and this theorem also enables u s to utilize the fact that in estimating B - BK the domain of the crucial integral is "punched" if Q is much smaller than I < - ' . With more work one may check that the difference pt(x)- v ( t ,p(z)) decreases rather rapidly when t 0 . \Ve could also estimate the more carefully. Solving Exercises 4.4-6 the reader integral of I E,v(t, p(x))lerIZI may become familiar with these methods. I n fact, the generalization of Theorem 4.6 holds, i.e., C1 can be chosen independent of r and C,(r) = c1 V cark then (the constants c1, c2 have nothing to do with the constants denoted by the same letters in the proof above, and a bit worse than the corresponding constants in Theorem 4 . 6 ) . Since p also satisfies ($ - 0:) p ( t , x ) = 6 (cf. the remarks after Lemmas 4.8 and 4 . 9 ) , the proof of Theorem E.10 works for dR(w") instead of dL(w") x must be used at the coralmost verbatim. In this case the substitution xy responding place, and it is not measure-preserving; but y can vary just in the fixed compact Q , so this causes no difficulty. Moreover, using ly-ll = Iy( we can see that simply e'lyl dy will replace e'lyl dy . Of course, the constants arising from Lemma 4 . 9 , Theorem E.18 and Theorem D.3 will differ; and consequently the constants of the result must differ, too. -+
-+
sQ-l
sQ
Theorem 4.11. If T satisfies the condition of Theorem 4.5 then the range of U ( t ) is contained in C"(T) for any 1 E S . Consequeritly, C " ( T ) is dense in X for such T 's. Furthermore, fixing a basis wl, . . . , W N in g aiid a closed subsector S1 of S U (0) we have a positive constant C independent of T such that the operators
It II 4 / k
8 T ( w a ) U ( t ) form an equicontinuous set if (Y varies freely and t varies in a clal . a ! bounded subset of S1 . PROOF: It follows from Lemma 3.7 and Theorem 4.10 by induction on la1 that 8T(wa)U(t) = T ( d L ( w a ) p t ) for any t E S and any a ,T h e asserted equicon-
87
Analytic Mollifying tinuity then follows from Theorem 4.10 and hence
{ $8T(wa)U(t)x
; cr
I
is
bounded for some positive s depending on t . Thus U ( t ) z E C”(T) by Proposition 3 . 1 8 . Since x is the limit of U ( t ) x when t 0 i n R,, the denseness of C”(T) follows by the trivial observation that there exist strongly elliptic elements. -+
Remarks. It is clear from this proof how one can formulate equicontinuity statements for the whole S1 , and using the more precise estimates mentioned after Theorem 4.10 we may tell rather exactly the depeiitleiice of these equicoritinuity relations on the same of T . The corresponding estimates on the left invariant derivatives of p make it possible to deal with the operators U ( t ) d T ( b ) for any b E U,. If one is just interested in obtaining good analytic vectors then the following modifications i n o u r method yield better results. I n the proof of Theorem 4.10 apply Cauchy’s formula with constant radius also to 6 ,then we obtain an estimate of the form IE,pt(z)l 5 Co(i)Clal for 0 < i 5 d and E E A’ , where A’ is some compact neighborhood of 1. Here Co(t) might be enormous for small t’s but at this price we have its independence of a . Since the factor Ilallk did not interfere with the estimate far from ( 0 , l ) (see (4-13)), we see that the ‘‘radius of analyticity” (i.e., the number s in the proof of Theorem 4.11 ) can be independent o f t for t > 0 and hence the corresponding s-analytic space is still dense; and this s is also independent of T and depends only on the basis of the Lie algebra! Moreover, it is not necessary to have a holoniorpliic semigroup for this, so the assumption on T can be relaxed considerably Clearly, it is enough t o find a dense set of I ’ S for which T ( p t ) z exists and belongs to this s-analytic space for small positive 2’s (this smallness may depend on a - ) and litnn+m T ( p t n ) x = 3: with some t,, 0 . We try to establish the first condition by seeking a continuous function p which is positive o n Go, such that E is ybounded and -+
for small 2’s. Observe that in Lemma 3.7 the condition ( i i ) about e was only necessary for the mollifying of $ @ ,so we can drop it now because pt is smooth. Thus the above ensures T ’ ( p t ) z is an s-analytic vector by the remark after (4-13) (since we have the C&) . CIaI estimate near ( 0 , l ) ) . Here the strongly elliptic element n and the compact neighborhood Q of 1 sliould be chosen independent of x if our ambition is to prove the denseness of s-analytic vectors, while they may depend on I if one just desires the denseness of C w ( T ) . Similarly, for the second condition we seek a coiitinuous Q such that e(1) = 1 , z is pbounded and $, = p t n / e satisfies the conditions of Lemma 3 . 5 . Observe that liml,o+ s p t = 1 (this may be checked most easily with the help of the space X = xr . Co(G0) from the proof of Lemma 4 . 4 , because the function 1 belongs to this space if 1.1 is Riemannian and r > 0 and we liave J u t . f = [ V , ( t ) f ](1) for any EX ). On the base of these considerations and Exercise 4 . 5 (applied with k = 2 )
Chapter 4
88
the reader can check the following. There is an s > 0 independent of T such that the set of s-analytic vectors is dense provided by the condition that for a dense set of 2's a constant n = ~ ( 2 1 ) 0 can be found such that
is bounded, where 1.1 is the distance from 1 with respect to a left or right invariant Riemannian structure. We close this chapter by proving the (joint) analyticity of p for t E S .
Theorem 4.12. The function p is analytic on the real analytic manifold S x G and for the variable in S this can also be considered in the complex sense (in other words, if 1 E S , 3: E G and p is an analytic chart around z then p ( . , p-'(.)) around ( 2 , cp(z)) can be extended in the "space variables" to become a holomorphic function on some open subset of CN+' ).
PROOF:First show that xrpt is bounded for any t E S and r 2 0 , where xp(z) = erlzl with an invariant Riemannian distance. Lemma 4.8 implies that (p(Ct,z)I 5 c . Ip(('s, y z ) l ds dy where c and the compact H o are independent of 2 if we fix t > 0 and C . Since the modular function grows a t most exponentially (cf. (F-19)), we obtain the stated boundedness from (4-5). In particular, p t E Co(C0) . We assert that if s1,sz E S and 1 E s1 sz S then
,,s
+ +
(4-14)
Pt(.)
= ( L ( z - ' ) P s , , ( V ( t - s1 - S 2 ) P s * ) " )
where (., .) denotes the usual bilinear function J f g over L'(G) x Co(G) , L is the left regular representation over L'(G) and V is the holomorphic semigroup from Lemma 4.1. First, V ( t - s1 - s 2 ) p s a = p t - s l follows from Lemmas 4.1 and 4.2 because p s , E Co(G) . Then the right hand side of (4-14), as a function of z , is nothing else than psl * p t - $ , and, moreover, it is the continuous realization of it because L is a continuous representation and (., .) is bounded. Then the continuity ofpt and Lemma 4.2 yield that (4-14) holds for every x , not just almost everywhere. Let U ( s ) = L ( p , ) (over L ' ( G ) ) . Then V ( s ) f = p , * f for any f E L'(G) (this can be checked by testing these L' functions with any p E C,(G) and applying Fubini's theorem). Therefore Lemma 4.2 and Theorem 4.11 imply that psl E C"(L) . Then we get the statement from (4-14) because f , g -* ( f , q ) is a bounded bilinear function from L'(G) x Co(G) into C .
EXERCISES 4.1. Assume that G/Go is a t most denumerable and V is a one-parameter semigroup over L2(G) such that its generator A satisfies R ( z ) A c A R ( r ) for all 2 E G . Suppose for a certain t the range of V ( t ) is contained in C(C). Show
89
Analytic Mollifying that a function u E L2(G) can then be found such that V ( t ) f = u+
f E L2(G) .
*f
for all
(Hint: apply the Closed Graph Theorem.) 4.2. Check that in the proof of Lemma 4.4 the fact that p t is function-like was not used a t all. Then infer from a Sobolev type statement that p , is smooth and check that the proof of Theorem 4.5 does not rely on Lemma 4.3 if it is circumvented this way. We note that the Sobolev type statement i n question can be obtained as a special case of Theorem 2.2.7 of [Hor] and it is much more elementary than the other theorem of [Hor] we mentioned t o be applicable when we just had Lemma 4.1.
4.3. Show that pl E L'(GO) for any t E S without using parabolic regularity. (Hints: Consider R over L'(G0) and check that C"(R) c C"(G,) . Then apply the Closed Graph Theorem and Theorem 4.5 .)
In the following two exercises 1.1 is a fixed invariant Riemannian distance from 1. The statements could suitably be generalized when 2 varies in a closed subsector. 4.4. Show that positive constants
for 2
5 "0yk-l , where e > 0
and
I'
CO,
. . . , c3 can be found such that
2 0 (in particular, for any t if
I'
= 0 ). Look
out for the negative sign before c3 (it is the heart of the statement).
-1'
4.5. Show the existence of a positive constant w and real constants
such that l p t ( z ) l 5 (1 V t - N ' k ) .exp
(bl
+ b2t - w .
for any 2 > 0 and I E G o . (Hint: Use the left variant of Lemma 4.8 .) 4.6. Let c
> 0 , k > 1,
t
>0
and r
20 .
b l , b2
Ixlk-ltk-1
Prove that
where the positive constants w1 and w2 depend on c , b and the dimension n . Here h is not necessarily an integer (if I' = o then we mean rk = o ).
This Page Intentionally Left Blank
91 5. The Integrability Problcin The problem in the title is essentially the following: how to decide that a certain given object is actually the tangent d T of some continuous representation T . There also is, of course, the question of unicity. Now a tangent is a n awkward object, it is a mapping from the Lie algebra of the group into some set of operators which are generally not continuous and each has its own dense domain. But we s a w in Chapter 3 that C"(T) is dense and, moreover, a n invariant core for any d T ( a ) , i.e., d T ( a ) C " ( T ) c C"(T) and d T ( a ) is the closure of its restriction to C " ( T ) . This is the motivation for the following definition.
Definition 5.1. Let g be a real finite dinirnsional Lie algebra, X be a sequentially complete locally convex space over C , Z be a dense linear subspace of X and Q, be a morphism (of Lie algebras) from g into End(Z) . Here End(Z) denotes the set of complex linear operators from 2 into 2 (without considering any topological properties); it is a complcx unital associative algebra with the usual operations and then the conimutator [ A ,B ] = A B - B,4 makes it a Lie algebra. Then we say Q, is 'integrable' if a Lie group G with Lie algebra g and a continuous representation T of G over X can bc f o i l r i d socli that ~ T ( Qis) the closure of @(Q) for all a E g . The requirement that @ ( u ) be dense i n d T ( u ) for all Q may seem too restrictive (it would be natural enough to require it j u s t for a Lie generating subset, cf. Exercise 5.1 ). However, in practice when we can show the validity of the weaker condition then the stronger one usually holds as well. In the other direction it is natural to require that Z be dense in C"(T) (with respect to the C" topology). Note this condition depeiids only on d T . If Q, satisfies this then we say it is 'strictly integrable'. Of course, ally integrable Q, extends to a strictly integrable one by enlarging Z into C"(T) and @ ( a ) into d T ( u ) . Choosing an arbitrary neighborhood of 1 in G the tangent of a representation is determined by the values of tlie representation in that neighborhood. We know that each Lie algebra has a Lie group, all Lie groups with the same Lie algebra are locally isomorphic, and there is a simply connected one among them (cf. Appendix F ) . These facts and the following proposition will imply that 'local integrability' would be an equivalent notion (see Proposition 5.3 ).
Proposition 5.2. Let G be a topological group and I€ be j u s t a group. Assume t h a t a local homomorphisni f : G 4 H is given, i.e., f is defined on some neighborhood of 1 in G and f ( z y ) = f ( t ) f ( y ) wlicriever z , y belong to another (smaller) neigh b orb ood . If G is simply connected and locally pathwise connected then we have a unique homomorphism F : G H H satisfying F = f on some neighborhood of 1 (the unicity of F is independent of this neighborhood). By 'simply connected topological space' we meau a pathwise connected space in which any two curves connecting tlie same pair of points are homotopic. Since
Chapter 5
92
the proof below merely uses from algebraic topology the very definition of homotopy (which we shall formulate), it is even accessible to readers who have not heard of it before.
PROOF:Let 7 : [0,1] H G be a curve (i.e., a continuous function) satisfying y(O) = 1 . Then we assert that a unique locally liomomorphic continuation o f f along 7 exists, i.e., we construct a mapping f : [0,1] H H such that r(0)= 1 and r ( t ) - l r ( s ) = f (y(t)-'y(s)) whenever It - sI is small enough; and prove that such a is unique (meaning that the smallness required for It may also depend on Denote by U a symmetric neighborhood of 1 in G which is so small that U . U is contained in the domain of f and we have the homomorphic property o f f satisfied for z , y E U . For the sake of the latter part of the present proof we do the construction for the following more general setting. Let P be a set of parameters, consider a family 7 ( t , p ) of curves parametrized by it, and suppose that the variation is small, namely we require that y(t,pl)-'y(t,pz) E V for t E [0,1] and p1,p2 E P with asymmetric neighborhood V satisfying V . V . V c U . Choose a po E P and, by the usual compactness argument, find an E > 0 such that y ( s , p o ) - ' y ( t , p o ) E V for 1s - 21 5 E . Then set xj = y ( ( j - l ) ~ , p o ) - ' y ( j ~ , p o ) and
SI
r r).
(5-1) rj(t,Y ) = f(z1) . f(xZ) . f ( x j ) f(y(.i&, P O ) - ' Y ( ~ , P ) ) for j~ 5 t 5 ( j + 1 ) .~ We infer from the conditions on the neighborhoods that fj(tl,pl)-'I'k(t2,p2) = f ( y ( t i , ~ l ) - ' y ( t 2 , p ~ ) )if Ik - j l 5 1 (observe that 7(j~,po)-'-y(t,p) E V . V and x, E I/ ). I n particular, rj and f j + l agree on the intersection of their domains (i.e., for t = ( j 1 ) );~ and so we obtain a mapping f : [0, I] x P H H satisfying the locally homoinorphic continuation property for each p . Returning to the case of one curve, the unicity of r is clear: f ( t ) = 1 f(yl)...f(y,) whenever n is large enough and yj = y ( 5 t ) - 7 ( i t ) . For brevity we call this f the lifting of 7 . It is evident that any homomorphism locally coinciding with f must take 7( 1) into r(1) if r is the lifting of 7 ; so the unicity part of the proposition follows. Now let y and d be homotopic curves connecting 1 and x ; this means that we have a continuous function x : [0,1]' H G such that ~ ( t0), = 7 ( t ), x ( t , l ) = d ( t ) , ~ ( 0 , s= ) 1 and ~ ( 1 , s )= 3: . Since x is continuous, any fixed s has a neighborhood P in [0,1] such that x ( . , p ) is a family of small variation in the sense we considered above. Then we see from (5-1) that f ( 1 , p ) for this family is independent of p (because x(1,p) = 2 ) and depends only on s . In other words, if we denote by f the lifting of the whole homotopy x ( . , s ) then f ( 1 , s ) is locally constant. Thus it is constant for [0,1] is connected. This and the simply connectedness of G implies that the end point of the lifting of a curve connecting 1 with E depends only on E , so we can define a function F ( x ) by this. We see that F ( E )= f(x) whenever a curve running in U connects 1 with x ; and such z's form a neighborhood of 1 because G is locally pathwise connected. If x ,y E G are arbitrary and 7 , d are curves connecting 1 with x and y then writing w ( t ) = y(2t) for t 5 1/2 and w ( t ) = x . d(2t - 1) for ' '
+
The Integrability Problem
93
we get a curve w connecting 1 with zy. ‘rhus with large enough n we have F( zy) = f ( r l ) .+ f. ( z n ) where the quotients r, (defined by w ) equal the corresponding quotients obtained from 7 and 1 9 . Therefore F is a homomorphism, the proposition is proved.
t 2 1/2
We remark that unicity (but not necessarily existence) of the continuation of a local homomorphism also holds under weaker assumptions. In fact, such unicity clearly follows if G is generated (as a group) by each neighborhood of 1, while it is evident that the subgroup generated by a neighborhood is open, and any open subgroup is closed. Hence if G is connected then unicity certainly holds.
Proposition 5.3. Let G be a connected Lie group and @ be a ‘locally integrable’ representation of its Lie algebra, i.e., we assume the existence of a mapping To defined on some neighborhood of 1 in G such t h a t To(g) is a linear operator for any g in this domain, To(gh)= To(g)To(h)for small enough g , h , TO(1) = I , the ‘local action’ ( g , ~-+) To(g)z isjointly continuous and dTo(a) = @ ( a ) for any a . Here dTo is taken analogously as in Definition 3.12 and @(a)E End(2) as in Definition 5 . 1 . Then @ is integrable; we have a unique continuous representation T of the for any a . A simply connected covering group 6 of G such that d T ( a ) = similar representation of G exists if and only if T ( y ) = I whenever p ( g ) = 1 where p : G H G is the covering function.
PROOF:To o p is a local representation of G , so it can uniquely be extended by Proposition 5 . 2 . Since a n y element of G is the product of several small elements, the extension takes its values from continuous operators. Then the continuity of the action follows from the continuity of multiplication i n the group. We see from Theorem 1.5 that T(exp2a) is determined by dT(a) for any t E R and consequently T is deterniined on exp g which is a neighborhood of 1. It is straightforward to check our stalements now. We mention that some authors define the ‘integrability of an operator Lie algebra’; this notion coincides with the special case when CJis injective (by identifying with @(g) then). In fact, the integrability of @(g) i n this sense is equivalent to the integrability of @ in our sense by Lie theory (cf. Exercise 5 . 2 ) . Therefore when seeking integrability conditions it would be superfluous to bother about such refinements as, e.g., requiring a certain condition just for a basis of @(g) instead of the image of a basis of g (for they are automatically valid). We have a variety of solutions for the integrability problem in the literature. They amply demonstrate the unfortunate circuiiistance that relatively simple conditions are either not enough for integrability (cf. Exercise 5 . 4 which is a variation of the famous counter-example from “el]) or, at the other extremity, are too restrictive t o be useful in practice. This is the cause of possible difrereiices in tastes as t o which complicated assumptions are “tolerable enough”. For unitary representations of general Lie groups the Nelson condition (the essential self-adjointness of a formal
Chapter 5
94
Laplacean, see below in more details) and the Simon condition (the unitary special case of the condition in Rusinek’s theorem we formulate as Theorem 5.14 below) are certainly among the generally accepted alternatives. In this chapter the emphasis is on variants which are applicable if the ground space is a Banach space or, if possible, a general sequentially complete locally convex space. We mention here that if X is a Hilbert space and @ is integrable then the unitarity of the representation of the simply connected group is equivalent to the extra condition on @ that each @(a)be skew symmetric and, evidently, if we have this for a Lie generating subset then for all a . If one has an integrable @ then the composition of the corresponding representation with any one-parameter subgroup shows the integrability of the corresponding restriction of Qi. In other words, each @ ( u ) is a ‘group pregenerator’. In Chapter 1 we considered some characterizations of these i n the contraction type case. In the sequel we consider @’swhose values are group pregenerators and examine what further conditions would then be sufficient for (or equivalent to) integrability. More precisely, it turns out that the conditions from which we prove integrability imply the denseness of @ ( u ) in the corresponding group generator; furthermore, in most cases we just need one-parameter groups corresponding to a Lie generating subset. Therefore we use the following notational conventions i n the remainder of this chapter.
Notation 5.4. T h e letter V (in itself or with any subscript) will denote a continuous representation of R over X . We denote its generator by A (or A with the same subscript) and u can only denote such an element of g that a V is fixed a t the outset satisfying A 2 @( u ) . Similarly, A ; 2 @(ui) or with any other subscript . It is generally easy t o find expressions for the hoped integral of Qi and if the homomorphic property is already established then continuity is most often checked by the almost evident Lemma 5.5 below (see Exercise 5.3 for another way). It is the proof of T ( g h ) = T ( g ) T ( h ) which poses the real problem. Lemma 5.5. Let v 1 , . . . , V N be a basis of the Lie algebra g of G and consider a product chart ‘p corresponding to it (so p( exp(llvl) . . .e x p ( l ~ v ~=) )t if t is small enough). Then the mapping ( g , x ) V 1 ( p 1 ( g ) ) . . . V N ( p N ( g ) ) r is continuous.
-
We used the notation v , instead of ui to emphasize tliat this lemma is valid even if the V’s has nothing to do with the v’s; but i n applications, of course, we have v , = a, .
PROOF:Since the continuity of means tliat the corresponding action is jointly continuous, we have an iterated composition of continuous functions here. If we assume each
a E g is a pregenerator then in the integrable case
The Integrability Problem
95
T(exp a ) = V( 1 ) must hold for small a’s. Therefore
also holds for small a 1 , 0 2 . On the other hand, if we know (5-2) for small a l , a2 then comparing it with Lemma 5.5 we obtain a continuous local representation - (also using that W ( s )= V(ls) is the one-parameter group generated by @(la)).So in this case integrability is equivalent to (5-2). Since the sides of (5-2) are continuous operators, it is enough to check their equality on a dense set. Maybe the most natural idea is then to seek a dense set of x’s a t which we could justify the formal manipulations with the power series involved. Let u l , . . . , U N be a basis of g and suppose that z is jointly s-analytic for the corresponding closures, i.e., with the usual multi-index notation we have c r ! - l t l Q I @ ( u ) ) 2: ; a } is bounded if 0 5 1 < s .
{
ra
It turns out that if, in addition, z E 2 then (5-2) holds a t z for small a 1 , whose needed smallness depends on the radius s of analyticity. Furthermore, the corresponding Taylor approximations tend to V(t)z. i n the C” topology, from which even strict integrability can be inferred if the jointly s-analytic vectors in Z form a dense set for some s > 0 . We do not go into the details because a sharper result is available (cf. Theorem 5.14 below) but mention that oiie must examine carefully the interplay between associative and Lie power series, and Dynkin’s version of the Hausdorff-Campbell Formula (see, e.g. Section IV/8 of [Serl]) is a useful tool here. A non-strict integrability condition is iiaturally obtained by weakening the into 2: E 2 where 2 is a greater linear subspace which is requirement z E Z invariant under each @ ( a ) and such that the corresponding extended @ is still a Lie morphism. Recall from Chapter 4 that the deriseiiess of s-analytic vectors for some s holds for fairly general representations (including any continuous representation over Banach spaces). Thus it is a quite reasonable hope that i n practical cases one may tumble over a dense subset of them in a suitable 2 or, better still, the space 2 happens to consist of such well-behaved vectors. An iinportant example is provided by the method of analytic dominance (see “el]): if the Nelson condition holds, i.e., x is a Hilbert space, we have for some basis 6 1 , . . . , b N of g that each @ ( b i ) is skew symmetric and the operator D = C,@ ( b , ) 2 is essentially self-adjoint then the &analytic vectors of its closure are jointly s-analytic for @ ( b l ) , . . . ,@ ( b N ) where s depends on t and on the structure of g . Moreover, these vectors are also analytic for any where b E g and one can take 2 = nF=lD(D”); eventually the Nelson condition implies integrability into a continuous unitary representation. For the reader’s convenience we single out what should be used from “el] here: Theorem 1 , Lemmas 5 . 1 , 5.2 and 6 . 1 . The fact the extended @ remains a Lie morphism follows by taking adjoints. The related Lemma 6.2 of “el] (about analytic dominance by higher order elements) is also notable as relevant to integrability problems. Here we mention that Lemma 5.2 as formulated i n “el] is probably not true (its proof is certainly wrong because its Formula (5.5) does not imply the q-invariance of fi as asserted there); but this nuisance drops if A is an algebraic combination of the X j ’ s as in the applications.
~
96
Chapter 5
The Nelson condition is almost necessary; more precisely, using Lemma 6.1 of "el] and our Theorem 3.32 it is not hard to verify the Nelson condition is equivalent to the following: @ is unitarily integrable and Z is dense in C 2 ( T ). In [BGJR] a successful attempt was made to create a non-unitary kind of analytic dominance and thereby extend the ideas above. Since the corresponding condition is also as necessary as it can be, it may aspire to become a 'generalized Nelson condition'; we prove these results i n the end of the present chapter (Theorem 5.15). The main problem with jointly analytic vectors is that this property involves so complicated operators that its checking may well be difficult or even impossible sometimes. This is the reason for seeking weaker conditions to ensure integrability. Our methods in pursuing this end merge into the following. If we can somehow obtain one-parameter representations V, corresponding to some basis al,. . . , a N then we set
where 'p is a product chart with respect to this ordered basis as i n Lemma 5 . 5 . Denoting by F the multiplication of the group in the chart cp , the locally homomorphic property can be written as
(for small I , y E RN ). We want to get simpler formulas ensuring (5-4). Since (5-4) holds for I = 0 and the operators Ai come forward by differentiating K , it seems a good idea to differentiate with respect to zj when 11 = . . . = "j-1 = 0 . Similar reasoning goes back (at least) to [Mool] . We show presently by this method that (5-4) holds whenever the V's intertwine the would-be pregenerat,ors in the way as they should.
Lemma 5.6.
We say that a E g is 'nice' (with respect to the fixed @ and
V ) if Z is invariant under any V(t) and (5-5)
V(t)cP(b)V(-t)= @(Ad(espta)b)
for any t and b . If g h a s a basis a1 , . . . , a N of nice vectors then 0 is strictly integrable. Note that if @ is integrable and Z is invariant under each T(exp t a ) for some a then a isnecessarily nice by (3-9). Also observe that for a nice vector a we have A = @(a) by Proposition 3.20 and so the espression 'with respect to V' is superfluous in the definition of nice vectors.
PROOF:The function F is defined by cp-'(F(z,y)) = cp-'(z)'p-'(y) . We shall differentiate this equation with respect to xj . Recall that if y , 29 are arbitrary smooth curves in a Lie group then (y . d ) ' = y' . 29 y '29' , where we denoted by
+
The Integrability Problem
97
prime the tangent of a curve and the products denote (left and right) translations (cf. Notation F.2). Comparing this with the facts that for any one-parameter subgroup y we have y' = y'(0) . y and g . v . g-' = Ad(g)v for any u E g and g E G , we obtain
Ad (exp(x1 a l ) ) . . . Ad(exp(xj - 1aj- 1 )) a, Now fix z E 2 and apply the left hand side of (5-4) to it with a fixed y and with xJ in the place of x where x j = ( 0 , . . . , O , z j , . . . , I N ). Denote the resulting vector by wj , then the invariance of Z enables us to apply the Leibniz Rule when differentiating wj with respect t o zj (cf. Proposition C.17) and obtain
where t k = Fk(t,y). Comparing (5-5) with (5-6) we can see that the formidable sum here equals @(aj)w,, and then Theorem 1.5 implies w, = vj(zj)wj+l . Since W N + ~ = Vl(y1) . . . VN(YN)Z, (5-4) holds a t t and hence everywhere for 2 is dense. Now Lemma 5.5 and Proposition 5.3 show the existence of a continuous representation T of the corresponding simply connected group C such that @ ( a i )c d T ( a i ) for all i and 2 is invariant under T . Since 2 is also invariant for dT(ai), we see from Proposition 3.18 that Z c C w ( T ) and the T-invariance implies 2 is dense there (by Proposition 3.20). Both Q, and BT are linear and a T ( u ) is dense in d T ( v ) for any u , thus the proof is complete. T h e above lemma strengthens as follows. Lemma 5.7. Assume that we j u s t have a Lie generating subset of nice vectors. Then @ is strictly integrable.
PROOF: Denote by S the set of nice vectors. We shall prove that the subalgebra generated by S is not greater than the linear subspace generated by it (even if the assumption of the lemma does not hold), aiid hence if S is Lie generating then it must contain a basis and Lemma 5.6 applies. Let a E S and D be a continuous operator from X onto X which has a continuous inverse. Then, obviously, DV(.)D-' is a continuous representation of R with generator D A D - ' , which contains DiP(a)D-'. We shall use this for special D's: suppose that D and D-' leave Z invariant aiid an automorphism 6 of g exists satisfying D@(v)D-' = @ ( 6 u ) for any v . Then @ ( & a )is contained in the generator of W = DV(.)D-l , this W leaves Z invariant and W(t)@(b)W(-2) =DV(l)~(S-'b)V(-l)D-' Q(Ad(expl6a)b)
= @(6 hd(exp ta)6-'b) =
Chapter 5
98
(the last equality can be inferred, e.g., from Ad oexp = ed , for 6 . ad v = ad 6v . 6 means exactly that 6 preserves the Lie product). So 6a inherits the properties of a . On the other hand, D = Vl(t) possesses the required characteristics with 6 = Ad(expta1) whenever a1 E S . Thus we can see that Ad(exp ta1)az E S for any a l l a2 E S and any t E R . Now the linear space spanned by S is closed (for g is finite dimensional) and therefore [all a21 = Ad(exp t ~ l ) a ~ ( belongs ,=~ to it; so this space is a subalgebra as we asserted.
8
Now we see that the integrability problem essentially reduces to seeking nice vectors in the sense of Lemma 5.6 (trying to enlarge 2 first if the invariance does not hold at the outset). A considerable part of [J-M] is devoted to this program. Further results are contained in [BGJR] and [Rus2]. We shall incorporate J . Rusinek’s proof in the present chapter as 5.10-14 (with slight modifications and extension to more general spaces it admits). Another proof of Rusinek’s tlieorein can be found in [G-J] though this other proof, it seems, sticks to Banach spaces. We mention [Pow] as a different kind of solution for the integrability probleni; we do not discuss it in this book. Now proceed with our program. Since Adg is an automorpliism of tlie Lie algebra g for any g and B DBD-’ is a n autoinorphism of End(Z) whenever D is a linear bijection from 2 onto itself, it is enough to require (5-5) for a Lie generating subset of b’s (and then it follows even for any b i n the complexified universal enveloping algebra, t o which @ extends). Thus Lemma 5.7 can be reformulated i n a more symmetric fashion. The invariance of 2 under V is a severe condition on nice vectors; for Frechet spaces and nice enough domains it will be enough by itself (cf. Proposition 5.9 below). If (5-5) holds for some b then @(6)V(-t)z is certainly a smooth function of t for any z E 2 (because v @(v)z is a linear operator from a finite dimensional space into 2 and 2 is contained i n C ” ( V ) ) . Now to establish (5-5) by a differentiation trick some “small bit” of this sniootlincss sliould be known beforehand (and smart methods might be found for proving this bit from other assumptions). Lemma 5.8 and Proposition 5.9 exemplify this; these results are essentially taken from [J-M] . We shall discuss another approach when proving Ilusinek’s theorem. Since the proof of Lemma 5.8 would allow several slightly different statements, we first present the proof in such a manner that the readers can formulate other versions if they please, and then state our version. Our notational conventions are assumed in the statement but not in tlie proof. So let - A be the generator of a one-parameter semigroup t ---* V ( - t ) over X (we use this queer notation in order to be in accordance with (5-5) i n the end). Choose a dense linear subspace 2 which is invariant under both A and V and fix a linear operator B in E n d ( 2 ) . Write v ( t , z ) = BV(-t)z for z E 2 and t E R+ . Now we need to interpret B ( t ) := er adABin some sense, for our goal is the formula v ( t , z ) = V ( - t ) B ( t ) z . Writing u ( s , t , z ) = V ( - s ) B ( s ) V ( s- t ) z the above certainly follows if u ( . , t ,z ) is constant on the interval [O,t]; and if all formal differentiations are justified then we have exactly that. Motivated by this (and with the Lie group setting in the back of our minds) we suppose that the (adA)-invariant linear subspace M of End(Z) generated by B is finite dimensional (where, obviously, we mean by a d A ---+
---+
The Integrability Problem
99
the following: ad A ( C ) z = A C r - CAz ). Then we consider B ( . ) as an entire function with values in M and the formula B’(t) = a d A ( B ( 1 ) ) is valid. Our aim will be achieved if for any z E Z and t > 0 we can find a w* dense set of functionals f so that (5-7)
a
as (f, V(-s)C;V(s - t ) r ) = - (f,V ( - s ) . a d A ( C , ) . V ( s - t ) z )
hold for a basis Ci of M and for 0 5 s 5 t (for then ( f , u ( . , t , z))’ = 0 obviously follows). Let 3 be a w* dense subset of D(A’) which is invariant under A’. Since V commutes with A , we can see immediately the following. If (5-7) holds with a certain C E End(Z) for all f E F , r E Z and for a fixed t then it also holds with A C and with CAI, instead of C (for the same f,z, t as above). So it would be enough to find a suitable 3 with which (5-7) holds for B instead of C; . Naturally, the purely algebraic assumptions we made about U so far do not suffice for this (counter-examples exist even if V is the regular representation of R over L2(R)). Since we just know the equicontinuity of the set of functionals { f o V ( - s ) ; s E [O, t ] } with respect to the original topology and not any weaker one (no matter how nice f is), the application of the Leibniz Rule requires the differentiability of v ( t , z ) = BV(-t)z in t with respect to the topology of X (cf. Proposition (3.17). On the other hand, if v(.,z)’ = v ( . , - A z ) does hold (as it should) then the Leibniz Rule is applicable even to V ( - s ) v ( t - s, z ) and (5-7) follows (with B instead of C; ) for any f . This is enough because G’( V ) is w* dense and A*-invariant. We want to replace the above mentioned differentiability of v with a smaller bit of it. First of all, since X is sequentially coinplcte, we can resort to Theorem C.18: it would be enough to prove that ,(! u ( . , z ) ) ’ = ( f , v ( . , - A T ) ) for a boundedness-showing set of f’s (which set is independent of z ) , because then these scalar functions will be smooth by tlie A-invariance of 2 (observe that a boundedness-showing set of functionals is separating). Using the same A-invariance we see the above is equivalent to the formally weaker assumption that (f,v(., z)) be locally integrable and (5-8)
(f, BV(-t)z ) - (f,B z ) =
L‘(f,
-l?V(-s)Az) ds
for all z E 2 and for a boundedness-showing set of functionals f . This is the smaller bit we mentioned, and it is certainly satisfied if D ( B * ) is boundednessshowing. Now the last condition is too severe on I3 ; we try to alleviate it. Bear in mind that eventually we are interested i n tlie case when B is a part of a generator. A much milder requirement is that 13 be closable in the completion of X (for any continuous representation extends to the completion). In this case if the i weak integral v(s, z ) ds exists and, moreover, belongs to this completion for any t and z then (5-8) follows for any f by the interchangeability of semi-weak integrals with closed operators (cf. Appendix C ). Krein’s theorem (see Theorem C.lO) implies that this is the case whenever B is such that S V ( . ) z is weakly continuous for all z E Z .
so
Chapter 5
100
This remark was just a digression; apply now the full force of the assumption ‘ B is contained in the generator of at least a semigroup’. If U is the semigroup in question then write B, = n ( U ( l / n ) - I) . Then B, is a continuous operator and limn+m B,z = Bz for any z E 2 . This shows first of all that v is a Baire-1 function (i.e.’ the limit of a sequence of continuous functions) on R+ x 2. If one writes a continuous operator instead of B in ( 5 - 8 ) then it evidently holds for any f ; apply this t o the continuous operators B, . Then to obtain (5-8) and the required local integrability it is enough (by Lebesgue’s theorem) if sup, I ( f l B,,V(-s)z) I is locally integrable for a boundedness-showing set of f’s (for all z E 2 ). We know that B,z = n . U ( t ) B zdt even as a Riemann integral for any z E 2 . Thus the above is certainly satisfied when I ( U ( t ) * f v ( ., z ) ) I has a locally integrable bound independent o f t (for small t ’ s ) . If A generaks a group then this applies t o both A and - A ; in other words, for positive and negative 2’s as well. I n particular, +
st’”
we have proved the lemma formulated below. Lemma 5.8. Using our notational conventio~is,suppose 2 is invariant under V and a certain a0 satisfies the following “slight regularity” with respect to V . The functions @ ( u o ) V ( . ) zare locally bounded ill 1 for all z E 2 . Then (5-5) holds with b = a0 . We call attention t o the fact that no measurability is assumed (for it follows). It is clear from the proof that the bit weaker requirement ‘there exist locally inte1 grable scalar valued functions u, such that llie functions -@(uo)V( . ) z are locally 212
bounded’ is enough as well. Furthermore, if the corresponding bouridedness holds just on a small interval J , around 0 then we have a similar condition on t J V ( f ) Z and so this implies our condition by the usual compactness argument. The union of Lemmas 5.7 and 5.8 yields a ratlicr general iiitegrability theorem which has the shortcoming that not only we need a \/,-invariant domain but we also have to estimate @ ( a j ) K ( t ) z for small t’s. The following proposition offers a bit better version (at the cost of requiring 2 to be even nicer).
+
Proposition 5.9. Let a1 ’. . . ,ah be a Lie generating subset satisfying Notation 5 . 4 . If 2 is invariant under each K ( t ) and, moreover, 2 can be endowed with a lre‘chet space topology which is finer thaii the subspace topology then Q, is strictly integrable.
In ‘finer’ we include ‘identical’ but that does not occur i n non-trivial applications. If fi is a Frlchet space then the condition is natural enough because if Q, is integrable then 2 = C m ( T ) with its C” topology satisfies it and even if the Z given at the outset is smaller, we know what Cw(T)sliould be (see Proposition 3.18 and also Corollary 3.24). Here the problem arises whetlier the extended @ is still a Lie morphism but the main difficulty generally lies i n checking the invariance of this subspace under K ( t ) ; of course, smaller Cw closed subspaces might also serve but the difficulty persists.
PROOF: Fixing i, j and
a continuous linear functional f on X we set
The Integrability Problem
101
(f,@ ( u j ) K ( t ) z ) for t E R and z E 2 . We saw in the proof of Lemma 5.8 that @ ( u j ) V , ( t ) z is a Baire-1 function, hence it is also Baire-1 if its domain is considered with any finer topology. Hence u is Baire-1 and since R x 2 as endowed with the finer topology of 2 is a complete metric space, a classical result about Baire-1 functions (see Proposition A.4 ) implies tlie existence of some ( t o , 2 0 ) at which u is continuous. Note that using the functional f we have circumvented the difficulty that X is not necessarily metrizable. Thus we have an E > 0 and a (finer) neigliborhood W of 0 in 2 such that l u ( 2 , z l ) - u(to,ro)l 5 1 if It - to1 < E and z1 - zo E W ; so by linearity we obtain l u ( t l z ) l 5 2 for these t’s if z E W . This neigliborhood W is certainly absorbing, so the situation is the following: there exist t o and E depending on f , and c depending on both f and z such that Iu(f,z)l 5 c for It - 201 < E . Since 2 is invariant under V, and t o is independent of 2 , we obtain the local boundedness of u(., z ) for any fixed z . Use the theorem about tlie equality of boundedness with weak boiindedness and apply Lemma 5.8 to complete tlie proof. u(t,z) =
Note this proof just used tlie existence of a finer linear topology such that R x 2 constructed from it is not meager in itself Another version is obtained if we require less from 2 but more from V , . Namely, if the other structure on 2 is just barrelled and finer tlian tlie weak topology of X then (using the fact that a continuous liiiear operator is also (w,w) continuous and the corresponding version of tlie Baiiacli-Steiiiliaus Tlieoreni) we obtain the continuity of @ ( a J )from this finer topology into (X, w) Assuming the local boundedness of K ( . ) z with respect to this barrclletl topology for each i and z we obtain the condition of Lemma 5 8 again. We turn to the other promised nietliocl of establishing (5-5). We can say informally that if (5-5) lias a good start then it propagates itsselfalong the real line. This observation is crucial in J . Rusinek’s proof we esplaiti here. The “propagation” mentioned above (similarly to many a method based on iuduction) works by finding a stronger induction step than the required final result (see the proof of Lemma 5.13 below). First we seek as strong a starting step as we can. Until the end of the proof of Tlieorem 5.14 let ( 1 1 , . . . ,a,, be a fixed Lie generating subset of g (and, of course, A , , V , be corresponding fixed objects as defined by Notation 5 . 4 ) . We also use the letters u , A , V in accordance with our notational conventions. We show presently how aiialytic vectors come into the picture. Recall that analytic vectors for an operator were defined i n Notation 1.11. Lemma 5.10. Fix a c E g , write C = @ ( c ) m i d s e t e t c x := for 111
< r and x
i.e., P,(X) =
M
2’ 7c3x
E d ( r , C ) . Denote the n’th Zij~lorpolyrioniial of e t x by P,
cJnz0 3x1 . Suppose z is sucl~that 23
@(6,)2
E d(v,C) for some
basis b, of g . Then for any It1 < r and any 6 E p we have lim @(b)P,&(C)z = e f C @ ( A d ( e s p- 2 c ) b ) t
n-o0
,
.
102
Chapter 5
PROOF:Denote by L, and R, the multiplication by @(c) from the left and right (considered as linear operators on End(Z)), they evidently commute and ad C = L, - R, . Hence R, = L , - ad C is a difference of commuting operators and the binomial theorem yields (5-9) for any b E g . Recall that (infinite) sums of non-negative numbers can be multiplied and grouped freely even if the result is infinite; while for vectors the absolute summability ensures this (in a locally convex space a set is called absolutely summable if for any continuous seminorm the sum of the corresponding non-negative numbers is finite; the sum itself belongs to the completion i n general). We assert that for any b E g the family
(5-10)
1" (')d@((-adc)9-j6)2 q! j
qEN,
j = O , 1 , . . . ,q
is absolutely summable whenever lil < r . Denote by 1.1 the sum-norm on g corresponding to the basis 6; , i.e., ICi ci . 6 i I = Ci I c i ) . Also denote by 1. 1 the corresponding operator norm. Let s 2 0 , 7c be a continusI' ous seminorm on X and u , , ( s ) = -inax;?r(C"S(b;)z) . We can estimate n,! 1tl4L-j
one term of (5-10) by Iadcl9-jIbl-
u.(ltl).
(.~ q - A!
'
Tlie sum of these estimates
equals Ibleltl.ladclC,"=o un(ltl) and it is finite (use u l , ( l t l ) = (!)"un(s) with 121 < s < r ). Grouping now this absolutely suminable family of vectors and using (5-9) we obtain coo;)
q=o
j = O n=O
and since ZI -+ Cj@(v)z is a linear operator from a finite dimensional space, the second sum on the right equals ~ ( t C ) j @ ( e a d - " 6 ) z .Finally observe that @(ZI)Z E d(r,C) for any v because d(r,C) is a linear subspace. We shall need the following concept. Denote by 4, the real linear span of those operators in End(Z) which are products of at inost n elements of @(g) (in particular, & is the set of real scalar operators). It is just a matter of taste to consider real linear spans, complex spans can serve as well. Then set dn(r,a) = { z E Z ; BZ E d ( r , A ) V B E
4, }
.
Since d ( r , A ) is a (complex) linear subspace, these sets are also linear subspaces and in their definition one may just consider a basis of U, . Obviously, B( d,+,(r,a) ) C d , ( r , a ) if B E U, .
103
The Integrability Problem Lemma 5.11.
d n ( r , a ) is invariant under A .
PROOF: ad@(a) leaves @(g) invariant, maps I to 0 and is a derivation of the associative algebra E n d ( 2 ) , therefore it leaves each &, invariant. Thus if B E U, then BAr = A B r C z with C E U, for any t E 2 and the statement follows from the A-invariance of d ( r , A ) and of 2 .
+
We shall also need the finer graduation corresponding to our Lie generating subset. For q = 0, 1 , 2 , . . . we denote by Uq tlie real linear span of products @ ( a i l ) " ' @ ( a i , ) , j 5 q (including j = 0 ) , and write 89 = @ - ' ( @ ( g ) n U q ) . It is evident that Up is a non-decreasing sequence of finite dimensional real linear spaces and UP . Up c UP+, . Then each 0, is a linear subspace of g , they form a non-decreasing sequence, their union is a subalgebra aiid each a; E g' . Thus g = gq for large q's because g is finite dimensional and tlie a i ' s generate it. Denoting by k the smallest such q we obtain tlie fortnula
The Cm space of the desired group representatioti slioultl be approached by the finer graduation because we do not know the closability of each 9 ( 6 ) yet (the importance of closability has already been reflected by Leninia 5.10 a i i d it is going to come up again when proving 1,emnia 5.13). So writre
X, = n{D(Ai, . . . A , , ) ; 1 5 ij 5 ? J L
(5-12)
Vj} ,
endow this space wilh the corresponding "Cq-topology" defined analogously as in Definition 3 . 1 9 , and continue the analogy by defiiiiiig X , ?'lieu we obtain sequentially complete spaces because X is sequentially cotriplctc and the A,'s are closed. continuously iiito X, Obviously, each A, maps If @ is integrable then the fact OT is a iiior1)Iiistii of unital associative , = C'-('f') even as locally algebras, Corollary 3.24 and (5-11) imply tliat X convex spaces. The continuity properties of tlic A,'s iiiiply t h a t (usiiig tlie notational convention (D-6) as usual) a n y operator of the forin C I _ c , A a maps X,+, continuously into X, for any q (including q = 00 ). T h e restriction of such an operator to tlie closure 2, of 2 i n X p then yields the unique extension of the corresponding operator in UP which is continuous from 2, into X . I n what follows this will be important for p = 6 , so denote by B tlie correspoiiding continuous operator from ZC into X for any n E U k .
Lemma 5.12. If z E d , ( r , a ) and Ill < r tlieii the corresponding Taylor approximations
En J=o
tj
.
-j! AJr
tend to V ( t ) z in the topology of
X, .
PROOF:For q = 0 the statement follows froin Theorem 1.5 and Exercise 1 . 3 . The condition becomes stronger when q increases, each X, is sequentially
Chapter 5
104
complete and their locally convex topologies are finer, thus it is enough t o show that the Taylor approximations form a Cauchy sequence i n X, . If I, is a sequence in a locally convex space and r > 0 then the condition that y , ( t ) = c y = o t j x j be a Cauchy sequence for It1 < r is equivalent to the requirement that t”x, be a bounded sequence for 121 < r . So we should check d , ( r , a ) c d(q)(r,@(a))where the superscript ( q ) means that we consider the analytic vectors with respect to the locally convex space 3 , . We prove this by induction on q . Assume it for some q and let z E d , + l ( r , a ) . Then @(b)z E d(q)(r,@(a)) for any b E g by the inductive hypothesis. Now Lemma 5.10 is applicable o n the locally convex space X, with c = a (our caution in proving that lemma for any c pays now: even though a satisfies Notation 5.4 in 2” X I the same does not necessarily hold in X, ). Thus - @ ( b ) A “ z is a bounded n! sequence in X, for It1 < r and b E g ; hence z E d(q+’)(r,@(u)).
Lemma 5.13. Let z E d k ( r , a) with some positive r . Then for all t E R and for any i we have AiV(t)z = V(t)@(Ad(exp-2a)ui)r
.
PROOF:The stronger requirement for the induction step we hinted at earlier will be to suppose the above at a certain 2 for any P ( @ ( u ) ) z instead of z where P runs through the set of polynomials i n one variable. So let S be the set of those real numbers t for which A,V(t)P(@(a))z= V(t)@(Ad(exp- t a ) a ; ) P ( @ ( a ) ) z
(5-13)
holds for all polynomials P for a fixed i . The lemma follows if we can prove that S = R . Obviously, 0 E S . We shall show t + s E S whenever t E S and Is1 < r (and consequently S = R ). Set
2,
=
Sj
Cj”=,? A J P ( @ ( u ) ) t
.
We infer from Lemma 5.11 that
P ( @ ( a ) ) zE d k ( r , a ) , and hence x,, -+ V ( s ) P ( S ( u ) ) z in X k by Lemma 5.12. Therefore if B E Uk then we can apply B to this relation and obtain limits in X . If B = @ ( b ) with b E g then we have another formula for this limit from Lemma 5.10 and get BV(s)P(@(a))z = V(s)@(Ad(exp-su)b)P(@(u))r (we could replace es@(a) by V(s) because of the q = 0 case of Lemma 5.12). Apply V ( t ) to this equation and take b = Ad(exp -tu)ai , then on the right we obtain the right hand side of (5-13) with t s because both V and T Ad(exp - T Q ) are homomorphic functions. Now observe that V ( t ) B is also continuous from Zk into X and so on the left we have limn+m V(t)Jr(Ad(exp - t a ) a i ) x , . Since I, = & ( @ ( a ) ) % with some polynomial Q and we assumed t E S , the expression
+
-+
The Integrability Problem
105
under this limit equals A i V ( t ) z , . But A; is a closed operator and V ( t ) z , + V ( t ) V ( s ) P ( @ ( a ) ) zbecause V ( t ) is continuous, so we obtain the left hand side of (5-13) and the lemma is completely proved.
Theorem 5.14. If a1 , . . . ,a, tion 5 . 4 ) such that the set
is a Lie gerieratiiig subset (satisfying Nota-
is dense then @ is strictly integrable. In particular, if Z integrable.
c A ( A j ) for all j then W = 2 is dense and @ is strictly
PROOF:We can extend the x ' s to the coinpletioii of X and if @ is strictly integrable with respect to the completion then all the more strictly integrable in X (for the smaller C" space is a topological subspace of the greater one). Thus we may and shall assume X is complete. Then each 1, is complete as well. The set W is a linear subspace and it is invariaiit under each A, , hence under U,UP = U,& . Then the condition of Lemma 5.13 is satisfied for a = a, and z E W because Uk is finite dimensional. Now we can iterate the result of this lemma because the elements under V , ( t ) on tlie right remain i n W . Thus L$(t)(W)c X, and V , ( t ) l w is a continuous operator from Xk, into X, by (5-11). Denote by W, the closure of W i n X, , then we have a ( k q , q ) continuous extension of Y(t)I,+,to Wk, by the complcteness of X, , and it equals the restriction of L $ ( t ) because x ( t )is continuous with respect to the coarser topology of X . We want to show that W , is invariant uiider any V, ( 2 ) . By the continuity it is enough to show & ( t ) ( W )c W , or equivalently tliat 6 ( 2 ) ( W )c W , for any finite q . So fix an a = aj , a finite q and z E CV and consider the set S of those real numbers t for which
for all polynomials P . We have some r > 0 s u c h tliat z E d k , ( r , u ) because Uk, is finite dimensional. Assume 2 E S and Is( < r ant1 let x , ~be the n'th Taylor approximation of V ( s ) P ( @ ( u ) ) z Since . P ( @ ( a ) ) zE d k , ( r , a ) by Lemma 5.11, we have the convergence of this approximation i l l X k q by Lemma 5.12. T h u s V ( t ) z , -+ V ( t ) V ( s ) P ( @ ( a ) ) zin Wq by the corresponding continuity of V ( 1 ) because t E S . Hence t s 6 S and S = R follows (for S is not empty, 0 E S evidently). The Aj's are continuous when considered i n 1 , and leave W invariant,
+
hence
@l(b)
=
defines a continuous operator from W , into itself for any W,
b E g . Their continuity implies that @ 1 is a morpliisin from g into End(W,) . We have @ l ( a i ) c Ai and we can extend the formula i n Lemnia 5.13 (for a = a, ) to any z E W , (with instead of @ ) because every operator involved is continuous in the topology of X, . So @ I satisfies the assunrptions of (the synimetric version of)
106
Chapter 5
Lemma 5.7 (see our remark not long after its proof), therefore it is strictly integrable. Denoting by T the corresponding representation of the simply connected group, we have C”(T) = X, (see the remark after (5-12)). The strict integrability implies that W is dense in this space. Now any combination of the @(a;)’s is a continuous operator on the linear subspace 2 pinched between W and 1 ,’ the aT(6)’s are continuous on the whole and 8T(6)IW= @I(b)l, = @(6)l, , so the theorem is proved. This theorem and Theorem 1.14 together yield a nice statement which is especially remarkable for isometric representations over Banach spaces: then we should just require ll(Z - 6@(6;))zII 2 IlzJJ for t E 2 and small real 6 for a Lie generating subset of b’s, and a dense set invariant under the @(6,)’s and consisting of analytic vectors for each of them. If one deals with skew symmetric operators over a IIilbert space then the statement of Theorem 5.14 can be proved by much less effort (cf. [FSSS] , [Sim] ). The great advantage of skew symmetric operators is that algebraic relations automatically extend to bigger subspaces by taking duals and tliis also enables us t o argue with - V ( - t ) B instead of B V ( t ) . If we want to generalize somehow the Nelsoti contlitioti to iioii-unitary representations then the absence of tliis extendibility of algebraic relations must be balanced by other conditions. Replacing self-adjoiiititcss by the generator property is a natural idea but probably not sufficient. We close this chapter by proving a relevant theorem for Banach spaces offered by [DGJIt] . Sitice i n some places we shall utilize that tlie inequalities expressing equicontitiuity has the same norm on their sides, possible extensioiis of tliis proof to iiiore getieral spaces are very lirnited, certain ‘ultra-equicontinuity’ conditions should be iniposed. We set up some notation before forinulatitig the tlieorcin for otherwise its composition would be too crowded. So let X be a l3anach space and suppose ~ g . Let that @(6l), -. . . , @ ( 6 ~ )are closable operators for soiiie basis 6 1 , . . . , 6 of B;= @(6,) and en(.) := maxlull,, IJBazll for x’s wliere it lias sense (if @ were integrable then tlie domain of en would coincide w i b h C”(7’)and en itself would provide its topology introduced in Defiliition 3.19). Denote by 2, the closure of 2 with respect to the corresponding projective topology (i.e., i t t the space n,D(P) as endowed with the family en of seminorms) and let this notation also refer t o this topology on 2,. Obviously, the restriction of a l l y 19;to 2 , is a continuous operator in this space and we obtain an extension of 9 to a Lie morphism @ I into End(2,). Theorem 5.15. Suppose the existence of a one-parameter semigroup S with the following properties. Denoting its generator by D we have 2 c D(D) and DO := DI, is of (at most) second order with respect to @(g), i.e., DO is a (complex) linear combination of operators whiclt are products of at most two elements from @(g) . If 0 < t 5 1 then for each i B i S ( t ) is densely defined and (5-14)
The Integrability Problem
107
for a dense set of x ’ s (which set may depeiid on 2 and i ) . Finally, the set of those x’s for which S ( t ) z E Z , for small positive 1’s (the s m a l l ~ ~ emay s s depend on x) is dense. Then the jointly analytic vectors of Q i l ( b l ) , . . . , @ p ( b ~ )form a dense set; if, in addition, we have a Lie generating subset satisfying Notation 5.4 then @ is strictly integrable. Some remarks may help t o understand why we formulated this theorem in this form. First of all, the conditions about (5-14) iniply that B ,S (t ) is, in fact, an everywhere defined continuous operator for 0 < t 5 1 satisfying (5-14) for any x (because if B is a closed operator and R is an everywhere defined continuous operator then one checks immediately B R is closed; and tlie closure of a densely defined continuous operator i n a complete space is everywliere defined and continuous). We refrained from writing this as an assumption to einpllasize that in applications it is not necessary to check (5-14) for all 2 . The upper bound 1 for t i n (5-14) is inessential: assuming any other positive number instead of it we get the same theorem by rescaling D . Theorem 4.11 shows that the conditions we formulated are valid for any strictly iiitegrable @ over a Banach space whctievcr I& corresponds to a strongly elliptic element of second order. This partly motivatcs tlie theorem; another piece of motivation came from some easy calculations w i t h self-adjoint operators over Hilbert spaces (cf. the Nelson condition). ‘ S ( t ) xE ’ seems tlie most inconveiiiciit of our assumptions for applications. We need this condition because i n tlie proof it will be vital that tlie structure of the enveloping algebra of g “works” on tlie UZ’swlicn they are considered at vectors S ( t ) z (for small positive 1).
z,
Since any algebraic conibinatioii of the @l(b,)’s is continuous in liere a t i t l it equals the restriction of D because D is closed over X and the topology of 2, is finer. On the other hand, 2 is dense in 2, whose topology is tlie would-be C” topology arid any A also contains Qil(a) by tlie same reasoning we just, applicd to D . Thus we can forget 2 or, in other words, we may assume Z = Z , and sliall drop the notations 2, and @ I accordingly. Let pn(x) := m a x t l n IIDkx)I . Evidently, on 2 we have p n / ~ 2 nis bounded for any n , so we have an inclusion between the correspondiug C“ topologies; what we shall need is exactly the opposite inclusion. We slirtll prove, i n fact, the boundedness of en/pn (on Z ) by induction on n and tlie itiduction step will easily follow from the following sharpening of tlie starting step: for any E > 0 we find C ( E ) such that PROOF:
2,
, we have a continuous extensioii of Do to
(5-15)
Ql(Z)
5 E l l D z l l + C(E) IlfII ’
’
for all z E 2 (and even for any z E D ( U ) but that is not important for us). The proof of (5-15) will essentially be a Laplace trailsformation of (5-14). Write f ( t ) = ~ i . t - ’ / ’ ; i n proving (5-15) just the following properties o f f will matter: f is non-negative, non-increasing, limt,o+ t f ( t ) = 0 and f is integrable on [0,1]. Since
108
Chapter 5
X is a Banach space, we find a non-negative w such t h a t { e-w'S(2) ; 1 E R+ } is equicontinuous, and then IIBiS(t)z(l = IIBiS(s)S(t- s)xll 5 M e w ( ' - 6 ) f ( s ). IIxll for any 2, 2 > 0 and for 0 < s 5 1 A t , where M is some constant (recall t h a t BiS(s) is a n everywhere defined continuous operator for 0 < s 5 1 ). T h u s the function ~ ( t=) IIBiS(t)xl(/ ( M . 11x11) can be estimated in the following way: u ( t ) 5 f ( t ) for t 5 t o and u ( t ) 5 f ( t 0 ) . e w t for t 2 t o , where t o E (0,1] is arbitrary. We know from Lemma 1.7 t h a t ( A 1 - D)-'z =
1"
e - x t S ( t ) z $2
>w
and this is an improper Riemann int,egral. T h e function B,S(t)x = is integrable for X > w , so we get another improper Riemann integral and the closedness of B; yields for any X
[B,S(s)]S(t - S ) T is continuous on the open half-line and e - " u ( t )
1
If X is large enough theti we can choose t o = - to estimate the right hand A - W
+ sofO
side, and get l o f ( t 0 ) f ( t ) dt as a crude estimate for the integral there. T h i s expression tends to 0 with t o , and substitut,ing T = ( X I - D ) z we obtain (5-15) by taking large A's. Turn to the promised induction. If Icy1 = n and /3 is tlie ( n - 1) long tail of a then for z E Z we have
1 1 ~ ~ L~ E1 . I 1I D O B ~ ~ I I +C(&)e,-i(Z) L E . I l [ ~ o B , ~ ] =+&etL-i(Doi)+C(&)e,-i(z) JJ by (5-15). T h e operator ad DO is a derivation of tlie associative algebra E n d ( 2 ) , therefore [DO, B @ ] zis a s u m of n - 1 terms of tlie form By' [ D OB,]BY2z , , and applying the same reasoning to [Bi,Do] we get a constant, ti independent of n a n d 4 such t h a t
(the fact
will be utilized later, that is why we were so 1 cautious though we d o not need this fact a t the tiioment). Choose E < - to ntc obtain en(%)5 C . ( e t l - l ( D z ) en-l(z)) with soiiie C (which, this time, depends on n ) and write the inductive hypothesis into this inequality to get the boundedness of en/pn on 2 . Let Z ( t 0 ) be tlie set of those vectors z E Z for wliicli S(2)r E 2 whenever t 5 t o . If 1: satisfies a similar assuniption just for positive 2's but with a greater t o then z = S ( E ) ~ belongs to this set whenever E is small enough, so the union of these Z(t0)'s is still dense. We shall prove that if t is sniall enough and positive then S ( t ) t is a jointly analytic vector for @ ( b l ) , . . . , @ ( b ~ )for any z E Z ( t 0 ) , thereby disposing of the first statement in the theorem. More precisely, we fix a tc
does not depend on
+
11
The Integrability Problem t E Z ( t 0 ) with llrll = I and show by induction number 6 such that the sequence
c, := Osup
tn/’e,
011
109 n the existence of a positive
(s(t)z) n!
consists of finite numbers and grows a t most exponentially. I n fact, we shall obtain a rate independent of z as well as t o and the same can be said of 6 (more precisely, we find a constant 60 having this independence and such that 6 = 60 A t o works). Unfortunately, this does not imply any essentially stroiiger statement than that of our theorem. We shall seek 6 5 1 . Then the finiteness of C1 is (essentially) one of our assumptions: C1 5 K V supt<’ IlS(t)ll witli the 1; froni (5-14). Now assume the finiteness up to the index n - 1-with a certain 6 5 1 . For a while fix a multi-index a with la1 = n and a t E (0,6] . Write I
= BOS(t)z ;
we have to estimate the norm of this vector. It turns out that splitting t in a smart way will be useful. The relation [S(.)z]’ = D S ( . ) z is also valid witli respect to any norm p k (because 2 is a D-invariant subspace of D ( 0 ) ) and Iicncc by the coincidence of topologies on 2 we liave this relation i n tlie space 2 if tlie variable is restricted to [ O , t ] . Writing p for the ( n - 1) long tail of cy we have Ufl1, and BOD0 are continuous operators i n 2 , thus B @ S.)z ( is coiitiiiuously differentiable on [0, t ] with derivative B P D S ( . ) z . This holds i n Z but we just use it witli respect to the coarser topology of X . Since 2 is a part of D ( D ) , Llie Leibiiiz Rule applies (cf. Proposition C.17) and we see that S ( r ) @ S ( t - 7 . ) ~ is contin~rouslydifferentiable with respect to r (in X ) with derivative S ( r ) [ & , D o ] , S ( t - - ~ * )on z [ O , t ] (for the continuity of this derivative we have also used t,lie continuity of tlie operator DO BPIz in 2).This implies S ( r ) [ D oBfl]S(t , - r ) dr~ = .5’(s)BaS(t- s ) z - BPS(t)z s E [ O , t ] . No\\! [ D O@ , ] S ( t - r ) z is a con(with a Riemann integral) for any B i S ( r ) is a poiiit\vise continuous and locally tinuous funct,ion of r on [O,s] and equicontinuous function on (0, s] whose norm is bountlcd by ail integrable function. Thus we obtain an improper Riemann integral and tlie intercliangeability of such integrals with closed operators yields
si
1’
B,,S(r)[Do, Bp]S(t - r ) z d r = B o , S ( s ) B ” S ( t- s)z - 2 .
Then we see from the inductive hypothesis, (5-14) and (5-16) that 11111 5 Ks-’/’(t - ~ ) - ( ~ - ‘ ] /-~ l)!Cn-l (n I i ~ n. r-’I’p,,(S(t - r ) z ) d r (here the measurability of tlie integrand follows from the contiiiuity of S ( . ) z in 2).This is true for any cy with la1 = n and for any 0 < s 5 t . If Icy1 5 n - 1 then we apply the inductive hypothesis and, writing s = At and substitut,ing 1’ = t u , we obtain
+
sos
Chapter 5
110
Since we want to estimate C,, , it is advisable to choose X so that the magnitude of n! turn up in the expression above. We may choose X = n-2 for this purpose. If s E (0,1] then st 5 6 and so we can write st instead of t above; introducing the notations
dp(u) = u - ' I 2 d u
if 0
< u < nW2
and 0 otherwise
+
nt1I2 h ( s (1 - u)) d p ( u ) ) with some constant c we have h ( s ) 5 c . ( C , , - ~ S - " / ~ for 0 < s 5 1 . We also know that h is continuous on [0,1] (for S ( . ) z is continuous in Z ) , and therefore bounded. The inequality can be applied to the integrand again and again, and writing c1 = c n t 1 l 2 J ( l - u ) - " l 2 dp(7r) we have
for all m . Immediate calculation shows IIplI = 2/71 , while (1 - u ) - ~ / ' < e on s u p p p if n 2 2 and h is bounded; hence if 2ect1/' < 1 then tending with m t o ~
00
we obtain h(1)
5
ccn-l
1 - 2ect112
and so, e.g., tlie choice
60
= 1 A (tic)-'
makes
the induction work. The second statement of the theorem follows froin Tlieoreni 5.14 by observing the simple facts that the set of jointly analytic vectors is a linear space which is invariant under any @ ( b i ) (consequently under any S ( u ) , too) and its elements are also analytic vectors for any @ ( a ) .
EXERCISES 5.1. Let Tl,T2 be two continuous representations of the connected Lie group G over X such that d T l ( a i ) c d T z ( a i ) for a Lie generating subset a l l . . . , a k of the Lie algebra. Show that T1= T2 . 5.2. Let IJbe an ideal of g and G be the siiitply connected Lie group with Lie algebra g . Prove that the connected Lie subgroup II of G corresponding to f~ is closed and G / H is simply connected.
5.3. Assume for any a E g we have a V (according to Notation 5.4)such that { V ( l ) ; a E W } is equicontinuous for some neighborhood W of 0 in g . Assume further that tlie jointly analytic vectors of @ ( u l ) ., . . , @ ( a ~form ) a dense set ( a l , . . . , aN is some basis). Show that the would-be action (exp a , z)+ V( 1). is (jointly) continuous a t 1. 5.4. Let ( M , p ) be the simply connected covering of R2 \ (0) (th'IS can be realized, e.g., by taking M = (0, +XI) x R and p ( r , a ) = ( r cos a ,r sin a) ).
T h e Integrability Problem
111
Endow M with t h e measure which locally coincides with the Lebesgue measure through p a n d consider the Hilbert space L 2 ( M ) then. If v is a non-zero vector in the plane and z E M is such t h a t p ( z ) is not a scalar multiple of v then we define T ( v ) r as t h e point in p - ’ ( p ( z ) v ) whose ‘angle’ (i.e.l the coordinate a in t h e above realization) is closest to t h a t of 2 . Then T ( v ) is defined almost everywhere; set U ( v ) f := f o T ( v ) and U ( 0 ) := I . Show the following facts. For any v E R2 the mapping 2 + U ( t v ) is a continuous unitary representation of R whose generator is the closure of 8,IC,m(M) (differential operators, being local, can be defined tlirougli p ) . U is not a represenis a niorpliisin of Lie algebras. tation of R2 even though v -+ (Hint: for a fixed v find a U(tv)-invariant dense subspace in C y ( M ).)
+
T h e next exercise could have been included i n Chapter 3 but, provides a nice illustration to the present chapter. 5.5. Let G be the ‘Ileisenberg group of frccdom n’, i.e., as a manifold G = R2”+l arid if ? I , < , 1 denote the first n , the secoiitl 71 and the last coordinates,
respectively, then
(here the f is a matter of taste, it simplifics soiiic foriiiulns). Consider t h e usual “Schrodinger representation” of G over Lz(Rn):
Check that C”(T) equals the Schwartz space (i.e., tliv w t of tliosc srnooth functions on Rn whose any partial derivative multiplied by iiiiy polynoiiiial is bounded). Let D be defined on C r ( R n ) by of(.) = A j ( x ) - llz112 . f(z) where A is the Laplacean operator. Show that D is esseiiti;illy sclf-atljoint (observe t h a t D - I is an operator onc encounters in the Nelson coiitlitioii for the representation above).
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113 6. Compact Groups
Compact topological groups, and especially compact connected Lie groups, play a central role in representation theory. On the one hand, their representations have much simpler structure than those of other groups (in a sense far surpassing even the commutative groups); on the other hand, it is often useful in studying the representations of more general groups or semigroups if we understand well the representations of certain ancillary compact groups (e.g., take a maximal compact subgroup in a semi-simple Lie group; or consider the so called almost periodic representations of topological semigroups, for an exposition of the latter see [Lyu] ). Furthermore, several concepts arising naturally in connection with compact groups have become the motivation for creating important analogous concepts for more general representations. N o wonder a great number of books about this or that aspect of representation theory treat the classical results on compact groups t o some extent. The author thinks he may as well present his own vision as to what are the most important results and how it is best to prove them. We shall prove, in particular, Weyl's formula about recovering the character from the highest weight.
Proposition 6.1 (The Unitary Trick). Let T be a continuous linear representation of a compact group G over a (real or complex) Rilbert space 'H. Then an invertible element B of the algebra B('H) of bounded linear operators can be found such that BT(.)B-' is a continuous unitary representation of G . Moreover, we can stipulate that B be a positive operator.
PROOF:Consider the Hermitian sesquilinear form (El
Y) =
J ( T ( t ) z T(t)Y)dt 9
where the integral is meant with respect to some right Haar measure on G , and its existence follows from the continuity of the integrand and from the compactness of G . We mention that this measure is also left invariant, though this fact has no bearing on the present proof. By the right invariance we have ( T ( s ) z ,T(s)y) = ( 2 ,Y) for any s , z, Y . Since T is locally equicontinuous and G is compact, T must be equicontinuous and we have a bound A4 such that I(z,y)I 5 M . IIzlI . llyll . Hence there exists a unique bounded operator P such that ( 2 ,y ) = ( P z ,y) and this P is self-adjoint and positive because (.,.) is Hermitian and ( 2 , ~2) 0 . Let B be the positive By) and hence square root of P , then (z,y)= (Ex,
(6-1)
( B T ( s ) z ,BT(S)Y) = ( B z ,BY) .
Since the operators T(t)-' = T ( t - ' ) also have a common bound, the number i n f l l r ~ ~ = l ( z , zis) positive, so B is not only injective but bounded from below. For a positive operator this implies that it has an everywhere defined positive inverse, and the operator BT(s)B-' is isometric for any s by (6-1). This is true for s-l,
Chapter 6
114
too, hence these operators are unitary. T h e continuity of the new representation is obvious. The above proposition shows, in particular, that a continuous representation of a compact group over a finite dimensional linear space is unitary with respect to a suitable scalar product. Why do we prefer unitary representations (of noncompact groups, too) to other ones? Because they are more tractable in many respects (as the remainder of this book might convince the readers), and also have wide applications. Perhaps the most obvious advantage of unitary representations is that they are ‘completely reducible’, for the following trivial lemma can be applied to them.
Lemma 6.2. Let ‘H be a (real or complex) Hilbert space, M be a closed subspace o f i t , and A be a set of bounded operators such that M is invariant under A and A is invariant under taking adjoints. Then the orthogonal complement of M is also invariant under A .
PROOF:If y E N and A
E
A then for any
2
E
M we have ( 2 , A y ) =
(A’t’y) = 0 .
Definition 6.3. We call a set of continuous linear operators over a locally convex space irreducible if no closed subspace (other than (0) or the whole space) is invariant under it. A continuous representation of a topological group is called irreducible if its range is an irreducible set of operators. In the opposite (‘reducible’) case we call the corresponding represen tations by operators restricted to a closed invariant subspace ‘subrepresentations’ of the original one. Many authors use t,he term ‘topologically irreducible’ for the notion described above. Since we shall not be concerned with the stronger property of ‘algebraic irreducibility’ (which some call simply irreducibility, and which consists in excluding the non-closed invariant subspaces, too), our notation seems more sensible in the context of the present book. In fact, the difference only arises for non-compact groups, because a compact group has no continuous irreducible representations over infinite dimensional locally convex spaces (see Exercise 6.1 or Proposition 6.14). Naturally, one tries to decompose a representation into a “sum” of irreducible ones in some sense or other whenever one can. If the ground space is finite dimensional and the representation is completely reducible then there is no problem. For infinite dimensional continuous unitary representations of locally compact groups there exists a general result of this fashion (the Gelfand-Raikov Theorem), decomposing the representation into a so called direct integral of irreducible ones. Unfortunately, the direct integral provided by this result may well be “more complicated” than the object it is supposed to simplify (even if it is improved by the smartest application of Choquet’s theory as proposed by S. Teleman and others). This is not an accident, further investigation revealed that the so called type I rep-
115
Compact Groups
resentations are those for which a more powerful decomposition theory is possible, especially if the group is ‘tame’, i.e., its each continuous unitary representation is type Z (see [Dix2], [Arv] ). We use the term ‘tame’ following [Kir] , it seems more meaningful than ‘type I’ or ‘GCR’ as this notion is often called. Of course, the other names convey important information to the expert. It is not very hard to find a wild group even if we just seek among connected Lie groups (see $ 1 9 of [Kir]). We shall touch again the question of tame groups in later chapters, now we show compact groups are the “tamest” for they require no direct integral whatever. In the sequel let each linear space be complex and each representation complex linear except when explicitly stated otherwise (though several proofs work in the real case, too).
Theorem 6.4. Let T be a continuous unitary representation of the compact group G over the Hilbert space 31. Then it is the direct sum of finite dimensional irreducible subrepresentations, i.e., a family 31, of finite dimensional T-invariant subspaces can be found such that 31 is the orthogonal sum of them and q(t)= T(t)I,, defines an irreducible representation for any j .
PROOF: A closed subspace generated by any collection of invariant subspaces is invariant, and its orthogonal complement is invariant (Lemma 6 . 2 ) ; so by Zorn’s lemma it is enough to prove the existence of one irreducible finite dimensional subrepresentation, and that will follow (by Lemma 6.2) if we can find a finite dimensional one, irreducible or not. Let P be a bounded operator, and consider the bounded sesquilinear form
It is invariant under any T ( s )by the right invariance of the measure, so if we denote by p the operator satisfying ( 2 , ~ = ) (Px,y) then we have T ( s ) - ’ p T ( s ) = P . Now if P is a non-zero positive operator then (., .) will-be Hermitian, non-negative for I = y and positive with a suitable x = y , i.e., P is also a non-zero positive operator. On the other hand, we assert that if P is a compact operator (i.e., the closure 2 of the image of the unit ball through P is compact) then p is compact as well. Let r be the supremum of the norms llT(t)ll when t varies in G , and denote by B the closed unit ball of 31, then the closure W of the set P T ( G ) B is contained in r Z and hence compact. Thus G x W is compact in the product space and its continuous image through the action must be compact, therefore Y := { T(L)-’PT(t)x; 2 E G, I E B } c T(G)W is totally bounded. T h e convex hull of a totally bounded set is again totally bounded, and the closure of such in a complete space is compact, so Y is included in a compact convex set. The closedness of this set and the Hahn-Banach Theorem implies that belongs
&PI
to it for any I E B , thus is compact indeed. Putting the pieces together, we have non-zero positive compact operators commuting with any T(s) . Applying the spectral theorem to such an operator we obtain a t least one finite dimensional eigen-
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space, and any eigen-space of an operator is, of course, invariant for the operators commuting with it. Considering that the spectral theorem for compact normal operators can be proved by more elementary means than the general one, the above proof (taken from [Hofl ) is perhaps the most elementary as well as the simplest. This theorem obviously implies that any irreducible continuous unitary representation of a compact group is finite dimensional. By a simple twist of the unitary trick this result extends not only to any locally convex space but even further (see Exercise 6.1 ). The problem of unicity of decompositions is closely connected with the fact that certain subrepresentations are "alike", while others are not. It is time to include the following fundamental definition.
Definition 6.5. Two unitary representations T and TI of a group over the Hilbert spaces 7-1 and 7-11 are said to be 'equivalent' if a unitary operator U from 7-1 onto 7-11 exists such that T I ( . )= U T ( . ) U - ' . Suppose that a certain unitary representation T of an arbitrary (not even topological) group G is decomposed into an (in general infinite) orthogonal sum T j of irreducible subrepresentations. Denote by J , the equivalence classes of the index set arising from the equivalence of subrepresentations. Choose a j , E J , for each a and a unitary operator u a , k : X,, H ' H k for k E J , establishing the equivalence of the subrepresentations. Denote by el: the canonical basis element in the IJ,I dimensional Hilbert space, i.e., ek = l { k } if this space is identified with !*(J,) (the !refers to the cardinality measure as usual). Then the mappings I @ ek -+ Ua,kx determine unitary operators U , from 7-1,- @ P ( J , ) onto 'Ha := ekEJ,7(where 1k we mean the tensor product in the sense of Hilbert spaces). These operators establish equivalences between the corresponding subrepresentations T, and the representations T j , @ 115,l where 1 , denotes the trivial representation l , ( t ) = I over the n dimensional Hilbert space (n is a cardinal). Now the unicity question of decompositions is completely solved by the following proposition.
ej
Proposition 6.6. Let S, be a family of pairwise inequivalent irreducible unitary representations of thegroup G acting on their respectivespaces X:, # (0) , nu be a non-zero cardinal for any a , and T be the representation e a ( S a@ I , , ) . Then the closed invariant subspaces for T are exactly the subspaces @,(Ka@Ma) where M a is an arbitrary closed subspace in !'(nu) for any a .
PROOF:It follows from Lemma 6 . 2 that a closed subspace is invariant if and only if the corresponding orthogonal projection commutes with each T(2). So the statement can be reformulated as follows: the projections in the commutant T(G)" are exactly those of the form C , I @ Pa where Pa is a projection over !'(nu) and the sum can be taken, e.g., in the so topology. More generally, we show that the von Neumann algebra T(G)" equals the set of operators of the form I @ A a , where
c,
Compact Groups
117
A , is a bounded operator on l z ( n , ) and sup, llA.11 is finite. These operators evidently belong to T ( G ) " . Now fix an element A E T(G)' . Let (Y # L,3 , PO and QO be one dimensional projections in 12(n,) and 1 2 ( n p ) ,respectively, and P = I K c9 ~ PO, Q = I'ia 8.0 . Write B = Q A P , then B E T(G)' and together with it the polar decomposition of B and the spectral resolutions of B'B and BB* also belong to this von Neumann algebra. But the restrictions of T to the range spaces of P and Q are equivalent to S, and Sp , respectively, so by the irreducibility of these the spectral resolutions above must be trivial, and then S, $ Sp implies B = 0 . Since this is true for any such PO,QO, the subspace H, = K, @ 1 2 ( n , ) is A-invariant for each a . Now fix an a and an orthonormal basis ej of l z ( n a ) . Let E j k be the partial isometry taking ek to e, and equalling 0 a t vectors orthogonal t o ek . Fix a p , write B = ( I @ Epj)A(Z@E k p ) and apply the polar decomposition and spectral resolutions as above; this time we have that B is a scalar multiple of a unitary operator in the range space of the projection P = I @ Epp . In particular, B is normal, and taking its spectral resolution we obtain B = C j k P with some constant cjk . Hence
For any finite set J C n , write PJ = I @ EjEJ Ejj and AJ := c j , k E J C j k E j k , then we can see P J A P J = I @ A J . We have llP~ll= 1 and the net PJ (we consider the set of J ' s ordered by inclusion) converges to the projection onto 3-1, in the so topology. Hence IlA~ll5 llAll , A J is a pointwise Cauchy net, and denoting = I @ A, . its limit by A , we obtain
If one just considers continuous representations of compact groups then the polar decompositions and spectral resolutions of operators of finite rank should only be examined in the proof above, and no knowledge of von Neumann algebras is necessary. On the other hand, a t this stage it is appropriate to point to the intimate connections between representation theory and the theory of von Neumann algebras. Namely, the latter one (especially its parts concerning decompositions) received a large impetus from its applicability to the decomposition problems of unitary representations, and also some concepts originated in representation theory has proved useful for other von Neumann algebras (not connected with group representations). It is debatable to what extent should a treatise on group representations retell the main facts in this other theory (and in the important link consisting in the theory of representations of C*-algebras). Since we have the excellent monographs [Dixl] and [Dix2], we might spare such pains; let us also offer [Arv] for those who want to become familiar with the essence by minimal effort, while for those who are interested in the Tomita-Takesaki theory (which was discovered after J . Dixmier had written his classics mentioned above) we may offer [S-Zs] . Definition 6.7. The 'spectrum' (some say 'dual object ),' of a topological group G is the set of equivalence classes of its irreducible continuous unitary
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representations (exclude the degenerate case when the ground space is (0)). I t is usually denoted by G . First observe this object is indeed a set, because the dimension of the underlying Hilbert space can not be arbitrarily large even for cyclic representations (a representation is called cyclic if it has a ‘cyclic vector’, i.e., a vector x for which the linear span of the set T ( G ) z is dense). It is customary (for locally compact groups) to endow the spectrum with a certain topology and, more importantly, with the structure of some measurable space (though the latter is usually done just for M2 locally compact groups). We return to this in later chapters, a t the moment we mention that for compact groups these things are uninteresting because the corresponding topology is the discrete one and each subset is measurable. For general M2 locally compact groups the measurable structure (as defined by G . W . Mackey) may be strictly finer than the Bore1 structure corresponding to the (Jacobson) topology; for tame M2 locally compact groups they coincide. We see from Proposition 6.6 that though the decomposition into irreducible subrepresentations is not unique, the subspaces ‘If,= K, @I t 2 ( n u ) are uniquely determined by the condition that they be the maximal invariant subspaces such that the von Neumann algebras generated by the corresponding subrepresentations be factors. For non-compact groups (even for G = R ) such a decomposition into a sum of ‘factor representations’ is not possible in general. However, we do have a nice enough (and as unique as it can be) disintegration over the so called quasi-spectrum for any M2 locally compact group (cf. Subsection 8.4 of [Dix2]); this quasi-spectrum (which is the set of quasi-equivalence classes of factor representations over separable Hilbert spaces) contains the spectrum in a natural way and equals it for tame groups. For compact groups any factor representation is the sum of equivalent irreducible ones, in other words, the ‘multiple’ of a certain irreducible representation, and Proposition 6.6 implies that both the equivalence class of this irreducible representation and the multiplying cardinal are determined by the representation. T h e same holds, in fact, for tame groups, while for wild groups exactly the other (non type I ) factor representations are responsible for the difference between the quasi-spectrum and the spectrum. We note that for wild groups the very spectrum is bad enough. The maximal factor subrepresentations established by Proposition 6.6 have the following properties: the linear span of the spaces of irreducible subrepresentations is dense, and all irreducible subrepresentations are of“the same equivalence class. We shall detect such ‘primary’ subrepresentations of representations of compact groups acting on more general spaces (see Proposition 6.14 below). T h e proof relies heavily on the so called matrix elements.
Definition 6.8. Let T be the representation of a group G over the locally convex space X . Then the complex valued functions T U , , = ( g )= ( u ,T ( g ) x ) (where x E X and u E X’ ) are called the matrix elements of the representation. If G is a topological group and T is continuous then these are, of course, continuous functions. If X is a Hilbert space then it is customary to single out an orthonormal basis e , and j u s t consider the matrix elements corresponding to them (i.e., when
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Compact Groups u
3: = ep 1; in this case for any fixed g the operator T(g)has the with respect to this basis, which is the origin of the notation.
= (. , e a ) and
matrix
T,,p(g)
Lemma 6.9 (The Orthogonality Relations). Let G be a compact group and Tl,T2 be two irreducible continuous unitary representations of G over the Hilbert spaces 3-11,3-12. Let rj ( j = 1 , 2 ) denote the corresponding matrix elements. Consider the scalar products
in the Hilbert space L2(G,p) where p is the normalized Haar measure (i.e., for which p ( G ) = 1 ). If T I $ T2 then f = 0 . On the other hand, if TI = T 2 then f = n-l . (tl,3:2) . (y1,y2) where R is the finite dimension of 3-11 = ‘If2 .
PROOF: We have I T ~ , ~ ( S ) I 5 11x11 . llyll for the matrix elements of any unitary representation, hence in our case (by the finiteness of p ) f ( ., y1; . , y2) and f(z1, . ; 2 2 , .) are bounded sesquilinear forms on 3-11 x 3-12 and they intertwine TI and T2 (this means the property (T1(s)rl, T2(s)z2) = ( ~ 1 ~ )~ by 2 ) the right and left invariance of p , respectively. So we have bounded linear operators A = A,, ,2 and B = Byl,ya from 3-11 into 3-12 such that f = ( B r l , x 2 ) = (y2,Ayl) and A , B intertwine TIand T 2 (i.e., ATl(s) = T2(s)A and the same holds with B ) . The usual polar decomposition and spectral resolution show that for TI 9 T 2 these operators (and f with them) must vanish, while for Tl= T 2 these are scalar operators. Then we have scalar valued functions a and b such that
f (211Yl ; 2 2 , Yz) = b(Y1, Yz)
’
(213 2 2 )
= 4.1
4. (YZ,Y1)
and fixing some z1 = x2 # 0 we can see b = c . (y2,y1) with some constant c . It remains to check that c = n-l . Let e l ,. . . , en be an orthonormal basis of the ground space and fj := f(el,ej;el,ej), then
for the integrand here identically equals 1
Remark. If G is not compact but at least unimodular then the above proof (except for the last argument with the dimension) works for an interesting kind of irreducible continuous unitary represent,ations, characterized by the condition that 2 + T ~ be, a~ bounded operator from the ground space into L 2 ( C ) for any y (since T = . , ~ ( S )= T ~ , ~ ( S -, ~this ) is equivalent to the similar condition with interchanged indexes for a unimodular group). It is known (see, e.g., Chapter 14 of [Dix2] ) that these representations are exactly the irreducible subrepresentations of the left regular representation (note the left and the “corrected right” regular representations are equivalent even if the group is not unimodular). T h e part of
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Chapter 6
G corresponding to these square integrable irreducible representations is called the 'discrete series' and, of course, for compact groups they comprise the whole spectrum. On the other hand, the discrete series of any non-compact Abelian locally compact group is empty for the irreducible representations are one dimensional and the corresponding matrix element has absolute value 1 everywhere (cf. our next chapter).
Theorem 6.10 (Peter-Weyl Theorem). Let G be a compact group, for ( E 6' denote by d(C) the dimension of the irreducible representations of this class and choose a representation Tc E ( acting on t?(d(C)) for all ( . Then the right regular representation R (where we consider L 2 ( G ) with the normalized Waar measure) is equivalent to @(Tc 8 I d ( < ) ) and for the space Mc of the primary subrepresen tation corresponding to C an orthonormal basis is provided by the continuous functions d ( < ) 1 / 2 T c ( . ) j k .
PROOF:Let S be an irreducible subrepresentation of R and C be its class. Then S rz Tc , so we have an orthonormal basis 2 1 1 , . . . , ud(c) of the space of S such that R(s)uk = C jT c (s )j k u j for all s E G and for all L . Choosing a (Baire measurable) representative for each u j we have u k ( t s ) = Te(s)jkuj(t) for all s and almost every t (the corresponding negligible sets may vary with s) and the sides of the equality are measurable, therefore we can infer by Fubini's theorem the existence of a t o such that uk is a linear combination of the functions g -+ T c ( t , ' g ) j k (we have also used that the translation of a negligible set remains negligible). Now the fact Tc is a representation means
Cj
Thus the space of S is contained in the asserted primary subspace. On the other hand, (6-2) also shows that for any fixed i the matrix of the restriction of R ( s ) with respect to the basis T c ( . ) i , l ,... ,Tc(.)i,d(c) equals T c ( s ) . Now multiplying these vectors by d ( < ) ' l 2 we get an orthonormal system by Lemma 6.9. Use Lemma 6.9 once more, together with Theorem 6.4 and Proposition 6.6 to complete the proof.
Several remarks may be interesting in connection with this theorem. Observe that if G is a compact Lie group then any finite dimensional matrix element is an analytic function by Cartan's theorem (also by our Proposition 3.18). Hence, by the theorem, the finite dimensional right invariant subspaces of L 2 ( G ) consist of analytic functions (one can prove the same for an arbitrary Lie group using Proposition 3.18 and Proposition 3.27). We see from (6-2) that the primary spaces of R are also left invariant, and the matrix of the restriction of L(g) with respect to the basis Tc(.)l,k,. . . ,Tc(.)~(~),~ equals Tc(9-l)' = , so L has the same primary spaces but their correspondence to the elements of the spectrum is changed (by conjugation). For any unimodular group certain bi-invariant subrepresentations of the regular representation correspond to the elements of the discrete series in a similar
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Compact Groups
fashion (see Proposition 14.3.5 of [Dixa]) but their sum does not yield the whole regular representation in general. Maybe the most popular consequence of the Peter-Weyl Theorem is the fact linear cornbinations of irreducible matrix elements form a dense set in L2(C) L r any compact group C . Since this set is right (and left) invariant, for a Lie group it is also dense in C"(R) (by Proposition 3.20) which coincides with the Rkchet space C"(G) for a compact Lie group (cf. the proof of Proposition 3.27; observe that Sobolev's lemma becomes much simpler for a compact manifold). We show a t once that this set is dense in the Banach space C ( G ) for any compact G (this result is usually called the Peter-Weyl C-Theorem).
Corollary 6.11. The linear space M generated by the subspaces Mc is dense in C ( G ) with respect to the usual maximum norm.
PROOF:Suppose p E C(C)* is such that ( p , f )= 0 for f E M c . Since Mc is invariant under translations, this implies p*f = 0 for any f E Mc (where the convolution is interpreted by Definition 2.1). The L2 topology is finer than the L' over any finite measure, so by the L 2 denseness of M we infer p * f = 0 for any f E L z ( G ) if p satisfies the condition for all (cf. (2-2)). In particular, this holds for f E C ( G ) and it is easy to construct a net fn E C ( C ) such that (1.1 * fn ,'p) ( p , 'p) for any (o E C ( G ) and p E C(G)* . Thus ( p , M ) = 0 implies 11 = 0 , hence M is dense in C ( G ) by the Hahn-Banach Theorem.
-
Observe that M contains the matrix elements of all finite dimensional continuous unitary representations of G by Theorem 6.4 (moreover, Proposition 6.1 shows that even those of the non-unitary ones). Since one can take the complex conjugate of a representation and the tensor product of two representations, we see M is a *-subalgebra of the C*-algebra C ( G ) , An interesting consequence of Corollary 6.11 is the fact each compact group is a Lie limit, i.e., any neighborhood of 1 contains a closed normal subgroup N such that G I N is a Lie group. In fact, the kernels of all finite dimensional continuous unitary representations provide such a family of normal subgroups. This family is closed under finite intersections (take the direct sum of the corresponding representations), so by the compactness of G it is enough to show the intersection of all these kernels is trivial. Now if g belongs to the intersection then f(g) = f ( 1 ) for any f E M so Corollary 6.11 implies g = 1 . ~
Fix the choices Tc E C for a while and introduce the notations r j k ( g )= . We compute the convolutions of these matrix elements on the base of (6-2) and the orthogonality relations:
Tc(g)jk
so we have a non-vanishing convolution only if
(6-3)
<=
Y
and j = k and then
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Chapter 6
Definition 6.12. Let G be a group and T be a finite dimensional linear representation of G . The 'character' of T is the complex valued function x(g) := trT(g) . The properties of the trace imply that any character is a central function (i.e., X(ghg-') = ~ ( h for ) any g,h E G ) and the characters of equivalent unitary representations are equal. Proposition 6.1 shows that the character of a continuous representation of a compact group equals some unitary character. On the other hand, the characters of inequivalent finite dimensional continuous unitary representations of a compact group are not equal (as we show presently), this is the origin of the notation (characters characterize the equivalence classes of representations). Theorem 6.13. Let G be a compact group and denote by xc the character corresponding to the class C E G . Let p be the factorization of G by inner automorphisms, X be the corresponding factor topological space. Then p is a continuous open mapping, X is a compact TZ space and if we endow it with the finite positive Radon measure induced by the normalized Haar measure through p then {xc ; C E C} becomes an orthonormal basis of L 2 ( X ) through the natural identification of central functions with functions on X . If x is the character of some finite dimensional continuous unitary representation T o f C then the multiplicities of the irreducible components in T are exactly the coefficients of the continuous function x on X with respect to the basis above. The linear space generated by the irreducible characters xc is also dense in the C*-algebra C ( X ) . In L ' ( C ) (considered with the normalized Haar measure) we have xc *x, = 0 for ( # v and xc * x i = d(()-'xc .
PROOF:This factorization is implemented by a group of homeomorphisms, therefore p is open. The equivalence relation itself is the continuous image of C x C in G x G (through the mapping (z, y) + (z,yzy-') ), hence compact. Observe for an open factorization the TZ property of the factor space is equivalent to the closedness of the equivalence relation. Since xc = rjj , the orthonormality and the convolution property are immediate consequences of Lemma 6.9 and (6-3), rcspectively. T h e statement x = ncxc (where nc is the multiplicity of ( in T by Theorem 6.4 and Proposition 6 . 6 ) is obvious. The basis property will follow from the denseness in the finer topology of C ( X ) . Consider the idempotent mapping @ : C(G) H C(X) defined by @ f ( p ( z ) )= Sf(gzg-')dg . It is contractive, thus @ ( M )is dense in C ( X ). Writing 1 = g z and s = g-' into (6-2) we obtain o p = E, * r& (for G is unimodular), and we see from (6-3) this function vanishes if z # k and equals d(()-'xc for i = k . Thus @ ( M )is the linear space generated by the characters.
cc
@.iCk
~j~
Proposition 6.14. Let G be a compact group and U be a continuous linear representation of G over the quasi-complete locally convex space X (if G is metrizable then sequential completeness is enough). We say that z E X \ (0) is of
Compact Groups
123
type C (where C E G ) if x belongs to the space of an irreducible subrepresentation of the class C (in a suitable basis of the corresponding finite dimensional subspace). Then the linear span Xc of elements of type C is closed and invariant under U ; writing x((g) = d(C)Xc(g-’) = d((‘)Xc(g) the continuous operator Pc = U ( T ~is an idempotent with range Xc and vanishing on each X u , v # C . The linear span of all Xc ’s is dense in X. For any x E Xc \ (0) the subrepresentation generated by it is finite dimensional and in a suitable basis has the form Tc @I l,,(,) with
4x1 5 4 0 .
PROOF : The quasi-completeness (or sequential completeness if G is metrizable) of X implies that U ( f ) is an everywhere defined continuous operator for any f E L’(G) . We infer from Lemma 3 . 3 , formula (2-3) and from the facts a( is central and G is unimodular the following: (6-4)
PcCJ(f)= CJ(xc * f ) = CJ(/)P, .
In particular, P: = Pc and PcPu = 0 for C # u (see the relevant formulas in the previous theorem). Let z E X \ (0) and define the operator A , : L’(C) H X by A,f = CJ (f)2: ; this is a continuous linear operator satisfying A,R(s) = CJ(s)A, for any s E G (recall that the notation f was introduced i n 2 . 4 ) . Since the L2 topology is finer, A , is all the more continuous when considered on L z ( G ) . If I is of type ( then we may view A , as a bounded operator from L’(G) into t 2 ( d ( C ) ) intertwining R with Tc . By polar decomposition we can infer that on the right support of A , (in L’(G)) R is equivalent to Tc (because A , = 0 is not possible: Azfn x for a net “tending suitably” to Dirac’s delta). This implies by Theorem 6.10 that A , vanishes on the orthogonal complement of Mc . For the moment designate this last condition (for an arbitrary 2: E X \ (0) ) by saying ‘3: is almost of type C’. Observe that the orthogonal projection onto Mc can be realized as convolution (no matter from which side) by the central function d ( C ) x c = i r e (this follows from Theorem 6.10 and (6-3)). Hence I is almost of type ( if arid only if A , = Apt, (when considered on L 2 ( G ); we have used (6-4) ). Tending to Dirac’s delta as above we can we this is equivalent to z = PCT . Collect our results obtained this far: Xc is contained i n the closed range of the continuous idempotent Pc and for any non-zero element z of the latter A , is a non-zero continuous intertwining operator between R and CJ supported by Mc from the right. Since Mc is finite dimensional, the range of such an A , must be finite dimensional and we can take a basis of this range so that the restriction of U be a continuous unitary representation (by Proposition 6 . 1 ) . The restriction of R to the right support of A , is equivalent to T C 8 l,,(,) with some n(x) 5 d(C) by Theorem 6 . 1 0 . Therefore C J I R ( A z l is equivalent to the same. But x E R(A,) because it is closed (tend to Dirac’s delta). Hence I E Xc and we have also proved the last assertion of the proposition. We now know that Xu is the range of P, , thus PcPv = 0 implies that Pc maps Xu to ( 0 ) . It remains to show the denseness of the linear span of the Xc’s. Since A,(f,,)-+ x for a suitable net f,, E C ( G ) and A , is continuous, the proof
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Chapter 6
is completed by inferring from (6-4) that A , ( M ) is a part of this linear span for any x . In the remainder of this chapter G can only denote connected compact Lie groups; and the word ‘representation’ will refer to complex finite dimensional continuous unitary representations of G whenever other thing is not stated explicitly. A nice exposition of the basic theory of these groups can be found in [Dy01. See also [Ada] for a somewhat different approach. Of course, this theory is quite classical but it seems that earlier literature had not achieved the clarity and elegance of the above mentioned sources. The topic is also covered t o some extent by a number of treatises of wider scope. The author’s suggestion is that one should compare [Dy-0] with the remaining material of the present chapter and then study [Tits] which is an excellent collection of results and d a t a (these things are rather thoroughly classified) without proofs. By Proposition 6.1 the Lie algebra g of G can be made a real Hilbert space such that Ad be an orthogonal representation. Then applying Lemma 6.2 one can decompose Ad into irreducible subrepresentations, and the connectedness of G implies that in this way we also get a decomposition of g into irreducible invariant subspaces for the family ad(g) of operators. Hence it is not hard to deduce this decomposition yields minimal ideals 81,. . . , g k and the non-trivial subrepresentations of Ad are mutually inequivalent, while if there exist trivial subrepresentations of Ad then the sum of the corresponding one dimensional ideals is t.he center a of g . In particular, g is the direct sum (as a Lie algebra) of a commutative and a semi-simple Lie algebra, the latter equals [g, g] and admits a scalar product with respect to which any ad 2, is skew symmetric (do not forget everything is real here). Naturally, one of these two components may be absent. It can be shown that -1 times the Killing form (we define it in (F-22)) can serve as this scalar product and if we have a simple algebra then any such scalar product is a negative multiple of the Killing form. On the other hand, if the Killing form of a real Lie algebra is negative definite (we call such an algebra ‘compact’, deviating from [Dy-0] where the Lie algebra of any compact Lie group is called compact) then the group of its inner automorphisms is compact and the Lie algebra of this group is isomorphic t o the Lie algebra we began with. Since R” is the Lie algebra of the compact group T” , we see that the Lie algebras of all possible compact Lie groups are exactly the direct sums of a commutative and a compact Lie algebra (of course, we omitted some proofs). Now if the Lie algebra of G is compact (in particular, if G is the inner automorphism group of a compact Lie algebra) then its homotopy group is finite. It is possible to give an ‘‘existence proof” to this result (which is relatively short) but the method in [Dy-O] (the main points of which we sketch below) enables us to construct the homotopy group, and not merely the finiteness but also the exact description of these groups is obtained. Some elementary considerations now show that the simply connected group corresponding to a compact Lie algebra is compact (while the simply connected groups of other Lie algebras are not compact; this is the motivation for the notation ‘compact Lie algebra’). So the center of the simply connected covering of an arbitrary compact G is identified with the direct product
Compact Groups
125
of (the center of the Lie algebra) and a finite Abelian group. Then the most basic theory of discrete subgroups of R” implies that the intersection of the homotopy group with a must be Z” in some basis (where n is the dimension of J ) , and
G = (T” x G ’ ) / N
(6-5)
where G‘ is t,he compact simply connected group corresponding to [g, g] and N is a finite subgroup of T” x Z ( C ’ ) . The central tool in studying G is the concept of a ‘maximal torus’. The ‘maximal tori’ are defined as the maximal connected Abelian Lie subgroups (these are necessarily closed in G and isomorphic to some T r ) . Their Lie algebras are exactly the Cartan subalgebras of g . Another key notion is that of a ‘regular element’. An element g E G is called regular if the subalgebra {v ; Ad(g)v = v } is commutative. It turns out this condition is equivalent to the requirement that the dimension of this subalgebra equal the minimum dimension of all such subalgebras (which integer is called the ‘rank’ of G and depends only on g). The other elements of G are called ‘singular’. It is important how small is the set of singular elements: except.ing the trivial case (when G itself is a torus and each element is regular) it turns out that the singular elements form the smooth image of a compact manifold of dimension less by 3 than the dimension of G (and if this number were 2 then the whole theory would be killed). For a regular element g the subalgebra {v ; Ad(g)v = u } equals the Lie algebra of the unique maximal torus containing g , while singular elements belong t o more than one maximal tori (all of their Lie algebras are contained, of course, in {v ; Ad(g)v = v } ). The group of inner automorphisms permute the maximal tori transitively. The intersection of all maximal tori equals the center of G . For the proof of these basic results see 5 1 of [Dy-0] . Note that relying on these facts one can give an amazingly simple proof for the important theorem that any element of the Weyl group (except the unit) moves each Weyl sector (see Lemma 5.15 of [Ada] ; versions of this result are usually referred to as Chevalley ’s lemma). In the sequel we fix a maximal torus in G and denote its Lie algebra by m . We have m = a @ m’ where m’ = m n [g, g] is a Cartan subalgebra of the semi-simple part. Let A be the complexification of this semi-simple part and M = m‘ i . m’ , then m‘ = i . M , with the notation of F.15 (because of the negative definiteness of the Killing form of [g, g] ). Clearly, the set of vectors v E m’ satisfying ‘ exp v is singular’ equals the Stiefel diagram as defined by (F-23), while the corresponding set in m is the direct sum of a with this Stiefel diagram. Let f = { v E m ;exp v = 1 } (this depends on G and is a discrete subgroup of the additive group of m ) and f , = { v E m ; expv E Z ( G ) } , then e x p f l = Z ( G ) , fl depends only on the Lie algebra and equals the direct sum of J with the set
+
{u E M ; eadu = I }
Chapter 6
126
(with the notations of Appendix F ) . Let P be one of the cells (cut by the Stiefel diagram +J ) whose closure contains 0 and consider the orbit p of this set under the action of the inner automorphism group Ad G . In fact, is the union of all such cells in all Cartan subalgebras, so i t does not depend on m and P . One shows (cf. $ 3 of [Dy-01) that the interior of the closure of is star-shaped with respect to 0 (thus simply connected) and p can be obtained from this set by throwing out the singular elements ( v is called singular if exp v is singular); furthermore, exp p equals the set of all regular elements in G . Then we invoke the lemma from topology asserting that if some “singular set” is so small that its codimension is 2 3 in a certain sense then it does not count when studying homotopy (see Lemma 1.8 in $ 3 of [Dy-01). This can be applied to both p and e x p p and some fiddling with exp and the inner automorphism group yields the following: let A = f n p and x E P , then for any v E A we have x - v E and choosing for any v a curve -y, connecting x with x - v in p we obtain a bijection from A onto the homotopy group of G a t exp 2 by mapping v to the homotopy class of exp oy,, . This implies in the semi-simple case the finiteness of the homotopy group, for is compact then. We also want to clarify the group structure of the homotopy group. Observe that we can replace y,, by any curve connecting I with I - v in g because g is simply connected and exp is continuous. Now define fo to be the set of points in i . M , which may be achieved from 0 by successive orthogonal reflections in the hyperplanes of the Stiefel diagram (the space i . M , is considered Euclidean with a negative multiple of the Killing form as scalar product). An equivalent definition is to take the additive group generated by the elements S x i H , where a may vary in R but considering just a simple set of roots we get the same (and in the latter case we have a t once a basis of f o as a 2-module). The equivalence of these constructions is proved in $ 2 of [Dy-0] as well as the fact fo c r . It is easy to deduce from the first definition that the closure of any cell intersects fa. Since v -s,(v) = a ( v ) H , for any v (where s, is the Weyl reflection in a ) , we have v - s,(v) E fo for v E fl 3 r . Hence we infer A + r o = f . The theory can be applied to the simply connected group corresponding to any compact algebra (because the compactness of factor and normal subgrogp implies the compactness of the original and the homotopy group of the inner automorphism group is finite), and then A must be trivial, so we have f = ro for such groups. The definition of fo depends only on [g, g] and m’, so the f o of our general G is the same as the corresponding ro for G’ and we obtain fo = { v E m ; exp* v = 1 } where exp* is the exponential mapping into the simply connected covering J x G‘ of G . Since exp = p o exp* with the canonical projection p and exp*lm is homomorphic, we have the following: if y , 19 are curves in m such that for their endpoints we have y(O)-r9(0), y( 1)-9( 1) E f , then exp oy exp or9 if and only if y( 1) 9(0) - y(0) - 3(1) E f o . This shows, in particular, that for v , w E A and u E f o the curve expy,,(s) is homotopic t o exp(z - w - s ( v u ) ) and therefore (denoting the operation of the homotopy group in A by * ) the class of w t v contains the curve exp(x - s(w v u ) ) . So if w v Q happens to fall into A then it equals w $ v . Since A fa= r , +) such that the isomorphism this must happen for some Q and ( A ,$) N (f/ro, is the natural one, i.e., mapping each element to the class where it belongs. Note
-
+
+
+ +
+ +
+
Compact Groups
127
we have exp* f1 = 3 x Z(G') and so the possible r ' s are exactly those discrete subgroups of fl which contain f, and a full lattice in a (we mean by full lattice a discrete additive subgroup of an n dimensional real linear space equalling Z" in a suitable basis). We define in Appendix F the Weyl group W as a set of operators on M but it is obvious how one should define the analogous object as some group of automorphisms of M and the mapping A + 1 @ ~dAl,,) ( is the natural isomorphism from the latter onto the former. Any point of G can be moved into M by an inner automorphism, so Exercise 6.2 shows that the set X of conjugacy classes (considered in Theorem 6.13) is identifiable with the factor of M under the action of the Weyl group. It would be pleasant to get closer information about the measure considered in Theorem 6.13. We show presently this idea is quite feasible.
Lemma 6.15 (H. Weyl's Integral Formula). Suppose G is not a torus, i.e., the semi-simple part of the Lie algebra is not absent. Use the notations A , M , etc. as above (in accordance with Notation F . 1 5 ) . For Q E Ro define & : M I-+ T by &(exp v) = ea(") where we extend Q linearly from m' to m by defining it to be 0 on a (since ea(") = 1 for any v E rl 3 , this definition is correct and yields an irreducible character of %I). If u is central and any of the integrands below is integrable then
r
where dh refers to the normalized Haar measure of the torus. PROOF: Let Z = G / n and p be the canonical projection p : C H 2 . itIis , Consider the analytic function (9, h ) -+ ghg-' on the manifold G x % a function of p(g) and h , so by the properties of factor manifolds we have an analytic function f : Z x M H C such that f(p(g),h) = ghg-' . If p , v and 0 are the normalized Haar measures on G and %I and the normalized G-invariant measure on 2 , respectively, then (see Proposition F . 1 2 ) we are a t a setting where Proposition F.13 applies (with any choice of the projection P in the Lie algebra). Since u is central, we have u(f(2, h ) ) = u ( h ) . The facts about regular elements imply that lf-'({g})l equals the number of elements in the Weyl group if g is regular and that the set of singular elements is negligible. It remains to check that now (for the F from Proposition F.13)
ldet F(g, h)l =
n
I&(h) - 11 .
aERn
The tangent vector (gPv, h(v - Pv)) equals +(O) if y(s) = (p(gexp s P v ) , h e x p s(v - Pv)) (cf. Notation F . 7 ) . Thus F ( g , h)v = i ( 0 ) with
. g exp S P V . h exp s(v - P V ) . (9 exp s~v)-' = g exp(s Ad(h-')Pv) exp s(v - Pv) exp(-sPv) .g-l
d ( s ) =gh-'g-'
Chapter 6
128
+
and hence F ( g , h ) = A d g . ((Ad(h-') - I)P I - P ) . We have det A d g = 1 because A d G is connected and is a part of the orthogonal group with respect to a suitable scalar product. Thus det F ( g , h ) = det P(Ad(h-') - I)lv where V is the range of P . Now choose P so that V be the intersection of [g, g] with the sum of the root spaces A , (this is possible because the conjugation corresponding t o [g, g] maps A , to A _ , ). This V is invariant under Ad M and t o compute the determinant we may switch to C @R V . There Ad(exp v ) ~ , = ead' 2, = eQ(')x, and the proof is completed by observing that -Ro = Ro . We want to explore the structure of representations of G ; by Theorem 6.4 and Proposition 6.6 this task is equivalent to the description of the irreducible representations. This problem is solved to a considerable extent by the theory of highest weights. Recall that a representation T of G is determined by its tangent (namely, T(expv) = edT(') ), a Lie morphism @ : g H ~ ( n is ) always integrable and hence it is a tangent if and only if e r p ( " ) = I for all v E r (because exp* = p-'({l}) in the simply connected covering), and T is irreducible if and only if d T is irreducible. Thus any result achieved in one of the twin theories of representing G and g has much bearing on the other (especially in the semi-simple case when G can be simply connected). There are two different approaches (often designated as 'analytic' and 'algebraic') endeavoring to place the main burden of the proofs in the first and second aspects, respectively. None of these approaches yields simple enough proofs to silence its opponents. We shall adhere more to the analytic approach (mainly because the work we have done this far makes this the more effortless) but also use some algebra as outlined in Appendix F . A completely algebraic treatment is found in [Ser2] (without proving Weyl's formula; for the algebraic approach to that see [Jac] ). First we include a simple statement which shall enable us to get rid of a whenever we want to. Of course, its proof can be found in almost any treatise on von Neumann algebras but it is so closely related to representations that it is worth retelling.
r
Proposition 6.16. Let A and B be *-invariant families of operators over the finite dimensional Hilbert space 1-1 such that their union is irreducible and A B = B A for A E A , B E B . Then we can write 1-1 = 1-11 @ 1-12 (as a Hilbert space) such that any A E A is of the form A0 @ I and similarly B = I @ BO if B E B . The corresponding families do and BO are irreducible over 'HI and 1-12 . On the other hand, if *-invariant irreducible families of bounded operators are given over two Hilbert spaces (not necessarily finite dimensional) then the union of the corresponding two families over the (Hilbert) tensor product is irreducible.
PROOF: First show the last statement. If a projection P commutes with the first family then consider the operators ( I @ E,j)P(I @ Ekp) as in the proof of Proposition 6.6. These commute with A which is *-invariant, hence their polar decompositions and the spectral resolutions of the corresponding positive operators also commute with A . By the irreducibility of do we obtain these operators are
Compac t Groups
129
normal; take spectral resolution again and consider strong limits exactly as in the proof of Proposition 6.6 to obtain P = I 8 Po with some bounded operator P O . Since P was a projection, PO must be a projection. Now if P also commutes with B then Po commutes with So and so the irreducibility follows by Lemma 6.2. Turning to the first statement let N be the commutant of B , it is a von Neumann algebra containing A . Let P I , . . . , P, be a maximal orthogonal family of non-zero minimal projections in N , we obviously have I = Cj P, . T h e usual argument with polar decomposition yields (by the minimality of the Pj's this time) that P,sPk is a constant times a partial isometry with right and left supports Pk and P, for any S E N (the constant may be 0 ) . Thus if PjNPk # (0) then Pj N Pk (two projections in a von Neumann algebra are called equivalent if they are the left and right support of a partial isometry from the algebra). Let Q be the sum of Pj's equivalent to P I , then we have Q N ( I - Q ) = (0) = ( I - Q ) N Q , i.e., Q belongs to the center of N . Thus Q is a non-zero projection commuting with A u B and therefore Q = I . This means each Pj is equivalent to PI ; choose a partial isometry Ejl E N with left support Pj and right support PI (for j = 1 choose P1 ). Then for any fixed S E N we have by the minimality of Pi that E;,SEkl = cjkP1 and therefore PjsPk = Ejl E3flSEklE;1 = CjkEj1E;l . Define l ? j k := EjlE;, (which is in accordance with its special case k = 1 because we chose E l l = Pl ), then these operators form a basis of N . Let 3-11 = C2(n) , 3-12 = Pl(3-1) and define an isomorphism from 3-11 @ 3-12 onto 3-1 by setting U ( t @ y ) = C;=,t j E j l y , then N corresponds to B ( X l ) @ C .Applying the first part of the proof we obtain that the commutant N " then goes onto C@B(3-12). Now A c N , B c N" and if one of the families do,BO were not irreducible then the corresponding projection in 'H would commute with A U B contradicting to our assumption. Observe that the finite dimensionality was only used in establishing a finite orthogonal family of minimal projections in N whose sum equals I . Now in the general case the irreducibility of A U B also implies that N is a factor but this factor is not necessarily discrete (or 'type I' if you prefer that name). If it is then we do have a similar (generally infinite) family of minimal projections and the same proof with easy supplements works. This implies that the irreducible continuous unitary representations of the direct product of two locally compact groups are exactly the tensor products of those of the components whenever at least one of them is tame (Proposition 6.16 as stated implies the same in the special case when both components are compact). Our proposition also applies to the (skew Hermitian finite dimensional) representations of g and hence to find all of the irreducible ones it is enough to solve this task in the cases when g = R or g is a simple compact Lie algebra. The former case is trivial enough, while studying the latter is equivalent to studying the representations of the corresponding simply connected compact group. This is worth a naming: N o t a t i o n 6.17. We say G is 'totally simple' if it is simply connected and its Lie algebra is simple (as is customary, (0) and R are excluded from the notion
130
Chapter 6
‘simple’). We shall develop the theory of highest weights for a totally simple G first; most arguments work in greater generality but this is not interesting because the final results will show what is the exact situation in the most general case. Definition 6.18. Let T be a representation of G , then TI, is a repre sentation of Im and so it decomposes into irreducible ones. Now the irreducible representations of a commutative group are one dimensional (because the second commutant of the range is commutative for a commutative group and the whole algebra of bounded operators for an irreducible represen tation). The corresponding primary subspaces (cf. Proposition 6 . 6 ) are called the ‘weight spaces’ of T (with respect to 332) and the one dimensional representations involved are called the ‘weights’ of T . Furthermore, we often call the ‘infinitesimal weights’ dw and (in the semi-simple case) also their complex linear extensions to M briefly weights. This practice can not cause great confusion for ‘Dl = m / r (as a Lie group) and w(exp u)= edw(u) . On the other hand, all one dimensional representations of 332 are called weights (independent of the representations of G ) and so are their tangents.
It is clear that the weights on m are exactly the real linear mappings 29 into
i . R satisfying s ( u ) E Z for all u E 2 Ti
r . If G is simply connected then
f = ro
and hence the condition is equivalent t o b(H,) E Z for all cr E 17 (with 29 extended complex linearly to M ). The infinitesimal weights of T can also be defined as the functionals Q for which dT(u)z = a(u)t if x belongs to the corresponding primary space and u ~ m . Notation 6.19. If G and 332 are fixed then we shall denote the set of characters of the Lie group ‘Dl by Ch . The elements of Ch are exactly the combinations of the weights with nonnegative integer coefficients; these coefficients are determined by the character because Theorem 6.13 is applicable to Im. The characters of G are determined by their restriction to 332 , so they can naturally be considered as a subset of Ch . T h e task is to single out this subset in a more manageable way. We shall see that the action of the Weyl group has much to do with this (as it should be expected, cf. Exercise 6.2 and Lemma 6.15). Observe that Ch is closed under multiplication and addition (the product of two weights is a weight) but, unfortunately, not closed under subtraction. The elements of the generated ring Ch - Ch are usually called the ‘virtual characters’ of 332. Definition 6.20. Let Chs := { f E Ch - C h ; f o @ = f V @ E W } (any character of G belongs to this set of ‘symmetric virtual characters’) and also consider the set of ‘alternating virtual characters’ Cha := { f E Ch - Ch ; f o @ = det@ . f V @ E W } where the determinant is computed with respect to the
Compact Groups
131
realization of W as a subgroup of the orthogonal group of m‘ . In other words, this number equals -1 i f @ is the product of an odd number of Weyl reflections and 1 in the opposite case. Observe Chs is a subring of Ch - Ch , while Chs .Cha c Cha . Since the set of weights is linearly independent (by Theorem 6.13) and the (finite) Weyl group acts on it, one can check Chs and Cha as additive groups are free Abelian groups. Look a t this more closely in the case of Cha. Call a weight w regular if dw( H a ) # 0 for all a E Ro (this is equivalent to saying that if a scalar product is chosen on dw = (., u ) g such that each A d s is orthogonal and v E m is such that 4. then i times the projection of v in [elg] belongs to some Weyl sector). T h e fact each @ E W \ { 1) moves each Weyl sector implies that the W-orbit of a regular weight consists of IWI elements, while if w is singular and dw(H,) = 0 for some a then the corresponding Weyl reflection leaves w invariant. Thus alternating virtual characters can not contain singular weights. For any weight w we consider the ‘elementary alternating sum’
A(w) =
[email protected]@ . O€ W
It is alternating, hence A ( w ) = 0 if w is singular; and the set { A ( w ) ; w is regular } provides fl times the elements of a basis for C h a as a free Abelian group.
Theorem 6.21. Let G be totally simple. Fixing a simple set of roots let e be the weight on M defined by the equations e ( H , ) = 1 if a E 17 , and set D = A ( @ ) where @ is the corresponding weight on 93’. Then f -+ fD is a bijection from Chs onto Cha and the (restrictions of the) irreducible characters of G go into a basis of Cha such that =tG corresponds to the set of A(w)’s described above. We note the requirement G be totally simple is not very substantial; the proof relies only on the condition that e be a weight (it is defined t o be 0 on a if G is not semi-simple). On the other hand, the theorem as stated yields by Proposition 6.16 a description of the irreducible characters of the group T” x G’ considered in (6-5); and the irreducible characters in the general case are singled out by the condition that the weights (of T” x G’) contained in them should annihilate the finite central subgroup N . We shall obtain even more definitive results for the general case from Theorem 6.22. The reasons for requiring total simplicity instead of just simply connectedness are the following: the scalar product to be considered on g is necessarily a multiple of the Killing form in this case, and the possible e’s are “personally” known from the classification theory of complex simple Lie algebras.
PROOF:First we check that Q = f CaCR+ a . If a E I7 , /3 E R+ and /3 # a then any root of the form /3 - ka must have a positive coefficient and hence belongs to R+ . Thus the Weyl reflection sa acts on the set of functionals R+ \ { a }
132
Chapter 6
and so leaves their sum invariant, consequently EPER+ / 3 ( H a ) = a(H,) = 2 . Now we want to prove D can also be expressed as follows:
where E varies in the set of sign functions { - l , l } R + and, of course, ec(v) = 51 CaER+ € ( a ) a(.) . . Each Q~ is a weight on m because Q - ec is a sum of roots but these weights are not necessarily different. Denote the second expression in (6-6) by 6 for the moment. Apply the invariance of R+ \ { a } under s, again t o obtain ~ O S ,= -6 for all Q E 17 . Thus, considering 6 as a function on 'JX = m / f (this is possible because the Q,'S are weights), we have 6 E C h a (because W is generated by the Weyl reflections corresponding to a simple set of roots). Since le0(u)/2 - e - a ( u ) / 2 1 = lea(") - 11 = 11 - e-a(")I for any root a and any v E i . M , , we can see that 1612/lWl equals the density function computed in Lemma 6.15. Hence its volume is 1 and therefore 116112 = IWI in L2(m)(considered with the normalized IIaar measure of the torus). Applying Theorem 6.13 t o the torus we = IWI . But 6 can be expressed in obtain that 6 = C ( n ( x ~such that the basis of Cha described before the theorem, so there must be a weight 6 such that 6 = A(29) . Then (6-6) will follow if we can show that e is regular and different from any ot.her e c . The regularity is most easily checked using the fact that, { H , ; Q E 17) is a simple set, of the dual system (this shows that e(H,) is the height of 11, in the dual system, so it can not vanish for a E Ro ). The proof of the other statement will be the first application of (a version of) the concept of highest weights. Let M i denote the set of those complex linear functionals on M which are real on M , (any weight belongs to this set). We define a partial ordering on M i by writing 'p < II, if 'p(v) < G(v) for all v in the Weyl sector of 17. Then for (Y E 17 we have a > 0 and consequently for any a E R+ , too; hence e > ec if E $ 1 . T h u s (6-6) is proved and we have also seen that Lemma 6.15 can be written i n the form
En:
(6-7) times isometric from This implies that the mapping x -+ x o p l m . D is L 2 ( X ) (described in Theorem 6.13) into L2(m)and any character of G goes into Cha (for the restricted characters themselves belong t o Chs). If x is an irreducible and so it must equal A(w) with character of G then its image has norm some regular weight w . We also want to go in the reverse direction, i.e., we should divide the elements of Cha by D (and then extend the result to G ) . Let w be a regular weight and u = A ( w ) / D on the set H where D does not vanish. H equals the set of regular elements in m by (6-6). Since A ( w ) and D are alternating, u is invariant under the Weyl group. Thus we can extend u t o a central function on the regular part of G . If h E H then the determinant considered in the proof of Lemma 6.15 does not
133
Compact Groups
vanish (and so the mapping ( p ( g ) ,h ) + ghg-’ is open on (G/m)x H ), thus the above extension will be continuous on its open domain. The singular part of G is invariant under automorphisms and negligible (e.g. by (6-7) but it also follows from its being of lower dimension). T h u s we identify u with a function on X = G / p which is defined and continuous on the complement of a negligible compact. Let x be an irreducible character of G , x . D = A(v) with some regular weight v and consider the integral xz = xii . It can be calculated by (6-7) (even though we do not know its existence in advance): we get (A(v), A(u1)) / I N ’ [ , i.e., f l if A ( v ) = f A ( w ) and 0 otherwise. The same reasoning yields that u E L 2 ( X ) and llull = 1 . Hence the second case can not hold for all x by Theorem 6.13, and therefore one of f A ( w ) equals x D with some x . Since Cha is generated by these, +-. $ D is injective the proof is completed by observing that the the mapping on the set of continuous functions by the denseness of II (since 337 = exp m , this follows from the denseness of regular elements in m).
,s
s,
$J
We mention there is an algebraic proof of the fact that dividing alternating virtual characters by D we get elements of Chs. Such a proof is found in [Ada], and then the integral formula for continuous functions is used there. Theorem 6.22 (Theorem of the Highest Weight). Let T be an irreducible representation of the totally simple G and fix 17 as above. Use the partial ordering on MC as defined in the previous proof before (6-7). Then T h a s exactly one weight w with the property that d w # du for each weight u of T . If u is any other weight of T then d w - du is a sum of positive roots (and so du < d w ). This ‘highest’ weight w h a s multiplicity 1 . Choose a non-zero element from its weight space and for any finite sequence a l ,... , crk from R- consider the element dT(z,,) . . . dT(z,,)< where z, E A , \ (0) and dT is extended to the complexification A of g complex linearly. Then these elements linearly generate the ground space and lie in weight spaces, so a basis consisting of weight vectors can be chosen from them.
<
The character of T equals the continuous extension of A(w’) (this is the
D
famous formula of If. Weyl), so it can be recovered from the highest weight. The possible infinitesimal highest weights are exactly the functionals cp E M’ satisfying cp(Ha) E N for (Y E 17 , and so 6 can be parametrized by N” where n is the rank of G .
PROOF:Choose a w from the weights of T satisfying the required maximal property (this is possible because T has finitely many weights and o u r < relation is transitive and irreflexive). Fix a non-zero weight vector for w and denote by X, the linear span of the elements <(crl,. . . , c r k ) = d T ( z , , ) . . .dT(z,,)( where k 5 p (including k = 0 ) and each crj is a negative root. If 7 is a weight vector of T with weight u and z E A , (where a E Ro is arbitrary) then the fact dT is a morphism of Lie algebras implies
<
(6-8)
dT(ii)dT(z)q= (du
+a
,ti)
dT(z)q
Chapter 6
134
for all v E M . Hence C(a1,.. . ,ak)lies in the weight space of the infinitesimal weight dw a1 +. . . (Yk and so the first passage of the theorem will be proved if we show that the union of the 'lip's is invariant under dT (for T is irreducible). This subspace is evidently invariant under dT(z,) if a is negative; and it is invariant under d T ( M ) by (6-8). Choose now an t, E A, \ (0) for each positive root a . We show by induction on p that each 31, is invariant under all 'positive' dT(t,)'s. T h e vector dT(cr)< would be, by (6-8), a weight vector for dw Q if it did not vanish; but dw < dw a and so dT(a)<= 0 for all a E R+ . This proves the case p = 0 . Now let p > 0 and C = <(PI ,... ,Pp) with Pj E R- . Write t = t, and y = zpI ; we have dT(z)dT(y)= d T ( y ) d T ( z ) dT([z,y]) . If a P1 is not a negative root then d T ( [ t , y ] ) C ( P 2 ,.. . ,Pp) E N,-1 by the inductive hypothesis or (in the case Q P1 = 0 ) by (6-8). If it is then obviously dT([z,y])C(P2 ,..., P,) E 'lip . On the other hand, dT(z)C(P2,..., P,) E 3tp-l by the inductive hypothesis and hence d T ( y ) d T ( z ) C ( P z ,... , P,) E 31, ; the stated invariance is thus proved. The second passage is a corollary of Theorem 6.21 now: since pc < p for any E f 1 by (6-6), the weight wb must have multiplicity 1 in the virtual character X D (where x is the character of T ) and therefore X D = A ( w b ) . Let Q E 17 and s, be the corresponding Weyl reflection. T h e weight dw o s, = dw - dw(H,)a is a weight of T , hence d w ( H a ) 2 0 . This shows one direction of the third passage. Now consider a 'p E M' satisfying cp(H,) E N for cr E 17 . Then 'p p is a weight taking positive integers at these H a ' s . Thus the element u of M , satisfying (., u ) = 'p p belongs to the Weyl sector of 17 (in particular, cp+ p is a regular infinitesimal weight). Now let w be the highest weight of the irreducible representation arising from this regular weight by Theorem 6.21. Then dw+p is a weight corresponding t o an element of this Weyl sector (for we have checked any such w has the property of cp), and A ( w 6 ) = f A ( + b ) by the second passage we have already proved. But the Weyl group moves the Weyl sectors, thus dw='p.
+
+
+
+
+
+
+
+
+
Now we describe G to some extent for a general compact connected Lie group G . Fix a maximal torus '532 as always. Let r, = a n r , it is a full lattice in 1; if 3 # (0) then we choose a basis el,. . . , e n of a so that Z', become 27r(Z") in this basis. T h e irreducible representations of G are naturally identified with those irreducible representations of T" x GI which map N to {I}(with the notation of (6-5)). Furthermore, N equals the image of r in m / ( r z I'o) N T"+' where r is the rank of GI. Since the tangents of the irreducible continuous unitary representations of R are simply the multiplications by imaginary numbers, Proposition 6.16 implies that the irreducible representations of T" x GI are exactly the representations
+
T ( z l , . . . , Znyg1,.. . , gk) = 2';
" ' z ~ ". Tl(g1) 8 " ' 8 Tk(gk)
z, E T , gl,.. . , gk are the components in the simple factors of G' , E Z and each T j is an irreducible representation of the corresponding t e
where pj
tally simple group G,
. Therefore the equivalence classes can be parametrized by
135
Compact Groups
2” x N‘; namely, if IZ is a simple set of roots and wj is the highest weight of with respect to the corresponding simple set 17 n 0, then this parameter is ( P I , . . . , p n , d w l ( H I ) , d w l ( H z ) ., . . , d w k ( H , - l ) , d w k ( H , ) ) if H I , .. . , H r is a list of 17 in a suitable order. The function
is the highest weight of T in the following sense: it is a weight of multiplicity 1 and any other weight is of the form w&1 . . . c i s with negative roots c r l , . . . ,a,(of A with respect to II ). Consider the basis e l , . . . , e n , i H 1 , . . . , i H , of m , then the coefficients of -i . d w yield the parameter of the class of T described above. Now let No be a generating subset of the finite Abelian group N and choose representatives E l , . . . , E m for its elements in f (we know that N = f / ( f z I‘o) through the exponential mapping of the group T” x G’). Then we assert that the representation ?’ above is a representation of G if and only if
+
for each E, , where w is the highest weight of T . The condition is obviously necessary. If it holds then we have w ( N ) = { 1) , and we know that ir = 1 even on the whole center, thus the relation between the highest weight and other weights implies that, indeed, T ( N ) = { I } . We can see that for a considerably good description of G we need nothing else than the coefficients of E l , . . . ,Em in the basis e l , . . . , e n ,i H 1 , . . . ,i H , . We remark that one defines a concept of maximal weights independent of simple sets of roots as follows. For arbitrary infinitesimal weights w1, w2 write w1 5 w2 if w1 is a convex combination of elements in the W-orbit of w2. This is a partial ordering satisfying ~1
5 ~2
A ~
2 ~1 5
U
3S€W
~1
=W ~ O S
and if w l , w2 are ‘non-negative weights’ with respect to a certain simple set of roots (i.e., exactly the weights described in the third passage of Theorem 6 . 2 2 ) then w1 5 w2 is equivalent to the condition W ~ ( Z I )5 W ~ ( V ) for any ZI from the distinguished Weyl sector. The simple proof of these facts can be found in Chapter 6 of [Ada] (for a general G ) . Since any weight can be “rotated” into the closure of the positive sector by an element of W , one easily checks now that the maximal weights (with respect t o this 5 ) of an irreducible representation form the W-orbit of the highest weight and yield all possible highest weights with respect to the different choices of )? . The other infinitesimal weights are convex combinations of these maximal ones (in particular, their representing elements in i . m lie at a smaller distance from 0 than those of the maximal ones).
Proposition 6.23. Let w be the highest weight of an irreducible representation of G (with respect to some IT) and e be the corresponding functional on
136
Chapter 6
M (i.e., p(H,) = 1 for cy E Z7 ). Extend dw to M complex linearly. Then the dimension of the represen tation equals
PROOF:Both the dimension and the expression above remain the same when switching to G' , and if a belongs to some simple ideal of A then a t H , the values of dw and p equal the values of the corresponding objects of this ideal, so we can assume (by Proposition 6.16) that G is totally simple. Let v be the element satisfying dw = (.,v) on M and denote the vector representing e this way by p , too, then Q = f CaER+ h , with the notations of F.15.If x is the character of the representation then, obviously, its dimension equals x ( 1). We want t o invoke Weyl's formula from Theorem 6.22but D( 1) = 0 , so we have to approximate 1 . For any z E i . M , we have x(1) = limt,o X(explz) ( 2 varies in R ) . We see from (6-6)that, denoting by m the number of positive roots, limt,o t-mD(exp t z ) = a(.) . Denote this product by p ( z ) , then Weyl's formula yields
naER+
x( l)p(z) = t-0 lim t-"'A(w6)(exp
12) .
+
By definition, A(wc)(exp t z ) = COGW det @ . exp(v Q , 2@z) . T h e limit above must exist, and this is an analytic function, so this limit equals the rn-th Taylor coefficient and we have
(6-9)
d e t @ .(v
m!.x(l)p(z) =
+ e , @ z ) ~.
OE W
This is true for any irreducible representation, in particular for the trivial (one dimensional) one, for which v = 0 . Thus the right hand side of (6-9)equals rn! . p ( z ) if = 0 . Writing z = i ( v e ) into this equality we obtain
+
r n ! . p ( i ( v + e ) )=
d e t @ . ( p ,i @ ( v + e ) ) " @€ W
and the right hand side of this equation equals that of (6-9)with 3: = i~ because W is a group of orthogonal operators with respect to the Killing form. Thus x( l)p(p) = p(v e) (for the function p is also defined on M and rn-homogeneous there). This amounts to the desired formula because the denominators in that do not vanish (in fact, (el H,) is the height of H , in the dual system).
+
EXERCISES 6.1. Assume T is an irreducible linear representation (by continuous operators) of the compact group G over the topological vector space X and a non-zero
137
Compact Groups
continuous linear functional u can be found such that the corresponding matrix elements u ( T ( . ) z ) are continuous for any 3: E X . Show that X is finite dimensional. (Hint: consider the sesquilinear form ( 2 ,y) = u(T(s)z)u(T(s)y)ds and use Theorem 3.8 and Theorem 6.4 .)
sc
6.2. Let !lJ bela maximal torus in the connected compact Lie group G . Show that if z , y E 9.X are such that y = gzg-' with some g E G then there exists h satisfying h9.Xh-l = 9.X and h3:h-' = y . Show that the factor topology on the space obtain from 9.X by factorizing with the Weyl group coincides with the topology of X considered in Theorem 6.13. (Hint for the first part: consider the unit component of the centralizer of y .) 6.3. Consider the group G = S O ( 3 , R ) of rotations with a common fix point in ordinary space (it is the "smallest" non-commutative connected compact group in a sense). Check the following facts. g = fio(3,R) equals the set of (real) scalar multiples of reduced rotations by angle 7r/2 around lines going through 0, i.e., these operators vanish on the corresponding line and rotate on the orthogonally complementing plane; let the mapping @ assign to each such operator its norm times the unit vector around which the corresponding rotation is positive, then Q, is an isomorphism onto ( R 3 , x ) (where x denotes vector multiplication) and Ad goes to the identical mapping, i.e., Ad(g)@-'3: = @-'gx for all g E G and 3: E R3 . On this realization of g the exponential mapping is the following: for
#
0 expz On the open yields almost reflections in 3:
density c .
(
X
is the rotation around - by angle 11x11 in the positive direction.
11~11
ball with radius R this exp is injective and regular and its image the whole G (the complement is the two dimensional submanifold of lines). A Haar measure in the corresponding logarithmic chart has where the constant c is the ji*(l) measure of the unit
2sin("x"'2))2 11X11
cube if p is the Haar measure in question (cf. (F-9) ). The space of conjugacy classes (defined in Theorem 6.13) can be identified with the interval [ O , R ] so that p ( g ) be the angle of the rotation g . 6.4. Show that SO(4, R) is not the direct product of two groups with simple algebras though its Lie algebra is isomorphic t o the direct product of two copies of sO(3, R) .
+
6.5. Let G = S ( U ( p ) x U ( q ) ) (i.e., the subgroup of S U ( p q ) leaving the subspace CP invariant). Show that the diagonal elements form a maximal torus (so the rank of G equals p q - 1 ), find r , f,-, , a and rl ; prove that if p , q > 0 then G can not be expressed as the direct product of a torus and a semi-simple compact group (in the degenerate case p = q = 1 we have G N T ).
+
6.6. Let n E N and consider the n + 1 dimensional complex linear space of homogeneous polynomials of degree n in two variables. Let T be the natural representation of G = S U ( 2 ) in this space, i.e., T ( g ) f ( X , Y ) =
138
Chapter 6
f( g;,X + g;,Y , g;,X + g;,Y ) . Considering the maximal torus consisting of the diagonal elements (then g + gll is an isomorphism from this subgroup onto T ) check that the weights of T are exactly w" w"-' , ' " , w2-" ' w - " where w is the identical weight in the isomorphism above, i.e., w ( g ) = gl1 . Show that G is nothing else than the set of the classes of these T's. 6.7. Check that the simply connected covering of G = SO(3,R) is iso-
morphic to S U ( 2 ) and G consists of the representations of odd dimension among the representations described in the previous exercise. This group acts on the unit sphere S2 = { 3: E R3 ; 11z11 = 1 } and the ordinary surface measure is a G-invariant measure; consider the regular representation U ( g ) f (z) = f(g-'z) over L'(S2). Prove that each ( E d has multiplicity 1 in U . (Hints: the rotations around the z axis form a maximal torus; determine the corresponding weights of the subrepresentation on the space of homogeneous polynomials of degree n in three real variables, and infer this subrepresentation is the direct sum of irreducible representations of dimensions 2n+ 1, 2 n - 3 , 2 n - 7 , . . . ; and observe the analogous subspace with n - 2 is contained in this one.)
139 7. Commutative Groups
As we observed in Definition 6.18 , the irreducible unitary representations of a commutative group are one dimensional (no topology is involved, and even the unitarity is just used through Lemma 6.2). So the main problem lies not in finding the set G but in disintegrating the reducible representations. This can be done with respect to a certain measurable structure on G which is rather natural as we shall see. First we establish the classical interplay between representations of groups and *-algebras. Observe that if x is a representation of a *-algebra A (i.e., morphism of *-algehras into the C'-algebra of bounded operators over some Hilbert space) then the 'singular set' of vectors a t which each .(a) vanishes is a closed invariant subspace, so by Lemma 6.2 its orthogonal complement should really be considered as ground space. T h e corresponding representations (for which the singular set equals { 0) ) are called 'nondegenerate'. Theorem 7.1. Let G be a locally compact group (not necessarily commutative) with a left Haar measure fixed, and consider A = L'(G) as a Banach *-algebra with convolution multiplication (as described in Proposition 2.2) and the involution f + f ( x - ' ) A ( z - ' ) (cf Notation 2.4). If T is a continuous unitary representation of G then f -+ T (f ) establishes a nondegenerate contractive representation of A . On the other hand, if a nondegenerate representation x of A is given then it is necessarily contractive, and if fn is an approximation of Dirac's delta in t h e sense that it satisfies (2-14) (but these functions can be chosen from A now, differently from the practice of Chapter 2), then U ( s ) := limn x ( L ( s )f n ) (limit in the so topology) defines a continuous unitary representation U . These two constructions invert each other (in particular, they are bijective). If ir is just a morphism of algebras (i.e., we know nothing of the behaviour of the involution under x ) but we a s u r n e it is contractive and the linear span of x ( A ) Z is dense in the ground space 31 then the construction of U still works and yields a continuous unitary representation and, in fact, such a x must be a nondegenerate representation of A .
PROOF:T h e completeness of the Hilbert space implies the existence of T(f) , and llT(f)ll 5 llfll follows from the well-known relation between integrals and continuous seminorms. The multiplicative property was proved in Lemma 3.3 (cf. the remarks after it), and the preservation of adjoints follows from the unitarity of T by the corresponding property of the left Haar measure. Lemma 3.5 shows that
in the so topology for any approximation fn of Dirac's delta, so we have a nondegenerate representation of A . We know (see (3-3)) that T ( L ( s ) f )= T ( s ) T ( f ), thus (7-1) also shows that the application of the second construction to this representation of A yields the group representation T as asserted.
Chapter 7
140
If T is a morphism of algebras then we have S p ( ~ ( a ) c ) S p ( a ) U (0) , while in a C*-algebra llbll = r(b*b)'12 holds ( r ( . ) denotes the spectral radius), so if T is a representation of a *-algebra then lla(u)l[5 r(a*a)'12 . If this *-algebra is endowed with a Banach algebra norm with respect to which t h e adjoint is normpreserving then r(u*a)'I2 I l ~ * c a l l ' / ~ = llall . Since A satisfies these conditions, its each representation is contractive. Fix an approximation fn of Dirac's delta, and observe t h a t f n * u + u a n d u * fn -, u in A for any u E A (the argument in t h e beginning of the proof of Theorem 2.1 1 yields uniform convergence and vanishing outside some compact for u E C,(G) , and then the Hausdorff-Young Inequality enables us to pass t o t h e general u in the same way as it is done in that proof). T h e operators a ( L ( s ) f n ) are uniformly bounded for a contractive x , so if a( L(s)f,,)i is Cauchy for a dense set of I ' S then a bounded operator U ( s ) exists such t h a t T ( L(s)f,)a: + U ( S ) I for all I (cf. Proposition C.17). We have L ( s ) ( f * g ) = L(s)f * g for any f , g E A by (2-3). hence limn x( L ( s ) f , , ) x( g )= a ( L ( s ) g ) in norm (for a is contractive). T h e assumption on the denseness of the linear span of a(A)IH is automatically valid if x is a nondegenerate representation (for the orthogonal complement of * ( A ) % is just the singular set), so U ( s ) exists indeed and
<
<
for all g E A and s E G . We know that L is a continuous representation of G over A , therefore U ( . ) p ( g ) I is a continuous function for any g a n d I . But t h e U ( s ) ' s are uniformly bounded, so ( s , ~ )-+ U ( S ) I is continuous with respect t o the product topology (cf. Proposition C.17). T h e fact U is a group representation follows from (7-2) because each U ( s ) is bounded. Multiply formula (7-2) by a n integrable function f ( s ) and integrate. Since x ( .)I is a continuous linear operator from A to 'H, we obtain U ( f ) r ( g ) z= ~ ( * fg ) x = r ( f ) r ( g ) z (the fact f * g = J f ( s ) L ( s ) g d s in A follows most easily by testing with p E Cc(G) and using Definition 2.1 ). This means U ( f ) = ~ ( f )because the linear span of a(A)IH is dense. T h u s we can get a back from U . We see from (7-2) t h a t U is independent of t h e choice of the approximation of Dirac's delta. We may obviously choose f n such t h a t f n 2 0 a n d Jf,, = 1 , and in t,his case IIL(s)fnlI = 1 and IlU(s)ll _< 1 for all s E G . But U is a group representation, so it must be unitary. Hence a is a nondegenerate representation for it is derived from the continuous unitary representation U .
If V : H ' I+ K is a unitary operator then V ~ f ( s ) T ( s ) V - ' ~ = ds J - f ( s ) V T ( s ) V - ' z d s for f E A and I E K , i.e., equivalence of two continuous unitary representations of G is the same as unitary equivalence of the corresponding representations of A . Observe that T ( A ) and the linear span of T ( G ) lie in t h e strong closure of the other one (any T ( f )can be considered as a Riemann integral with respect to the measure f p ) , and hence the generated von Neumann algebras coincide. T h i s leads to the coincidence of irreducible and factor representations a n d of decompositions and disintegrations, too.
Commutative Groups
14 1
We saw in the proof above t h a t ~ ~ a (5a r) ( ~a ~ * ~ ) ' for / ~ any representation a of an arbitrary *-algebra A . Hence if the spectral radii in A are finite (in particular, if A can be endowed with a Banach algebra norm) then p ( a ) := supn ~ ~ a ( a ) ~ ~ is finite. Using direct sums one easily constructs a representation A such t h a t p ( a ) = ~ ~ a ( ufor) all ~ ~ a E A . T h e closure of the corresponding a ( A ) provides a realization of the completion of t h e normed *-algebra ( A / J , p ) where J = { a E A ; p ( a ) = 0 } , so this completion is a C'-algebra, called the enveloping C'algebra of A . I n the case of A = L'(G) we have p(f) 5 l l f l l and J = (0) (consider the left regular representation and observe L(f)gn + f in L ' ( G ) if gn is an approximation of Dirac's delta, hence if L(f)= 0 as a n operator on L 2 ( C ) then f = 0 as a n element of L ' ( G ) ) . T h e representation theory of a *-algebra is equivalent to that of its enveloping C'-algebra because the representations of the latter are contractive. T h e most elementary properties of C'-algebras and the Krein-Milman Theorem imply t h a t llall 2 = sup, ~ ( a ' a )where p varies in the set of pure states (i.e,, the extremal points of the set of 'states', which are the positive linear funct,ionals of norm 1 on the C'-algebra). But pure states generatmeirreducible representations by t h e GNS construction, so we get an equivalent definition of p by taking the supremum over just the irreducible representations. As we said in Chapter 6 , generally we d o not discuss the representation theory of C*-algebras because we can rely on [Dix2]. In this spirit we should have omitted the foregoing material of this chapter (except for the part proving t h a t certain algebra morphisms arc automatically representations). But it seems the key position of Theorem 7.1 i n representation theory justifies retelling.
In the sequel until after Theorem 7.6 below G can only denote commutative locally compact groups. Then the operation of G is written additively and its unit element is denoted by 0 . We fix a IIaar measure on G and d o not indicate it in the integrals. T h e irreducible continuous unitary representations of G are one dimensional, so they are nothing else than the continuous morphisms from the group G into the group T and the different ones are also inequivalent. Theorem 7.1 now shows t h a t G is naturally identified with the set of non-zero one-dimensional representations of t h e commutative *-algebra A = L ' ( G ) and this set can also be characterized as the contractive non-zero multiplicative linear functionals on A . Moreover, any multiplicative linear functional on A is automatically contractive because A is a Banach algebra (this is a standard result but we offer to those who are not familiar with it t o prove it as Exercise 7 . 2 ) . Now the restriction of t h e w* topology of A* makes G a locally compact space and adjoining the functional 0 we obtain a w* compact set. We note this topology on G coincides with the Jacobson (some say 'hull-kernel') topology. In fact, even in the non-commutative case the natural mapping from the pure states onto 6' is continuous and open from w* into Jacobson, cf. Chapter 3 of [Dix2] ; and the w* topology of the enveloping C*-algebra on bounded sets of fiinctionals coincides with the coarser w* topology of A . Since different pure states may generate equivalent representations in the non-commutative case, this factor topology can be rather pathological in general. We shall not need this
Chapter 7
142
connection with the Jacobson topology and mentioned it just as a curiosity. On the other hand, the following description of G will be important.
Proposition 7.2. The restriction of the w* topology ofA' to G equals the restriction of the usual topology of C(G) (i.e., uniform convergence on compacts) where in the latter case the elements of G are considered as mappings from G into T c C . This topology makes G a topological group with the natural operation (C + .I(.) = I ( s ) v ( s ) .
PROOF:The second topology is finer because the elements of G are functions L'(G) one can find a sequence of compacts of absolute value 1 and for any f such that f . l ~ , f in L ' ( G ) . We have v(s) (v,f) = (v,L ( s ) f ) for any v E d and f E A for v ( s + t ) = v ( s ) v ( t ). Now fix ( E G , s E G and a function g E L'(G) satisfying ((',g) = 1 (this is possible because (' # 0 ). Then if v is close to (' a t L ( s ) g and t is close to s then the continuity of the regular representation L and the uniform boundedness of the v's imply that (v,L ( t ) g ) is close to ((', L ( s ) g ) ; now if v is close to (' at g , too, then we can see that v ( t ) is close t o ('(s) , i.e., -+
(7-3)
s,('
-+
('(s)
is a continuous function
cn
if we consider the w* topology on G . Hence if a net tends to (' in this topology then In -+ ( locally uniformly (as functions on G ) . The continuity of the subtraction i n G with respect to the topology of locally uniform convergence is obvious. We note that in the non-commutative case the w* topology on the states equals the topology of locally uniform convergence on the corresponding 'positive definite' functions (see Subsection 13.5 of [DixS]), so these topologies all the more coincide on the pure states. We can see from Proposition 7.2 that G is again a commutative locally compact group, so one can take its spectrum H and (7-3) shows that
(7-4)
J : G HH ,
.IS(('):= ( ' ( s )
defines a morphism from G into H (considered as topological groups). This J is, in fact, an isomorphism of topological groups (Pontryagin's famous Duality Theorem, for a proof see, e.g., Subsection 1.7 of [Rudl]). For a connected commutative Lie group G the one dimensional representations can be described through their tangents wich are certain functionals on the (commutative) Lie algebra, so in this case one easily finds G (see Exercise 7.3) and the Duality Theorem obviously holds. Observe that the classical Fourier transforms equal the Gelfand maps if T and R are identified with 2 and R in suitable ways. More precisely, if f E L2(T) then it is traditional (and reasonable enough) to call its coefficients with respect to the basis of the irreducible characters cn(t) = t" the 'Fourier series' of f
Commutative Groups
143
and, of course, to compute f^(C,,) we must take the scalar product (f,Cn) in the Hilbert space and this is exactly the value of the Gelfand map at = C+, if the identification of irreducible representations of G and L'(G) is made via Theorem 7.1. Sticking t o this identification we define the 'Fourier transform' in the general case as F f ( 6 ) = Sf(s)C(s)ds and observe it is the composition of the Gelfand map and the action of the automorphism C -+ -C of G . Of course, the Fourier transform (as well as the identification in Theorem 7.1) depends on the normalization of the Raar measure. The question is intimately connected with the 'inverse Fourier transform' (in which we have a positive sign instead of the negative one here, cf. Theorem 7.6 below). Proposition 7.3. The norm p of the enve1opingC'-algebra of A = L'(G) equals the spectral radius in tlie algebra A and the C- algebr a itself can be identified with C o ( G )so t h a t the Gelfand map be the canonical imbedding of A into it. PROOF: For any *-algebra we have p ( u ) 5 r ( n * a ) ' I 2 , so in our case (denoting the Gelfand map by and using that each multiplicative linear functional is also a representation of the *-algebra A ) we have p ( a ) 5 (maxc lii(C)12)''2 = lliill . On the other hand, C,(G) is a C-algebra and so it h a s a n isometric representation, hence p ( u ) 2 11ii11 . It remains t o check that tlie image of A through the Gelfand map is dcnse in CO(G),and this follows from the Stone-Weierstrass Theorem (see C.2).
Theorem 7.4 ( Stone-Nairnark-Ambrose-Godement Theorem). Let U be a continuous unitary representation of G . Then there exists a unique spectral Radon measure E from G into the projections of the ground space (cf. Notation C.16) satisfying (7-5)
JsdE
for all s E G
and each spectral Radon measure defines a continuous unitary representation by this formula. The von Neumann algebra generated by t h e range of E equals U(C)"" and we have U ( f ) = S f d E for all f E L ' ( G ) . Note that if G = R then this theorem reduces to Stone's theorem: the continuous unitary representations of R are exactly those of the form U ( s ) = e i 3 A where A is an arbitrary (in general unbounded) self-adjoint operator and the exponential mapping is interpreted through spectral theory. In this case i A equals the infinitesimal generator of U as defined in Chapter 1 .
PROOF:We have a nondegenerate representation T of Co(G) corresponding to CJ by Theorem 7.1 and Proposition 7.3. By the spectral theorem we have a unique spectral Radon measure E such that ~ ( f = ) f dE for all f E Co(G) , and E generates the same von Neumann algebra as T does.
s
144
Chapter 7
T h e formula C(s) (C,g) = (C,L(s)g) can be reformulated as J s . g = L ( s ) g and therefore we obtain from (7-2) h
Since U ( L ' ( G ) ) R is dense, we can see this E satisfies the assertion. If F is another spectral Radon measure satisfying U ( s ) = J s d F for all s E G then ( U ( s ) i , y ) = J-C(s)dFz,y(<)and hence for any g E L'(G) we have by Fubini's theorem
i.e., a(g) = S g d F . This extends to Co(G) because the image of L'(G) is dense there, thus F = E . Now let E be an arbitrary spectral Radon measure on G and define a n operator valued function U by (7-5). T h e properties of spectral integrals show at once U is a unitary representation, for J s . J t = J ( s t ) and Js(C) E T . Continuity will follow if we can show U ( . ) z is continuous, and this is equivalent to saying t h a t s + J s is continuous from G into L 2 ( E z ) . But this mapping is continuous from G into C ( G ) by (7-3) and lJsl 5 1 , so the theorem is completely proved.
+
There is another version of this theorem using the following variant of t h e spectral theorem: if a is a cyclic representation of the commutative C*-algebra C o ( X ) (where X is a locally compact space) then x is unitarily equivalent to a multiplier representation (i.e., we can find a finite non-negative Radon measure p on X and a unitary operator V from the original ground space onto L 2 ( X ,p ) such t h a t Va(f)V-'g = f g ), while any nondegenerate representation is the direct s u m of cyclic ones. T h e spectral measure corresponding to a multiplier representation is described by E ( A ) f = l A . f . Applying this to X = 6' we obtain VU(s)V-'f = J s . f for f E L 2 ( G , p ) . It is known t h a t two multiplier representations are equivalent if and only if each of t h e corresponding two measures is absolutely continuous with respect t o the other one. If the Hilbert space is separable then one can find a sequence p1, p 2 , . . . , pu of mutually singular finite non-negative Radon measures (some of them may equal 0 ) such t h a t T N $,(n, @ 1") where a, is the multiplier representation with the measure p, , and the equivalence class of each a, is determined by the equivalence class of T . If the Hilbert space is non-separable then also holds a similar resolution of but we must allow cardinals R > w and the an's are perhaps not quite as nice (cf. Exercise 7.4). T h i s 'multiplicity theory' extends in a sense to tame groups, see Subsections 5.4 and 8.6 of [DixZ] (also cf. [Arv]). Similarly to compact groups, application of t h e decomposition theorem to the regular representations yields interesting results. Consider the left regular representation L on L 2 ( G ) (in our commutative case the right regular representation
145
Commutative Groups has a simple relation with this one: 'p E Cc(G) one easily checks
R ( t ) = L ( - t ) = L(t)-' ). Testing with
in t h e sense of (2-3) (the same holds even for non-commutative locally compact groups; the fact f * g E L 2 ( G ) was proved in Theorem 2 . 1 2 ) . Hence if E is the spectral resolution of L by Theorem 7 . 4 then
(7-7) Now let gn be an approximation of Dirac's delta chosen from L2(G) and set p,, = E,,, . We shall show that the limit lim,, ( p n , u ) exists for u E Cc(G) a n d defines a Naar measure on 6'. We have by (7-7) ( f * g,, , gn) = f dpn where on the left we consider the scalar product in L 2 ( C ) .Consider this for f E Cc(G), then the net f * gn tends uniformly to f which is continuous a t 0 , hence (using the fact + 1 ) we obtain
s
sz
Let S be the square of the *-subalgebra Cc(G) of L ' ( G ) (i.e., the linear space spanned by a * b , a , b E C,(G) ). Denote by B the Gelfand image of S . We know S is dense in any LP(G) with finite p because aj * b + b uniformly and inside a compact with suitable a j . T h u s B is a dense *-subalgebra of Co(C). T h e polarization identity of sesquilinear mappings shows t h a t B is spanned by B+ := { Id? ; E CC(G)1 . If )t E L z ( C ) then the net 111*g,, is bounded in L 2 ( G ) ,thus for cp E B+ we obtain that cpp, is a bounded net in Co(G)*. Then the same holds for cp E B . This net converges a t each function from B by B 2 c B and by (7-8), hence a t any function from Co(G) (cf. Proposition C . 1 7 ) . Denote by pip the limit (it is a finite Radon measure). If u E Cc(G) is arbitrary then choose a p E B close to 1 on s u p p u and set v(C) := u(C)/cp(C) for p(C) # 0 and v(C) = 0 otherwise. Then v E Cc(G), thus we have the limit
*
Since this is true for any u , the limit of p,, defines a positive linear functional on Cc(G) and thereby a non-negative Radon measure p . For all v E Cc(G) and (D E B we have J v v d p = IimJvpdp,,
= Jvdp,
,
Chapter 7
146
i.e., 'pp = p v . T h u s B c L 1 ( G , p ) . We assert t h a t G d p = $(O)
(7-9)
for $ E
s
= 11, E B . If h E C,(G) then (7-8) is applicable to f = h * $ , while 'ppn p v = cpp a t because it belongs to Co(G).T h u s h ' p d p = h*$ ( 0 ) . If IS is a compact in G then (7-3) implies t h a t for any E > 0 we find a neighborhood U of 0 in G such that IC(s) - 11 5 E for C E It' a n d s E U . Hence if h 2 0 , h = 1 and supp h c U then Ih(C) - 11 5 E on IS and lhl 5 1 everywhere. So if hj is a suitable approximation of Dirac's delta then (by the integrability of 'p)
-
Write
'p
s,
J
cpdp = lim I
J hj
. ' p d p = l i m h j *11,(0) I
and (7-9) follows. For any f E L1(G) and v E G we obviously have = j ( v + ( ) . Since v ( 0 ) = 1 , we can see from (7-9) that p is translation invariant when considered on B (observe t h a t is t h e translation of 141'). Now if u E C,.(G) then express it in the form u = vcp as above and choose a sequence ' p k E B tending to v uniformly. Then R(v)pk + R(v)v uniformly and R(v)'p E B is p-integrable, hence R ( v ) u d p = liin R ( v ) [ V k ( P ] d p= lip cpkcpdp = k Judp
G(C)
Iv31z
J
J
J
(for v k ' p E B ), so we have proved the invariance of p . We must also check p # 0 but this follows from (7-9) applied to 11, = f * f * with f E Cc(G)\ (0) (then 11,(0) = ( f , f ) ). This also shows that p is independent of the choice of the net y, (for a Haar measure is determined by its non-zero value at one function), and depends only on the choice of the regular representation (i.e., on the choice of t h e Haar measure on G ) . Convolution by a function commutes with translations, hence S is invariant under translations. We obviously have L ( s ) f = J s . f , so (7-9) implies t h a t h
for 11, E S . In other words, on S the Fourier transform is inverted by the Gelfand m a p of G if we use the Haar measure p (constructed above) on G . It is customary to call this Gelfand m a p 'inverse Fourier transform' on the whole L 1 ( G , p ) . To go further we need the following. Lemma 7.5. Let v be an arbitrary Radon measure on G . If u E L' (GIv ) and f E L'(G) then
147
Commutative Groups
where the bounded continuous functions J s are considered as bounded linear functionals on L1(G, v) , and f is the Gelfand map o f f .
u(()
PROOF: This amounts to Fubini's theorem applied t o the function f(s) . . ((s) . Now the last factor is continuous on the product space and bounded,
so the integrability of u and f is enough to justify this application.
We shall use this lemma with v = p in the following two-cases: u(C) = F g ( - ( ) = g ( C ) with g E S , or f E S and then write 'p = F f = . We calculate by the inversion formula that in the first case ( J s , ~ = ) , and in the second case = f (using that S is invariant under conjugation). When this f varies in S then 'p sweeps B , so we obtain: ~-
s(s)
i
The first line shows that the Fourier transform is isometric on S between the Hilbert spaces L2(G) and L 2 ( G , p ) . We saw S is dense i n L 2 ( G ) . As regards B approximate u/cp uniformly as usual, then limk'ptcp = u in L 2 ( G , p ) because 'p belongs to this Hilbert space. Thus a unitary operator, called Plancherel transform, arises between L 2 ( G ) and L 2 ( C , p ) which continues 31k9. Denote this Plancherel transform by V . The formulas above imply tthat, Je(3f - V f ) =0 for f E L ' ( G ) fl L2 ( G ) and 1/, = Vg E V ( S ) = B , while J,(ii - V - ' u ) . $ = 0 for u E L ' ( 6 ) fl L 2 ( G ) and $ = V - ' p E V - ' ( B ) = S . Now the methods by which we proved the denseness of S and LI i n the Ililbert spaces yield more: the elements of C,(G) are approximated froni S uniformly and vanishing outside a compact, hence in any L P space at the saine time, while the elements of C,(G) are approximated from B by ' p k p where the Pk'S converge uniformly, so this approximation is LP whenever cp belongs to Ihe corresponding L P space. Since 21 and Ff are bounded, we see the formulas extend to compactly supported continuous $'s; therefore u = V - ' u and 3f = V f . From (7-7) we now obtain V ( J f d l ? ) g = 3 1 . Vg for any g E
.q
L1(G) fl l I 2 ( G ) and f E L'(G) , so the denseness of V (L' fl L 2 ) implies V ( J ' p d E ) V-' = @ where on the right we mean the operator of multiplying by this function. Collect our results into a theorem.
Theorem 7.6. Our choice of the Zlaar measure on G distinguishes a Haar measure p on G such t h a t the restriction of the Fourier transform to L ' ( C ) f l L 2 ( C ) is isometric into L z ( G , p ) . This isometry extends to a unitary operator V from L2(G)onto L 2 ( G , p ) ,called the Plancherel transform. For u E L 1 ( G , p ) f l L 2 ( C , p ) we have V-'u (s) = sd u ( ( ) .((s) dp(C) . If P is the "multiplier" spectral measure ) VL(s)V-' = J x d P and VR(s)V-' = on L 2 ( C , p ) (i.e., P ( A ) u = 1 ~ . u then
sJ s d P .
148
Chapter 7
We close this chapter by discussing the basic idea of the 'normal subgroup analysis' of G . W . Mackey, especially when the normal subgroup is commutative. This will give us motivation for the Imprimitivity Theorem we treat in the next chapter. We include this topic in the present chapter because it is essentially a n application of t,he SNAG Theorem (in the commutative case; in t h e general case the unicity theorem of t h e so called 'central disintegrations' plays the same role). We mention that the idea (in a n important special case) was already used in t h e classical paper [Wig] and even the Stone-von Neumann Theorem (dating back to 1930) is a related result (cf. [Mcl]). In the remainder of this chapter G is a locally compact group and N is a closed normal subgroup of it. We pose the following problem: if G is M2 and N is tame then find a manageable criterion ensuring t h e tameness of G , and if it holds then try to obtain information about G . We emphasize the case when N is commutative and G is a semi-direct product of N by some locally compact group, in this case quite definitive results will be obtained (in the next chapter), while these conditions are wide enough to encompass a lot of important applications. For a while we shall not suppose G is M2 in order to make it clearer where this condition becomes really necessary. By a theorem of S. Sakai (see [Saka]) for G t o be tame it is enough t h a t its factor representations be type I , i.e., one should check whether the following holds: for each continuous unitary factor representation U of G the generated factor U(G)" is discrete (i.e., isomorphic to B ( K ) with some Hilbert space K). If G is A42 then we could restrict our attention to separable Hilbert spaces because any representation is the direct sum of cyclic ones, and then the classical theory described in [Dixl] is enough to prove the above statement (such a proof can be found, e.g., in [DixP]). We mention that in [Sak2] even more is proved: wild groups must have factor representations of type IIZ . T h e equivalence of several descriptions of tameness is a major achievement in mathematics, and the classic [Dix2] provides a nice exposition of this topic (see especially Chapter 9 of [DixZ]). Since N is a normal subgroup, G acts on it by ( s , t ) + sts-' . T h i s leads to actions on some structures whose elements are functions on N . First of all, observe that this action on the locally convex space C ( N ) is continuous. Consequently, the actions on the topological spaces of positive definite functions and pure positive definite functions are also continuous. A very important case is the action of G on the continuous unitary representations of N , and we denote it simply by writing the element t o the left, i.e., if V is a representation and s E G then SV is the representation s V ( t ) = V ( s - ' t s ) . T h u s t h e ground space and the generated von Neumann algebra of sV coincide with those of V , and if V , V1 are two representations then an operator intertwines them if and only if it intertwines SV and sV1 . Now two irreducible continuous unitary representations are equivalent if and only if they have a non-zero intertwining operator, while for factor representations this condition is the same as quasi-equivalence (see Subsection 5.3 of [DixZ] ). T h u s G acts on the spectrum and also on the quasi-spectrum of N . T h e elements of N act trivially on these objects (for they m a p a representation into an equivalent other one), so one can talk about the action of t h e group G I N .
149
Commutative Groups
T h e action on the spectrum is continuous because it is continuous on the space of pure normalized positive definite functions and the topology of ii' is obtained by open factorization. Of course, if N is commutative then N coincides with this set of complex valued functions and no factorization takes place. One may consider the natural action of G on L ' ( N ) b u t it is not necessarily isometric; correcting it by the corresponding extension of t h e modular function A of N we obtain the representation
(7-10)
s f ( t ) := f ( s - ' t s ) A ( s )
(note this A is not the modular function of G i n general). Then s V ( f ) = V ( s - ' f ) holds, i.e., the two possible definitions of defining the action of G on t h e nondegenerate representat.ions of L'(G) yield the same notion. I t is not too hard to check that. the corresponding act,ion of G on the representations of the enveloping C*-algebra of L ' ( N ) over a fixed Hilbert space is continuous (where the set of these represenhtions is endowed with the usnal weak type topology). Hence if N is A42 (when the measurable structures of G. W. Mackey are defined on the spectrum and quasi--spectrum of N , see Subsection 3.8 and Chapter 7 of [Dix2]) then these measurable st,ructiires are invariant under each s E G . If N is also tame then the measurable st,ructure is the Bore1 structure of the Jacobson topology on N , so the above also follows from the continuity of the action then. If N is commutative then (s-'C, f) = ((, sf) for C E ii' , i.e., sf = s .f and F ( s f ) = s 3f (for the inversion of the group N commutes with any automorphism). So the action of G 011 C o ( N )is compatible with the identification of this C*-algebra with the group C*-algebra of N (see Proposition 7 . 3 ) . Let. U be a continuous unit.ary representation of G over the Hilbert space 3-1 and 2l := U(G)" be the corresponding von Neurnann algebra. Since the restriction of U t o N will play a key role i n the analysis, we give it the separate notation T (so T ( s )= U ( s ) if s E N ). Let 332 = T ( N ) " be the von Neumann algebra generated by T and denote by 3 the center of M (if N is commutative then 3 = 332 ). If s E G then A U ( s ) A U ( s ) - ' is an automorphism of B(X) leaving T ( N ) invariant, hence it maps 3 onto itself. If P E 3 is a projection and Q = U ( s ) P U ( s ) - ' then U ( s ) P is a partial isometry est.ablishing the equivalence of the subrepresentations ( s T ) p anti TQ. If the ground space 3-1 is separable and N is Mz then the theorem about central disint.egratfioris (see Theorem 8.4.2 of [DixP] ) is applicable to the representa.tions of the C*-algebra of N corresponding to these subrepresentations. On the other hand, one defines a "natural" action of s on central disintegrations (we omit t h e details), and eventually one obtains t h a t if P is represented by t h e scalar field 1~ with some measurable subset H of t h e quasi-spectrum then t h e scalar field 1,N represents Q (in any central disintegration; note t h a t exactly the "almost unique" nature of these disintegrat,ions moves t h e proof). Now we show a slight variation for the case of commutative N , which works without any separabilit,y restriction (neither on 3-1 nor on N ) . h
-+
Proposition 7.7. Let N be commutative a n d E be the spectral resolution
150
Chapter 7
of T by Theorem 7 . 4 . Then (7-11) for any bounded Bore1 measurable function
'p
on N and any s E G .
Note that (7-11) implies U ( s ) E ( H ) U ( s - ' ) = E ( s H ) for any Borel set H and all s (and, in fact, is equivalent t o this formula). In the other direction, by the properties of spectral measures, (7-1 1) also holds for unbounded measurable functions.
PROOF:We have U ( s ) T ( . ) U ( s ) - ' = s-'T as representations of N , hence U ( s ) T ( f ) U ( s ) - ' = T ( s f ) for any f E L ' ( G ) . We have already checked that the Gelfand map intertwines the action defined by (7-10) and the natural action o n Co(N), so (7-11) holds for 'p = f . Since these are dense in the C*-algebra Co( N) , we obtain (7-11) for all 'p E CO(N) . Now F l ( H ) := U ( s ) E ( H ) U ( s ) - ' and f ' 2 ( 1 { ) := E ( s H ) define spectral Radon measures (for Fz use that G acts continuoiisly on N ) , and their integrals are equal on Co(fi); therefore F1 = F2 and the proposition is proved. If a projection P commutes with each U ( s ) then P E QC . If, in addition, P E then P belongs to the center of Q . Therefore if Q is a factor and N is commutative then "G acts ergodically on E", i.e., if H is a Borel set satisfying E ( s I 1 ) = E ( H ) for all s E G then either E ( H ) = 0 or E ( H ) = I . Similarly, if is a factor, N is M2 and 'H is separable then G acts ergodically on p where p is the measure on the quasi-spectrum arising in a central disintegration of T .
Definition 7.8. Let N be M 2 . We say that G acts 'separably' on the quasi-spectrum of N if a countable family of G-invariant measurable sets can be found separating the orbits (i.e., if J 1 ,52 are different orbits then a member of the family contains one of them and is disjoint from the other one). Recall that if N is also tame then the measurable space considered in this definition equals the Bore1 structure of the topological space N . Note that G may fail to act separably on fi for quite nice N and G I N (cf. Exercise 7.6). If G acts separably on the quasi-spectrum of N then we can find an orbitseparating sequence H I , H 2 , . . . of G-invariant measurable sets which contains the complements of its elements. Then if Q is a factor and N is commutative (or X is separable) then setting X = n { H k ; E ( H h ) # 0 ) (or X = n { H k ; p ( H k ) # 0 ) ) with the corresponding spectral measure E (or measure p ) we obtain that X is a measurable orbit whose complement is E-negligible (or p-negligible). The unicity part of the SNAG Theorem (or the almost unicity of central disintegrations) implies that X is already determined by the equivalence class of T , all the more by that of U . In particular, if G is MZ and acts separably on the quasi-spectrum of N then G splits according to these orbits (of course, some orbits may not correspond to any representation of G).
Commutative Groups
151
Separable action just excludes one half of the possible pathologies; the action might be bad enough inside the orbit X . If G is M2 and N is tame then we are dealing with a n action of the form ( s ,C ) + p ( s . q ( 6 ) ) where q is measurable into a suitable Polish space, the . refers to a continuous action and p is continuous (cf. Chapters 3 and 4 of [Arv]). This is something; but if N is commutative (which is in the focus of o u r attention) then we have much more even if G is not M2 : the action is continuous and X is Hausdorff (of course, the action is also continuous in general but the topology of N may be too coarse to get much profit by this). It would be enough for the path we shall proceed on if X were just T I , and this is certainly satisfied if N is ‘CCR’, i.e., for each irreducible continuous unitary representation V of N and for all f E L ’ ( N ) the operators V ( f ) are compact (see 4.4.1 of [DkP]; cf. Subsection 1.5 of [Arv]). By more effort it can be shown that any orbit must be TI whenever N is just tame. We just hint at the proof then fi is ‘‘almmt Hausdorff” as described in 4.5.6 of [Dix2] ; hence one infers that N can not contain a copy of certain very simple non-Tl spaces and this fact excludes the possibility of non-Tl orbits of continuous actions even of the group 2. Choose ( 0 E X and denote by H its stabilizer in G , i.e., I€ = { s E G ; sCo = Co} . Note N c H . Assume N is tame and H is closed (the second condition follows from the first one for that implies X is TI ). Then we can ] ) S
152
Chapter 7
group of G . Choosing a subgroup H from this orbit we have a spectral Radon measure P over 31 on GIH such that the range of P generates the yon Neumann algebra 3 and
(7-12)
U(s)( J p d F )
u(s)-’ = Jq7(s-1+fP(r)
for any bounded Bore1 measurable function cp on GIH and any s E G . If N is commutative then X is equivalently defined by the property that E ( X ) = I with the spectral resolution E of T ; and if E X is chosen so that H be its stabilizer then P can be such that
c
(7-13)
T ( t )=
( ( s - l t s ) dP([s])
for all t E N
where [s] is the class of s in GIH . P is uniquely determined by (7-13) in this case, and this construction still works with non-separable Ifilbert spaces. Notation 7.10. If an arbitrary locally compact group G its closed subgroup H , a continuous unitary representation U of G over 31 and a spectral Radon measure P on GIH (and over 3c) satisfy (7-12) then one says that a ‘system of imprimitivit,y’ is given.
It is unfortunate that this notion was christened to this long and ugly name but we must live with it for the notion has become widely known under it. A few words are in order about left vs. right cosets. For reasons unknown to the author G. W. Mackey chose to consider H\G instead of our GIH above. Though this is a symmetric matter, once a choice is settled a great number of subsequent arguments must conform with it. It seems that the majority of authors have followed Mackey’s choice (including the present author in [Mall), probably in order t o spare the pains of checking a lot of formulas anew. In this book we do otherwise, partly for the sake of variety and partly because thus we act from the left (which seems more in accordance with practices in the rest of mathematics). We call the readers’ attention to the precedent in [Kna] . In the next chapter we show that a system of imprimitivity is necessarily a n ‘induced system’ (this result is called the Imprimitivity Theorem and in the case of Mz groups it is due to G . W. Mackey). On the base of this theorem and related results we shall proceed with our program. At present we show another result in this tame group extension circle. Lemma 7.11. If U is a factor representation and GIN is finite then 3 is finite dimensional. If, in addition, 332 is a discrete von Neumann algebra then 2l is a discrete factor.
PROOF:Let S be the spectrum of the commutative C’-algebra 3 i.e., the set of its non-zero multiplicative linear functionals endowed with the w* topology. If @ is an automorphism of 3 (asa (?-algebra) then ~ ( u:= ) uo@ is a homeomorphism
Commutative Groups
153
of S onto itself and @ can be recovered from it by the Gelfand isomorphism: & = a o T . Now GIN acts on this C'-algebra and the set of common fix points equals the center of Q , so if Q is a factor then this set consists of scalar multiples of the identity. Let A be the group of homeomorphisms of S corresponding to the action ,f GIN over 3 , then A is finite and for any non-constant p E C ( S ) we have a r E A such that 'p o T # 'p . The equivalence relation implemented by A is the continuous image of S x A in S x S (where A is considered discrete), hence compact. The corresponding factorization is open because A is a group of homeomorphisms, hence the factor space is compact Tz . The continuous functions on this factor space yield A-fixed elements of C ( S ), so they must be constant. Since the factor space is compact T2, it consists of one point then. Therefore S is a single orbit and, by the finiteness of A , S is finite. If M is discrete then it contains a non-zero Abelian projection Po. Since PO is Abelian, PornPo = Po3 is finite dimensional and therefore M also contains a non-zero minimal projection P . Choose a representative s, from each class of G I N , then UjCJ(sj)T(N) = U ( G ) and therefore P U ( G ) P = UjU(sj)CJ(sj)-'PCr(sj)T(N)P . Now Qj := U ( s j ) - ' P U ( s j ) is a minimal projection of '332 because U ( s j ) implements an automorphism of '33, hence Q,MP is one or zero dimensional. The finiteness of GIN now implies that the linear span '23 of P U ( G ) P is finite dimensional. On the other hand, '23 = P A P where A is the linear span of U ( G ), A is a *-subalgebra because U ( G ) is a *-invariant group, and so A is strongly dense in Q by von Neumann's density theorem. Thus the finite dimensional 2 3 '. is strongly dense in PQP and therefore equals it. So PQP is finite dimensional and hence Q contains a non-zero minimal projection. For a factor this means discreteness.
Theorem 7.12. I f GIN is finite and N is tame or CCR then G is also tame or CCR, respectively.
PROOF:If N is tame then any factor representation of G is type Z by Lemma 7.11. Hence G is tame by the theorem of Sakai we mentioned in the beginning of the discussion of normal subgroup analysis. We emphasize that in the Mz case this theorem is quite classical. Now let CJ be an irreducible continuous unitary representation of G (assume no tameness a t the moment). Then = B('H) , so it follows from the proof of Lemma 7.11 that each minimal projection in M is finite dimensional. This implies by the finite dimensionality of 3 that M is a finite direct product of factors with the same property. Now 'CCR' is stronger than 'tame' also in the non-separable case (see Theorem 5.5.2 of [Dix2]), and discrete factors with finite dimensional minimal projections are of the form B ( K ) @ C with ' C ' acting on a finite dimensional space. Hence if N is CCR then T is a finite direct sum of irreducible representations and so T ( f ) is a compact operator for any f E L ' ( N ) . N is an open subgroup for the finite Hausdorff space GIN must be discrete, thus the restriction of a left Haar measure of G to N is a left Haar measure for N . Let g E L'(G) , then g is the finite sum of functions of the form L ( s ) f with f E L ' ( N ) and s E G . By ( 3 - 3 )
154
Chapter 7
we have CI(L(s)f)= U ( s ) T ( f ) and this is enough because the compact operators form an ideal.
EXERCISES
7.1. Show that in Theorem 7.1 the group representation can also be recovered by the formula T ( s ) = (so) limT( n A(s-')R(s-')fn)
7.2. Let A be a Banach algebra and 'p : A --* C be a morphism of algebras (so no continuity is assumed). Show that 'p is contractive. b=
(Hint: if a" .)
xrzl
Q
E A were such that
llall
< 1 and
V ( Q ) = 1 then consider
7.3. Let G be a connected commutative Lie group. Show that GN R" x T k with some n , & and then G 21 R" x Zk in the following way: ((s) = exp
(~j"=
i<jsj)
.
nj":,
s j .~
7.4. Let IH = t?(R) (so its Hilbert space dimension is the cardinal of continuum), A = Co(R), and A : A H B(X) be the representation r ( f ) g = fg . Show that the commutant of A ( A ) is Abelian (so A is 'multiplicity-free'), and A is not equivalent to a multiplier representation (not even with a non-finite Radon measure, though this statement is not more general for R is A42 ). 7.5. Let G be a commutative locally compact group and f E L'(G) be such that F f is integrable (with respect to a IIaar measure on G ) . Show that the inversion formula holds for f , i.e., f = Ff o J if L ' ( G ) is considered with the Haar measure distinguished in Theorem 7 . 6 . h
7.6. Let N = C 2 N R4 as a Lie group and consider the following action @ of R over N : @ ( t ) ( z w, ) = ( e i z z , e i a z w ) where Q is a fixed irrational number. Let G be the corresponding 5 dimensional semi-direct product (this is the simpler one of the examples of wild Lie groups from Chapter 19 of [Kir]). Show that G does not act separably over fi . (Hints: consider a Haar measure p on the torus T2 c N and check that if f E L ' ( N , p ) is invariant under the action above then f p is invariant under the translations of the group T2 ; infer that p is G-ergodic.)
155 8. Induced Representations
The concept of induced representations is a “twist of the regular representation”, and enables us to create representations from representations of a smaller group. First consider the algebraic core of the matter. Let G be a group, H be a subgroup of G and S be a linear representation of H over the complex linear space K . Then the linear space
is invariant under the left regular representation L of G (which we consider on vector valued functions now). The restriction Llx0 might be called ‘the representation induced from S’ but this ‘Ho is too big to be useful in most cases; to bring analysis into the concept we should intersect ‘Ho with a suitable “analytic” subspace of K G . Unfortunately, this is not simple. We shall consider the unitary induction of G. W. Mackey and prepare ourselves by surveying the simplest case when G is finite and K is finite dimensional. Fix a scalar product on K so that S be unitary. Then for f1, f 2 E ‘Ho the function ‘ p ( t ) = (fl(t),fi(t)) depends only on p ( t ) where p : G H G/H is the factorization, so if q is any cross section of p (i.e., a function satisfying p o q = id ) then ‘ p ( t ) = IHI . cp(q(z)) . On the other hand, we can identify ‘Ho with K G I H by taking f to f o q , for f can be recovered by f ( t ) = S [ q ( p ( t ) ) - ’ t ] - ’ f [ q ( p ( t ) ) ] and writing any function in the place of f o q here we obtain an element of ‘Ho by this formula. Thus, denoting by ‘H the Hilbert space whose linear space is ‘Ho and the scalar product is ( f l , f 2 ) = C ,~ ( t ,) we can identify ‘H with Li(G/H,p)where p is the
c,
c,
h
normalized cardinality measure p ( A ) = IAl . Since L is obviously unitary over
P/HI
the Hilbert space K G , the induced representation U = LI, will be unitary. Now let r : C / H H C be a function and denote the corresponding multiplier on ‘H by [ r ] ,i.e., [ r ] f ( g )= r ( p ( g ) ) . f(g) . Then we have U ( s ) [ r ] U ( s ) - ’= [sr] where, of course, s r ( z ) = r ( s - ’ z ) is the natural action of G on the functions on G / H . Thus U and [.] form a system of imprimitivity according to Notation 7.10. We want to extend these ideas to the case when G is a locally compact group, H is a closed subgroup of it and S is a continuous unitary representation ( K may be infinite dimensional). The first difficulty is that the function ~ ( t=) (fl(t), f 2 ( t ) ) may not be integrable with respect to a left Haar measure for too many f l , f2 E ‘Ho, so it is not a good idea in general to derive a scalar product from L2(G)@ K as we did for a finite G (but this works very well if G is compact). Since ‘p just depends on p ( t ) also in the general case, it is natural enough to try to integrate ’ p o q (which is independent of the cross section q ) with respect to a suitable measure. Now a G-invariant positive Radon measure exists on G/H if and only if the restriction of the modular function A of G to H equals the modular function 6 of II , so if this does not hold then one should modify either U or 310 somehow to get a unitary representation. We present both versions and the relation between them. This is a typical case of the situation when a definition contains a number of enabling
Chapter 8
156
lemmas whose formulation outside of the definition would be awkward. We shall “prove the definition” instead, i.e., include a ‘proof’ to check the validity of these enabling lemmas.
Definition 8.1. Let G be a locally compact group, H be a closed subgroup of G and S be a continuous unitary representation of H over the Hilbert space K . Denote by A and 6 the modular functions of G and H , respectively, and let p : G I+ GIH be the factorization. Use the letter q to denote cross sections o f p . We fix left Haar measures on G and H and an integral on G or H without indicating the measure is meant with respect to these left Haar measures. Version (a). Let 3-11 be the following “correction” of 3-10 :
(8-la) 3-11
:= { f E
K G ; f ( g h ) = 6 ( h ) ’ / 2 A ( h ) - 1 / 2 S ( h ) - ’ f ( gfor ) hE H , g EG} .
This is a linear space invariant under the left regular representation L . If f 1 , f 2 E are continuous then we have a unique Radon measure t 9 ( f l , f 2 ) on GIH such that
3-11
for any u E C J G ) (here E is the conditional expectation, see Notation F.10). The set of continuous f’s with finite t9( f , f ) is a linear space 3-11, . Set
for f l , f 2 E 3-11, , then 3-11, becomes a pre-Hilbert space. The operators L ( s ) are unitary over this pre-Hilbert space and their unitary extensions yield a continuous unitary representation U, over the completion 3-1, of 3-11, . This U , is called the representation induced from S . Version (b) , Choose a continuous positive e-function e (see (F-15) in Notation F.lO) and denote by m the corresponding quasi-invariant measure on GIH (see Notation F . 9 and (F-17)). Let xb be the set of those continuous functions f from 3-10 for which Ilf(q(.))ll E L 2 ( G / H , m ) and for f 1 , f ~ E z b set
(8-2b) Then ‘?& becomes a pre-Hilbert space and
defines a unitary representation over 3-1b . Its unitary extension to the completion 7f of3-1b is a continuous unitary representation, called the representation induced from S (in this version). The mapping f -+ f is unitary from ? t b onto 3-11, and intertwines the representations U and U, .
Induced Represent a tions
157
PROOF:T h e constructions of this definition use Notations F.9 and F.10, so we suggest reading these carefully before the present proof. A key step is the existence of continuous positive @functions. If u E C,(G) and u 2 0 then e(s) = u(sh)A(h)6(h)-' dh yields a non-negative continuous pfunction and we can then obtain a positive one by an argument using the paracompactness of G / H (for a more detailed explanation see Proposition 1 of [Mall). We also need some easily checkable facts about the conditional expectation E . We just consider it on Cc(C) , then sEu = E L ( s ) u and E( ( r o p ) . u ) = r . E u hold for s E G and r E C ( G / H ) . This E maps the set of non-negative compactly supported continuous functions on G onto the same on G / H (see Theorem 15.21 in the first volume of [H-R]). We mention that (F-16) is an immediate consequence of Fubini's theorem and the properties of the Haar measures, and on the base of (F-16) and the last mentioned property of E one easily checks that any pfunction 'p gives rise to a Radon measure on G / H by (F-17) (the main step: if Eu = 0 then choose a E C,(G) such that Ev = 1 on the compact set ~ ( s u p p u ) and infer u . 'p = 0 ). Now if 'p is non-negative then d ( G / H ) = SUPo<Eu
sH
sG
sG
i.e., dl is finite if and only if r is &integrable. Applying this to 'p = e and r ( p ( t ) ) = llf(t)112 with a continuous f E 3i0 we can see that the mapping f -+ e'l2f maps Xb onto (the fact that this mapping is a linear bijection from the continuous functions from '?lo to those from ' H I follows from the assumption that e is a continuous positive &function). Furthermore, if f E ?tb then (f,f) as defined by (8-2b) equals ( ~ ' / ~~f' ,/ ' f ) as defined by (8-2a) and we evidently have the intertwining property between U and U , . It is easy to see that wb is a linear space, hence so is 3-11, . It is sesquilinear to assign d(f1, f 2 ) t o the continuous functions from R 1, therefore we have a polarization identity and the complex Radon measures in (8-2a) are bounded. In we also have a polarization identity, so we have the asserted equivalence of the two versions. It remains to check that U is a continuous unitary representation. Since Q is a pfunction, the function p ( s t ) e ( t ) - ' just depends on s and p ( t ) , so we have a continuous function F : G x G / H H R such that F ( s , p ( t ) )= e ( s t ) e ( t ) - ' (also use that p is an open mapping). By the properties of E for any u E Cc(G) and s E G we thus have
zl,
Chapter 8
158
and therefore F ( s , .) . rn equals the corresponding translation of rn . This implies that U(s-’) is an isometric operator, and this holds for all s E G , so we have a unitary representation. Hence it is enough to show the continuity at a dense set (e.g. by Theorem 3.8 but, of course, this is much simpler, Proposition C.17 is enough). If f E H b and r E C,(G/H) then ( r o p ) . f E and if cp(p(t)) = Ilf(t)ll then [If - ( r o p ) . f l l = 11(1- r)cpll where on the left we mean the norm in 31 and on the right the norm in L 2 ( G / H rn) , . Therefore 03-51
H,, := { f E H b ; p(supp f )
is compact }
is dense
in ?i . Let f be a fixed continuous function from G into K , then we can define U ( s ) f by (8-3) and if s, s then (also using the continuity of e ) U(sn)f -.+ U ( s ) f locally uniformly. If f E H O then the net v, defined by v,(p(t)) = I I U ( S n ) f ( t ) - U ( s ) f ( t ) l l converges to 0 locally uniformly on G / H because p is an open mapping. If f E H ‘, then vn(z) = 0 for n 2 no and I 6 KlK where (I = p(suppf) is compact, Ii1 is a compact neighborhood of s and no is large enough (to this K1 ). So in this case v, 0 in any LP space of any Radon measure, hence U ( . ) f is continuous for f E H,, and the proof is complete. --f
--f
Observe that the set H,, defined by (8-5) is independent of the choice of e , As regards relations between different e’s it is obvious from their definition that if el is another continuous positive qfunction then el/e = r o p with a positive r E C ( G / H ) and any such r defines a continuous positive qfunction this way. A continuous positive qfunction is, of course, determined by the corresponding quasi-invariant measure. We are going t o list several facts without proof. They are not essential for the subsequent development but may help t o see better these induced representations. If p is any non-negative Radon measure on G / H then its translations are again non-negative Radon measures. If a non-zero p is just ‘weakly quasi-invariant’, i.e., each of these translations is absolutely continuous with respect to p then one can find a positive locally rn-integrable function r such that p = r.rn . Conversely, any positive locally rn-integrable function gives rise to a weakly quasi-invariant measure . this way. p can equivalently be derived from the positive qfunction ( r o p ) ~ A Borel subset of G/H is rn-negligible if and only if its inverse image in G is negligible for a Haar measure (recall that we define Haar measures as Radon measures, cf. Notation B . l ) . It follows from a Riesz-Fischer type argument that the elements of 31 and 31, can be realized as the equivalence classes of certain ‘locally strongly measurable’ functions from HO and 311 , respectively (two functions are equivalent here if they equal almost everywhere). Then f e’/’f still yields the unitary intertwining between the two versions and the definitions of the scalar product and the induced representation itself extend verbatim. If G is M2 then one finds a Borel measurable cross section q of p and the mapping f + f o q maps ‘H unitarily onto L W / K rn). --f
Definition 8.2. The ‘system induced from S’ consists of the induced representation U and the multiplier spectral Radon measure P on G / H , i.e., for
Induced Represen tations
159
which
for r E C c ( G / H )and f E
3-11,
or equivalently (in Version (b)) for f E ' H a
.
An obvious calculation and the properties of spectral measures show that any induced system is a system of imprimitivity. The converse of this statement (called the Imprimitivity Theorem) was first proved by G . W. Mackey in the case when G is A42 ; L. H . Loomis verified the general case and, at the same time, simplified the proof considerably. The improved version we present below is essentially taken from [Jar] (also cf. [0rs] ).
Theorem 8.3 (Imprimitivity Theorem). Let ( U , P ) be a system of imprimitivity over the Hilbert space 'H (see Notation 7.10). Then it is an induced system in the following way. Let p, E C,(G) be a net satisfying p, 2 0 , p, = 1 and supp p, + 1 . Let K O be the linear span of U ( C c ( G )3-1 ) , then ( I ,y) = lim,, E p , dP,,, exists for I , y E K O and defines an Hermitian form with ( I ,z) 2 0 . Let K be the Hilbert space obtained by the usual factorization and completion from (KO,(., .)) . If [ z ] E K is the class of z E K O then set S ( h ) [ z ]= 6 ( h ) ' / 2 A ( h ) - 1 / 2 [ u ( h ) ~ ] for h E H . This S is correctly defined and extends unitarily to a continuous unitary representation of H over K . Set P : K O H K G , 3 1 ( s ) = [ U ( S ) - ~ I, ] then this 9 is the densely defined restriction of a unitary operator (also denoted by !P) intertwining ( V ,P ) and the system induced from S if the latter is considered according to Version (a). Equivalently, if e is a continuous positive @-function then !&,z(s) = e ( ~ ) - ' / ~ [ U ( s ) - 'defines z] a unitary intertwining operator between ( U , P ) and the induced system as defined in Version (b) with this e . We have @(KO) C 'HI, (equivalently, !&(KO) C '& ) with the notations of Definition 8.1. If we replace the spectral measure P by a positive linear mapping Q : C c ( G / H )H B(3-1) satisfying S U ~ ( Q~ ( r <) z z) , ~ = ~11z11~2 for all I E 'X then the constructions above (writing, of course, ( Q ( E p , ) z , y) instead of E p , dP,,,) still work except that P(X) may be a proper closed invariant subspace for the induced representation; this subspace is not necessarily invariant for the induced system but if? is the orthogonal projection onto it then !PQ(r)= ( S r d P S )P holds for all r E C,(G/H) (denoting by Ps the spectral measure from the induced system). 9 intertwines the representations in this case, too.
s,
SGIH
sGIH
160
Chapter 8
because U is a unitary representation and Q satisfies the imprimitivity condition. Since E is a positive linear mapping from C,(G) into C,(G/H), Q o E is also a positive linear mapping. Hence it is bounded from “ C ( K )fl Cc(G)”into B(X) for any compact I< c G . Thus we obtain a Radon measure T on G x G by setting
for h E Cc(G x C) (cf. Notation B . l ) . Then we can write
and Fubini’s theorem (more precisely, its special case concerning compactly supported continuous functions) applies. Right translation by 2,’ in the inner integral (of the reversed order) and changing the order back yields p((o) =
J
cp(s)F(s)ds with
F(s)=
u( st,
’) (st, ZI
t 1)A(t
’) d r ( t
1
,t 2 )
This F depends on u , v , x,y and s , and we shall just need the following consequence of the explicit expression above: F ( L ( g ) u ,v , x,y; s) is a continuous function of g and s . Thus for any x,y E K O we have a unique continuous function FZ,,such that
for all (8-7)
(o
E C,(G) , and the function g , s -+ F ~ ( s ) z , y ( sis) continuous.
The continuity of FZ,, implies the existence of the limit limn (Q(E(o,)z , y) and we obtain ( ~ , y = ) FZ,,(l). T h e fact (., .) is an Hermitian form satisfying ( 2 ,I) 2 0 for all I E K O is an obvious consequence of the positivity of the operators Q(E(o,) . The imprimitivity condition now implies that
hence for all s E G we have Fu(s)z,u(s)y = L(s)Fz,, for these are continuous functions. Consequently,
Induced Represen tat ions
161
If h E H then E(R(h-')cp) = 6 ( h ) . Ecp for any cp, so by the same reasoning as above we obtain R(h)F,,, = 6(h)A(h)-' .FZ,, , i.e., each F,,, is a gfunction. Now it is clear from (8-8) that each S ( h ) is well-defined and isometric on the pre-Hilbert space obtained from ( K O , (., .)) by factorization. Obviously, S is a representation of H over this space and, of course, over its completion, hence each S ( h ) must be unitary. Since 6 / A is a positive continuous function on H , for the continuity of S it is enough to check that ~ ( s = ) [U(s)z] is continuous from H into K for any z E K O . We assert K
(8-9)
is continuous from G into
K
By (8-8) we have ( U ( t ) z ,U ( s ) z ) = ( U ( s ) U ( s - ' t ) z , U ( s ) z ) = F~(s-~t)z,y(~-') , and this is a continuous function of s and 2 by (8-7). This amounts to (8-9) by obvious calculation. Having the continuous unitary representation S , we can consider its induced system ( U s , P s ) (use Version (a) from Definition 8.1). Since U ( g h ) - ' z = U ( h ) - ' U ( g ) - ' z , we have 9 z (gh) = 6(h)'/2A(h)-'/2S(h)-1[U(g-')z] for h E H , g E G and 3: E K O , i.e., @(KO) c 3tl . Evidently, 9 (considered on K O ) intertwines U and L . (8-9) means that 9 z is a continuous function for any z E K O , and (8-8) means that F,,, is nothing else than the gfunction to be considered in Definition 8.1, so the Radon measure t9 = t9(9t,Py) satisfies Ecpd29 = c p . F,,, and therefore, by (8-6),
SGIH
s,
(8-10) for any r E C,(G/H) . Then our assumption on Q implies that !P maps K O isometrically into the pre-Hilbert space 'HI,,, so it extends to an isometric operator on 3t (for KO is dense by Lemma 3.5). For brevity write [r] = r dPS for r E C,(G/H) . Compare (8-4) and (810) (and use the polarization identity and express r as a combination of non-negative functions) to get ( [ r ] S z, !Py) = (Q(r)z,y) for any r E C,(G/H) and z,y E KO , hence also for any z,y E 3t because the operators involved are bounded. In other words, Q ( r ) = 9*[r]!P. Since !P is isometric, the only thing we have not yet proved is the surjectivity of 9 in case Q is derived from a spectral Radon measure P . In this case Q is multiplicative and from (8-6) we obtain for any z E K O and cp E C,(G) that
s
J,(P. (Ecp
0
P)
*
FZ,,= (Q (IEcpI')
, 2) = (Q(Ecp)z,Q(Ecp)z) .
= 11Q(r)z112 for r = E p , i.e., for all This means, by (8-4), that 11[r]9~11~ r E C,(G/H) . Since 9 is isometric and 9 Q ( r ) z = y [ r ] S z even in general, in the multiplicative case we obtain @ Q ( r )= [ r ] 9 by the denseness of K O . We switch to Version (b) for it is more convenient for the rest of the proof. We know from (8-5) that X,, is dense. Let f E 3tcp . For s E G and E > 0 we
162
Chapter 8
choose an x E K O such that Ilf(s) - [x]11 < E . Then y := e ( s ) ’ / 2 U ( s ) xE K O W= and @by(s) = [z], hence by the continuity of f and @by the set { 2 E G ; 11 f ( 2 ) - !&by (t)11 < E } is an open neighborhood of s . But f - @by E 3i0 , hence W = WH . Let I< be a compact neighborhood of p(supp f ) in G / H and let s vary in G , then the open sets p ( W ) n i n t ( K )and K\p(supp f ) form an open covering of the compact T, space K , hence a partition of unity can be found subordinate to this covering (cf. Definition A.2 and Proposition A.3). This partition of unity is finite because Ir‘ is compact, and if r is a member not vanishing on p(supp f) then supp r c p(W) n i n t ( K ) for some W . In particular, these members belong to C e ( G / H )if defined t o be 0 outside K . Thus f ( 2 ) rj(p(l))@byj(i!)ll 5 E
1
1
cj
2
for all i! and f - cj[rj]@byjII 5 E~ . m ( K ) for a suitable finite sum. Since I< does not depend on E and this finite sum belongs to @b(%) if Q is multiplicative, the proof is complete. We remark that the paracompactness argument above can be replaced by a simpler partition into Borel sets if one realizes [r]f as ( r o p ) . f also for compactly supported bounded Borel measurable functions r (the corresponding modification of Definition 8.1 is a bit easier than the one involving strongly measurable vector valued functions we mentioned after its proof). For a n alternative proof of the fact @ intertwines P and P s see Exercise 8.1.
Theorem 8.4 (Intertwining Theorem). Let S, S’ be two continuous unitary representations of the closed subgroup H of a locally compact group G over the Hilbert spaces K and K’ . Consider the set of intertwining operators 72 = { A E B ( K , K ’ ) ; S’(h)A = A S ( h ) Vh E H } and the analogous object for the induced systems ( U , P ) and (U’, P’) (considered with the same e in Version (b) or, equivalently, both should be considered in Version (a)): Rind =
{ A E B(%,%’) ; U‘(g)A = AU(g) V g E G and P ’ ( X ) A = A P ( X ) V Borel set X c G / H } Then the natural mapping@ defined by @ ( A ) f= A o f (for A E 72 and f E %b or, equivalently, for f E 8 1 , ) establishes an isometric linear bijection from 72 onto R i n d . Its inverse can be obtained as follows. If B E R i n d then consider this relation in K x K’: x x’ if an f E Zb can be found such that B f E 7-i; , f ( 1 ) = x and B f ( 1 ) = x’ . Then, in fact, is a densely defined bounded linear operator, and its closure equals @ - l ( B ) . Consider @ as a “functor”, i.e., a mapping defined on intertwining operators of all pairs of continuous unitary representations of H . Then @ preserves products and adjoints. If S = S’ then @ yields an isomorphism of von Neumann algebras between 72 and R i n d .
-
-
-
PROOF:If A E 72 and Bo f := A o f for f E KG then, evidently, Bo(R0) c 3 i b and if f is continuous then Bo f is continuous. Since llB0 f (g)ll 5
Induced Represen tations
163
llAll.llf(g)ll , we have & ( ‘ H b ) C 31; and there exists a unique bounded operator B from H into H’ agreeing with Bo on Ha. We also see that llBll 5 IlAll . T h e relations B U ( s ) f = U ’ ( s ) B f (for s E G and f E Hb ) and B ( J r d P )f = ( J r d P ’ ) Bf (for r E C,(G/H) and f E Hb ) are evident from the definitions of the objects involved. Hence B E Rind . Since B(3-Ib)c 3-l; , to show that the asserted method recovers A from B it is enough to check the denseness of the set { f( 1) ; f E Hb } in K . We show that even { f( 1) ; f E H c p } is dense, where H,, is the subset of functions with compact “factor support”, see (8-5) . The following vector valued “conditional expectation” (essentially taken from [Blal]) is the tool enabling us to circumvent the difficulty that the factorization p : G H G/H might not have nice cross sections. If 3: E K and ‘p E C,(G) then set
4P9 x)(g) = s (L(L7-l )cpIH)
(8-11)
2
=
J, ‘p(gt)S(t)z dt
for all g E G . It is not hard to check that e(cp, x) E H,, for any cp and x . Now if cpn is a net of non-negative functions from C,(G) which are positive a t 1 and their supports tend to 1 then we have ,J anpn = 1 with suitable positive numbers Q, and e(a,‘p,, x)( 1) + x by Lemma 3.5. Thus the asserted denseness holds indeed. We are going to prove that U ( ‘ p ) ( X )C xb for any ’p E Cc(G) (this is the key observation for the harder half of the theorem). Fix such a ‘p and for f E H,, consider the function g : G H K (8-12) It is easy to check that g E H,, and testing with h E H c p we obtain U(’p)f = g (by applying Fubini’s theorem to the compactly supported continuous function ~ ( 2 ) . ( U ( 2 ) f ( q ( x ) ), h ( q ( x ) ) ) on G x G/H , where q is a cross section of the factorization; recall that the coincidence of the product topology in G x G/H with the factor topology follows from the openness of p ) . Make the substitution t + st-’ in (8-12) and infer from Holder’s inequality that
5 e(s)-’
lls(s)112
(4
A(l)-zdt) .
lv(st-1)12e(l) V(W dt
,
SUPP ‘p)-’s
If
is fixed and s varies in a compact then the second integral here has the form drn(3:) with u’s having a bound and vanishing outside a certain compact. This implies that U(’p)lxcp is a continuous linear operator from the Hilbert space topology of 3-1 into the topology of locally uniform convergence on C(C, K) n Ho . The latter is a complete locally convex space, denote its topology by 7 and, for the moment, denote by F the corresponding continuous extension of U(cp)IxHcp to H . Now let f E 3.1 and fn E H,, be a sequence such that Ilf - frill 5 2-” . Then ‘p
SGIHE u (x). Ilf(q(x))llz
164
Chapter 8
is finite (where the norms are meant in L 2 ( G / H , m ) because ) U ( p ) is a bounded operator in X . On the other hand, U ( ' p ) f , -, Ff in T , i.e., locally uniformly. and U ( ' p ) f , Ff also in 3.1 (cf. Proposition c . 4 ) . Hence we infer that F f E Thus -+
U ( p ) is continuous from 31 into ( % , r ) .
(8-13)
Fix B E R i n d , then BU('p) = U'('p)B for any 'p E C c ( G ), and hence B(&) c Eck where these notations refer to the constructions in Theorem 8.3 applied to U and U ' . Thus, by (8-13), for any f E K O we have f(1) Bf(1) . Now if t E Ec then one can find fo E X c p by (8-11) such that fo(1) is close t o 3: and then, by (8-la), a 'p E C,(G) can be found such that f(1) is close to t with f = U ( ' p ) f o E K O . Therefore is a linear relation with dense projection in Ec . Now let t y and f E 31) a function realizing this. Since P is the multi2 plier spectral measure, we have Pj = (Ilfll o q ) . m and (1t112= lirn, r , dPj
-
-
-
SGIH
whenever r,, is a net from C c ( G / H )satisfying r, 2 0 ,
SGIHr , dPh, . SGtH6 d P '
supp r, p(1) . Similarly, llyll = limn and writing F, = &dP and F,!, = 2
-+
-
sclH
SGIHr,dm
= 1 and
Choosing such a net r, we have
is the dense restriction of a bounded operator A and IlAll 5 IlBll . If f E KO then Bf E Ecb c 'Hi and since B intertwines U and U ' , we obtain f ( s ) e(s)-1/2e(l)'/2BU(s-1)f (1) = Bf (s) , i.e., B f = A o f . Thus B = @ ( A ) on a dense set, hence everywhere for these are bounded operators. For f E K O we have A o f E Xi c 3 i b , and evaluating the definition of 7th a t 1 we obtain S'(h)AS(h)-' = A on the set { f ( 1 ) ; f E K O } . We already checked the denseness of this set, so we see A E R . For f E ? f b the relations @(XA1 A2)f = X @ ( A l ) f @ ( A z ) f and @(AlAZ)f = @ ( A l ) @ ( A Z ) f are obvious, as well as ( @ ( A ) f , g )= (f,@ ( A * ) g ) for f E Xb and g E . Hence, by the denseness of and X i , @ preserves products and adjoints. Finally, it is well-known that a bijective morphism of *-algebras between two von Neumann algebras is automatically a morphism of von Neurnann algebras (i.e., isometric and also homeomorphic between the ultra-weak topologies).
Thus
-
+
+
a)
Observe that the Intertwining Theorem shows, in particular, that a system of imprimitivity determines the inducing representation up t o equivalence (and we may distinguish the realization constructed in Theorem 8.3 which is, of course, independent of the choice of the approximation p ' , of Dirac's delta). Induced representations are widely discussed in the literature for they are the main tool of constructing representations (note that the more classical method of considering regular representations is a special case of this). We now return to the normal subgroup analysis but emphasize that induced representations are also
Induced Represen tations
165
important for other groups including those having no non-trivial normal subgroups at all. Let GIN , U , T ,a,m,3,X be as in the previous chapter (cf. especially Proposition 7.9). Assume GI N and U are such that Proposition 7.9 is applicable and choose a subgroup H accordingly. Then we can see by the Imprimitivity Theorem that U is an induced representation such that the spectral measure of the induced system generates the commutative von Neumann algebra 3. Hence the von Neumann algebra generated by the induced system equals for 3 c M = U(G)"" and the Intertwining Theorem shows that 2l is a discrete factor if and only if the inducing representation S generates a discrete factor (because a von Neumann algebra is discrete if and only if its commutant is discrete). T h e unfortunate thing is that H is not necessarily much smaller than G . We can comfort ourselves with the fact that SIN is far from arbitrary, cf. Sections 13.3 and 14.1 of [Kir] . Now suppose G is the semi-direct product of N by a locally compact group A , i.e., a closed subgroup A 5 G can be found such that the mapping s , t --* st is a homeomorphism from A x N onto G . Denote the restriction of the action of G over N to A by @ i.e., @ ( s ) ( t )= sts-' , Write B = H n A then H = B N because N c H . One easily checks that the natural bijection s B + s H establishes a homeomorphism between the locally compact spaces A / B and G / H , and this identification intertwines the actions of the group A . Observe that the argument before Proposition 7.9 about the isomorphism of the measurable spaces X and G / H was independent of the origin of X and just used that G is M2 N is tame and X is a measurable orbit containing an element with a closed stabilizer H . Accordingly, in the semi-direct product case s B -+ sC := C o @ ( s ) - ' is an isomorphism from the measurable space A / B onto X then. Hence if P is any spectral measure on A / B then it defines a spectral measure E on fi (concentrated on X ) by this isomorphism. Now the SNAG Theorem implies that if N is commutative then E defines a continuous unitary representation and a unitary operator intertwines two such representations if and only if it intertwines the corresponding spectral measures on A / B . Let ( U , P ) be an induced system, then it is a system of imprimitivity,
<
U ( a ) ( J A I B .It(sC)dP(sB))U(a)-' = J A I B J t ( a - ' s C ) d P ( s B ) , i.e., U ( a ) T ( t ) U ( a ) - ' = T(a2a-') if T is the representation of N corresponding t o P by the above. Thus U and T are the restrictions of a continuous unitary representation of the semi-direct product G . Comparing Proposition 7.9 Theorems 8.3 and 8.4 we obtain the following.
so one has
Proposition 8.5. Assume A and N are M 2 , N is commutative and the dual action of A arising from @ on N is separable. Let X be a measurable @-orbit, C E X and let B be the stabilizer of C in A . If S is an arbitrary continuous unitary representation of B then its induced system ( U s , P s ) defines a continuous unitary representation U of the semi-direct product G by
(8-14)
U(st)= Us(s) .
J,,,
z('(t) d P S ( z B )
Chapter 8
166
for s E A and t E N , and we have U(G)' 21 S(B)" . Furthermore, if S1 is another continuous unitary representation then U1 N U if and only if S1 N S . In Version (a) the formula above can be written as
U(st)f ( 2 ) = ('(l!-'sts-'z)
(8-14a)
*
f(s-12)
f
E 3c1, . Now consider an arbitrary factor representation U of G . Choose C E X where X is the @-orbit in N on which the spectral resolution of T = U I , is concentrated, and let B 5 A be the stabilizer o f ( . Then there exists a continuous unitary representation S of B such that U is equivalent to the representation obtained from S by (8-14).
for
Note that any orbit is a-compact because A is an Mz locally compact group and the action is continuous. Hence each orbit is Borel measurable for fi is Tz . Since tameness can be decided by examining just factor representations, one obtains that if G and N satisfy the conditions above then G is tame if and only if all possible E's (i.e., the stabilizers of the elements of k ) are tame. Note that A itself occurs among these, for it is the stabilizer of 0 . If ('1 = o @(Q)-' with some Q E A then its stabilizer equals QBQ-' . Observe that Version (a) in Definition 8.1 is such that if we "change the names" of the elements of G then nothing happens: if z is an isomorphism of locally compact groups from some G' onto G , H' and S' are the corresponding closed subgroup of G' and its representation, and we fix the left Haar measures so that z be measure preserving for both the whole group and the subgroup then X I , 7.1, and the induced system transform accordingly. Apply this to G = G' = A , H = B and z(s) = Q - ' S Q , then H' = QBQ-' and, denoting by ( V , P ) and ( V ' , P ' ) the induced systems, we have V ( Q - ' S Q ) V ' ( s ) and P ( Q - ' Z Q ) P ' ( 2 ) where the subsets of factor spaces are identified with their inverse images in A , and means the unitary equivalence implemented by the natural identification of 7.1, and 7.1;. Thus the representation U' defined by (8-14) with 1'( and QBQ-' instead of C and B is equivalent to U o 2 where we set z(s) = a-'sa also for any s in the semi-direct product. Since U ( Q )is a unitary operator implementing U N U o t , we have U 21 U' . On the other hand, the orbit X is determined by the equivalence class of U . So if G and N are as in Proposition 8.5 then G can be obtained as follows: from each @-orbit in fi choose an element ( , denote its stabilizer by BC , and apply the construction (8-14) to (representatives of) the elements of the disjoint union of the spectra BC. Furthermore, if S E E B c and Q E A then, writing ('1 = (' O @ ( Q ) - ' and z(s) = Q-'SQ , the class [ S o 21 E b,-, would yield the same class in C by (8-14) as [Sl does.
<
-
-
-
[q
In case G is tame and Mz the representation theory of G is rather completely reduced to the knowledge of G as a measurable space (Theorem 8.6.6 of [Dix2]), this structure is standard and equals the Borel structure of the usual topology of G (Proposition 4.6.1 of [DixZ]), and the canonical mapping from the set of
Induced Represen tations
167
normalized pure positive definite functions (endowed with the topology of locally uniform convergence) onto G is continuous and open (Theorems 13.5.2 and 3.4.11 of [DixP]). We emphasize that by 'Bore1 structure' we always mean the a-algebra generated by the topology, i.e., by the open sets (equivalently, by the closed sets) in an arbitrary topological space. The following question is thus very important: given a normalized pure positive definite function on B , what can we say about the pure positive definite functions arising in the corresponding representation of G . Let S be an irreducible continuous unitary representation of the stabilizer B in A of a certain C E fi (where N is assumed commutative), and z b e a unit vector in the space of S . Let cp be the corresponding positive definite function, i.e., cp(b) = ( S ( b ) z , z ). For u E Cc(A) write w(u) =
J, u(ab)6(b)-'/'A(b)'/'
. S(b)zdb
,
then w E 711 , it is continuous and ~ ( s u p p w )is compact. If the support of u is small enough, u 2 0 and u(1) > 0 then w # 0 , hence we find a u such that llwll = 1 (in 81~). Putting (8-14a) together with this, we have ( U ( s t ) w(a) 4.) = ) 1
((a-'sls-'a)
.
J
u(S-'abl)qiiq/+(blbz)
. ' p ( b , ' b l ) d b l dbz .
BxB
It is known that if W c A / B is open then E(C,(p-'(W))) = C c ( W ); hence if 6 is a finite Radon measure arising from a p-function which vanishes on p-'(W) then I6l(W) = 0 . Choosing a v E C,(A) such that Ev = 1 on ~ ( s u p p w )we obtain that the pure positive definite function $(st) = ( U ( s t ) w w) , can be computed by (8-15) .(a) .C(a-'sts-'a) . u(s-'abl)
.
. cp(by'b1) dadbl dbz
AxBxB
(note the integrand here is continuous and compactly supported). Observe that p(suppw) c ~ ( s u p p u ) and so we can fix v when u varies under the condition that its support be contained in a certain compact neighborhood of 1 . Then the construction of z1 depends only on B and its Haar measure, and independent of S, 3: or 6 . Set (8-16)
Z ( B )= { C E
fi ; B is the stabilizer of C }
.
The following is clear from (8-15) : if Sn,zn is a net of pairs consisting of irreducible continuous unitary representations and unit vectors in their respective spaces such that the corresponding positive definite functions v n ( b ) = (Sn(b)zn,t n ) tend 4 C in Z ( B ) then we have to 9 locally uniformly and, on the other hand, In
Chapter 8
168
$, $ locally uniformly where $ , ( s t ) = (U,(st)w, , w , ) with U , constructed from S, and by (8-14) and with w, constructed from Sn,z, with the same 1 and hence the normalized u . Since $ ( l ) = llw112 = 1 , we have $,(l) net = $,(l)-l$, also converges locally uniformly to $. Thus the mapping from B x Z ( B ) into G implemented by (8-14) is continuous. One can also obtain information from (8-15) concerning the case of variable B's sometimes but if we can do with a denumerable collection of B's (as is the case in practice) then this is not very important. Turning to the other direction of the problem, consider a net U, of irreducible continuous unitary representations of the semi-direct product G over a fixed Hilbert space tending to the irreducible continuous unitary U in the usual weak type topology (cf. Subsection 3.5 of [Dix2]). We still assume N is commutative. By Theorem 13.5.2 of [Dk2] and by the polarization identity, for any z , y (U,(.)z ,y) -+ ( U ( . ) z ,y) locally uniformly (this property is also equivalent t o Un U by 3.5.4 of [Dix2] ). In particular, we have these convergences for T, and for U , l A . Applying the same theorem from [Dix2] to the former we have that Eg,y E,,y in the w* topology of Co(fi)"if En and E stand for the spectral resolutions of T, and T . In other words, sfi'pdE,>, is a continuous function of U for any 'p E CO( fi) . Consider a fixed 3: # 0 and let 2 be an open subset of fi . Then
<,
4,
-
-
-
and so this depends semi-continuously (from below) on U . In particular, it is Borel measurable. Hence if 21 c 2 is another open set then E,(Z \ 2,) = E,(Z) E,(Z1) is also Borel measurable. Now the family of subsets 2 of fi satisfying that E,(Z) is a Borel measurable function of U is a monotone class invariant under forming finite disjoint unions, and the differences above form a semi-ring, hence this monotone class contains all Borel subsets of N (see Chapter 6 of [Hall]). In particular, if 2 c fi is a Borel set then { U ; E,(Z) > 0 } is a Borel subset of the space of irreducible continuous unitary representations of G over our fixed Hilbert space. But each such E is @-ergodic (as we observed after Proposition 7.7), so if 2 is also @-invariant then the set above equals {U ; E(2)= Z } . This set is obviously invariant under unitary equivalence, so by taking the disjoint union corresponding to the Hilbert spaces t 2 ( n ) ( n = 1 , 2 , .. .,w ) we obtain the following. Lemma 8.6. If A and N are Mz and N is commutative then for any @-invariant Borel subset 2 of k the set a(2) := { [U] E G ; E ( 2 ) = I )
is (Mackey-) measurable, where E is the spectral resolution of T = UIN . Now suppose that, in addition to the conditions of this lemma, A acts separably over and the @-invariant Borel set 2 is such that a Borel subset J of 2 can be found intersecting each orbit in 2 a t exactly one point and J c Z ( B )
N
Induced Represen tations
169
with some closed subgroup B (see (8-16)). Then (8-14) implements a continuous injective mapping from B x J onto a ( 2 ) by Proposition 8.5 and the considerations after it. If G is also tame then the Borel structure of the topology of G coincides with the Mackey structure and it is standard, hence so is the Borel structure of the topological subspace a ( 2 ) by Lemma 8 . 6 . The mapping above, being continuous, is measurable. Since G is tame, B must be tame by Proposition 8.5 (if we exclude the degenerate case Z = 8 ), hence B is standard, while J is standard because N is even Polish (for the one-point compactification of N is the space of multiplicative linear functionals, including 0, over the separable L ' ( N ) and hence metrizable in its w* topology). Some easy and routine arguments with a-algebras and Polish spaces show that the product measurable structure on the product of two standard measurable spaces is again standard. On the other hand, the product of the Borel structures of two M2 topological spaces equals the Borel structure of the topological product (as is well-known and, in fact, not difficult to check). Since B is M 2 , the space of pure states is also A42 and hence B is M2 for the corresponding factorization is open. So we see that the Borel structure of the topological space B x J is standard. Since we saw a ( Z ) is also standard, Corollary 2 in Subsection 3.3 of [Arv] applies again (we have used it before 7 . 9 ) and shows that the two measurable spaces are isomorphic through our bijection. Thus we have the following important theorem. Theorem 8.7. Let A and N be as in Proposition 8 . 5 . Furthermore, suppose that an at most denumerable family J1, J 2 , . . . of Borel subsets of N can be found with the following properties. The intersection of any @-orbit with any J k consists of at most one point; the union zk of the @-orbitsintersecting J k is a Borel set for any k and N is the disjoint union of these Zk 's; and the stabilizers of the elements of J k are the same (for a fixed k ) , denote these by B k . Finally, assume that each Bk is tame. Tfien t h e semi-direct product G is tame; and G as a measurable space (in the sense of G. W. Mackey) is the disjoint union of the spaces B k x J k (considered with their Borel structures) through the bijection implemented by (8-14). The Borel structure of B k x J k equals the product measurable structure arising from the Mackey structure of B k and the Borel structure of J k C fi .
EXERCISES 8.1. Show that if Q1,Qz are (orthogonal) projections over a Hilbert space such that QlQ2Ql is a projection then they commute. Infer that if 9 is an imbedding of a IIilbert space 'H into a Hilbert space 'H' and P, P' are spectral measures on the same measurable space over these Hilbert spaces such that P$z,py = Pz,y for any x,y E 'H then 9 P ( . ) = P ' ( . ) 9 . 8.2.
With the notations of Definition 8.1 let
X be the locally convex
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170
space C ( G , K )n considered with the topology T of locally uniform converinduces a r-continuous linear functional by gence. Any element h E ? f C p ( h , g ) := J G I K ( g ( q ( z ) ) ,h ( q ( z ) ) ) drn(z) . Show that the weak topology generated by these functionals on X is Hausdorff. 8.3. Let A = N = R , @ ( s ) ( t ) = es . t and G be the corresponding semi-direct product (observe that G is isomorphic t o the Lie group described in Exercise 3.6 through the change of coordinates a = ea , b = e s t ). Show that G is tame and describe G as thoroughly as you can. (Results: G = nu{(, where z E R is identified with the one dimensional representation U,(s, t ) = e i Z * and C contains the representation U ( s ,t)f (2) = exp(ie’-”t).f(x-s) over L2(R).The topology of G is the following: if Z n R # 8 then Z is open if and only if C , E Z and Z n R is open, while the one point sets { C } and {<} are open.)
r}
8.4. Let A = O(2,R) , N = R2 and @ be the natural action of A over N . Determine the Mackey structure of the spectrum of the corresponding semidirect product G (this G is nothing else than the congruence group of the Euclidean plane).
171 9. Projective Representations If X is a linear space over F then one considers the ‘projective space’ of X . This is defined as follows: on X \ {0} consider the equivalence X-y
:-
3XEF\{O}
:
~ = X Z
and let P be the set of equivalence classes; and call the subsets of P corresponding to the two dimensional linear subspaces of X the ‘lines’ of P . If u is a nonzero linear functional on X then the affine subspace P,, = ~ ‘ ( { l } is ) naturally identified with the subset of P consisting of the classes intersecting P,, and the ‘ideal subspace’ P \ P,, is nothing else than the projective space of the linear subspace Ker u . The intersections of the lines of P with P,, yield the lines of the latter (as an affine space). An ‘isomorphism’ between two projective spaces is a bijection preserving the lines ( i n both directions). Naturally, this notion is uninteresting if the projective spaces consist of a single line, so the concept of projective spaces is usually restricted to those where d i m X 2 3 (though it is also custornary, for F = R a t least, to consider a different definition for the isomorphism of projective lines using the concept of ‘double ratios’). One usually calls a representation of a group over a projective space ‘projective’ if its values are autornorpliisms of the projective space. If A’ is finite dimensional then it is customary to endow P with the factor topology of the Euclidean topology of X \ (0) (wliicli coincides with the factor topology obtained from the unit sphere if a scalar product is fixed), then P becomes a compact iiietrizable space and the subspace topology of P,, in P is the same as its subspace topology in X . One may talk about ‘continuous projective representations’ of a topological group over P in this case. The study of these representations was fashionable enough in the last century (of course, the notion of a topological group did not exist then; concrete groups were examined). Now the aim of the present chapter is not the exposition of this but the analysis of a related matter arising in quantum physics. The central assumption of quantum physics is the following: one should take an infinite dimensional separable complex IIilbert space 71 (so it may be identified with f?(w) which was called “the” Hilbert space in early days), certain ‘probability measures’ on the set of orthogonal projections in ‘7-l (see detailed definitions below) describe the possible states of the physical system under consideration, and an ‘observable real valued quantity’ is determined by a spectral measure P on R in the sense that the “probability” of the statement that the value of the quantity falls in the Bore1 set H equals p ( P ( H ) ) if the system is in the state p . Observable vector valued quantities correspond to spectral measures on Rn accordingly. The task of the theory is to determine the actual states and observables (of course, just the relation between them has a physical meaning). We put “probability” between double quotes to emphasize that this is not mathematical probability but a concept based on the assumption that tests can theoretically be repeated indefinitely and, in the limit, they would yield some “probability”. Now h i s assumption is very doubtful (the universe may not be deterministic even in this vague sense) but
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172
without some sort of an assumption like this we could hardly make physics; and if the difference between computed results and reality is small enough then the job is well done. In most physics books the above central assumption is formulated somewhat differently (but equivalently). First of all, by ‘probability measure’ on the set of projections we mean a function p with values in [0,1] such that p ( 0 ) = 0 , 4 1 ) = 1 and p Pn) = C,“==, p(Pn) whenever PI, P 2 , . . . is a sequence of mutually orthogonal projections. Now Gleason’s theorem (see [Gle2] ) asserts that such a probability measure is necessarily implemented by a positive operator T of trace class: p ( P ) = t r ( P T ) (not just for e 2 ( w ) but also for 12(n) with 3 5 n 5 w and, in fact, the main burden of the proof lies with the case n = 3 ; for n = 2 the analogous statement would be false). If 2 : 1 , 2 2 , , . . is an orthonormal basis consisting of eigen-vectors of T and An denotes the corresponding eigen-value then the above can be reformulated in the form
(x:=l
00
00
n=l
n=l
Thus any probability measure is a 6-convex combination of the so called pure ones: if 2: is a unit vector then p ( P ) := l l P ~ l 1is~ called the ‘pure state’ corresponding to 2:. Since the 6-convex combination of any sequence of probability measures is again a probability measure, an equivalent definition of states is t o describe them as the 6-convex combinations of pure states. The other difference is that most physicists prefer talking about self-adjoint operators to spectral measures on R (and refer to the spectral theorem when they, eventually, have to resort to spectral measures). Now if E is a spectral measure on R , A = J t d E ( t ) is the corresponding self-adjoint operator and 2 is a unit vector in D ( A ) then ( A x ,2:) evidently equals the expectation of p o E where p is the pure state corresponding to 2:. If 2: happens t o be an eigen-vector of A with eigen-value X then E,({X}) = 1 and so if the system is in the state corresponding to 2: then the value of the quantity corresponding to A equals X with probability 1 . Thus an observable is most tractable if its self-adjoint operator has a discrete spectrum. For vector valued observables physicists like considering the self-adjoint ‘coordinate’ operators J,,, t j d E ( t ) . The central assumption of quantum physics may seem weird enough but the theory has certain results and therefore it is worth studying. I t turns out that such physical quantities as mass, distance, velocity, etc. can be set in correspondence with objects connected with the representations of the Galilean and PoincarG groups (and also other groups) in some models. Since the ‘logic’ of the system is the set P(31) of orthogonal projections in X , one should consider actions of groups over this set. Let !J3 be the projective space of 31 and if Q E P(31)\ (0) then let [Q] be the subset of !J3 whose elements are the classes contained in the range of Q . This [Q] is a projective subspace of !J3, and the one and two dimensional Q’s yield the points and lines of !J3. Let @ be an automorphism of P ( X ) as a lattice. Then one easily checks that @ permutes the one dimensional projections among themselves and the corresponding bijection f from !J3 onto itself is an automorphism
Projective Represen t a tions
173
of this projective space. We clearly have [Q,(Q)]= { f(z) ; L E [Q]} for any Q . A classical theorem of projective geometry asserts that any automorphism of a projective space (whose defining linear space X is at least three dimensional) is implemented by an a l m a t linear bijection F from X onto X , where ‘almost linear’ means that an automorphism (Y of the ground field exists such that F ( h + y) = a ( X ) F ( x ) + F ( y ) . We note the proof of this theorem is easily reduced to the case of minimal dimension, i.e., when X is three dimensional and we consider a projective plane. Since C has many automorphisms, Q, can be ugly enough. Now suppose, in addition, that Q, preserves orthogonal complements, then one checks immediately that if Q1, Q2 are orthogonal one dimensional projections then @(Ql)IQ,(Q2) . Let F be an almost linear operator implementing f , x1,x2,23 be an orthonormal triple in ‘H, yj = Fx,/ IIFzjII and V be the operator defined by V x j = yj (this V is a unitary operator between the corresponding three dimensional subspaces because f preserves orthogonality). Then the restriction of F to the linear span of ~ 1 ~ x is2 of~ the ~ form 3 FoV where FO is an almost linear bijection of the linear span of yl, y2, y3 such that the elements of the projective space corresponding to this basis are fixed points for the projective automorphisrn implemented by Fo . This projective automorphism also preserves orthogonality because V is unitary. One can show by a moderate effort that in this case t,he automorphism connected with the almost linearity of Fo commutes with conjugation, and it is well known that C has no other such automorphisms than the identical one and the conjugation (the key step is that R has no non-identical automorphisms). Also omitting further (easy) details, we arrive a t Wigner’s theorem: @ is implemented by either a unitary or an anti-unitary operator(an anti-unitary operator U is a conjugate linear bijection satisfying ( U x , U y ) = (x,y) for all z , y E 31 ). Wigner’s theorem gives profound motivation for the following definition.
Definition 9.1. Let G be a group, 31 be a llilbert space of dimension 2 2 and P(3-I) be the set of orthogonal projections in 31. In this book a ‘projective representation of G over 31 ’ means a representation A of G over P(31)satisfying ‘ such that that for any g E G a unitary or anti-unitary operator F exists on H A(g)Q = F Q F - ’ for all Q E P(31). We exclude the case of one dimensional IIilbert spaces because it would only yield trivial representations. If we also excluded the two dimensional spaces then our projective representations could be identified with the orthogonality preserving actions over the projective space J‘3 of 31, with the actions over P(3-I) as an ‘orthocomplemented lattice’, and also with the actions over B(31) as a von Neumann algebra if we also allow conjugate linear automorphisms (this last identification persists over two dimensional Hilbert spaces, too). We mention that one may consider models i n quantum physics whose logic is not P(31) but the set of projections in a smaller von Neumann algebra. Neither Gleason’s theorem nor Wigner’s theorem holds in this generality, and it is an interesting question what are the “domains” of these theorems. Those probability measures and lattice automorphisms which can not be extended linearly (or conjugate linearly) to the von Neumann algebra do not seem interesting; on the other hand, automorphisms or conjugate linear automor-
174
Chapter 9
phisms of a von Neumann algebra which are not spatial might be tractable enough. We do not pursue this line further in this book. If a bounded linear operator commutes with each projection then it must b e a scalar. On the other hand, an anti-unitary operator V can not commute with each projection (if dim31 2 2 ): if z,y are orthogonal unit vectors such that V z , Vy and V(z + y) are scalar multiples of t,y and t+ y , respectively, then V(z iy) is not a scalar multiple of z + iy . Hence if F is an operator realizing A ( g ) in Definition 9.1 then the set of all possible F ' s equals { z F ; t E T } . Let U'(31) denote the union of the sets of unitary and of anti-unitary operators, it is a group and, in fact, it is the semi-direct product of the normal subgroup U(3-1) of unitary operators by the group { I , J } if J is any anti-unitary operator satisfying J 2 = I (such a J always exists, e.g., one can take the conjugation when identifying 31 with ['(n) where n is a cardinal). The canonical mapping a defined by a ( F ) ( Q )= FQF-' is a projective representation of the group U'(31) over 31 and Ker 7r = T . T h e study of projective representations is reduced in a large measure to the study of (almost) linear representations with the help of the following ancillary object.
+
Definition 9.2. Let A be a projective representation of the group G over 31 and a be the canonical mapping described above. Set GA := { (9, F ) E G x U'(31) ; a ( F ) = A ( g ) } , denote by p the mapping p ( g , F ) := g on G A and let taking of the second coordinate.
i? : G A H U'(31)
be the
The following facts are obvious: G A is a subgroup - of G x U*(IH),p is a surjective morphism, Kerp = { 1) x T and A o p = 7r o A . Since scalar operators commute w i t h linear operators, each element of Kerp commutes with the elements of A - ' ( U ( Z ) ) (this is a normal subgroup of index 2 or 1 ). Thus the possible projective representations of G can be obtained as follows: take those extensions of G by a normal subgroup isomorphic to T in which this T is either central or contained in a normal subgroup of index 2 and central in that; and consider the morphisms from these groups into U'(31) which are such that they take the elements of the extending T to the corresponding scalar operators. If we consider a non-central extension then the action of the elements outside that normal subgroup r of index 2 must be conjugation on T . In the first case the representation will be unitary, while in the second case the anti-unitary part can be separated neatly enough as follows. Choose 7 E GA \ r , let @ be the automorphism of r implemented by 7 (i.e., @(s) = 7s7-' ), then if J is the anti-unitary image of y then the unitary part U of the representation must satisfy
U ( @ ( s ) )= J U ( S ) J - ' for all s E r and U ( 7 ' ) = 5 ' . On the other hand, if J and U satisfy these conditions then they define a morphism of the whole group into U*('H). If G is such that for any normal subgroup of index 2
Projective Represen tations
175
there exists an element of second order in its complement then for any extension above one can choose y satisfying p(y2) = p(y)' = 1 , i.e., y2 E T . We show that in this case 7 ' E { l , - 1 } . We assert that if J is any conjugate linear operator then the set of eigen-values of J 2 is invariant under conjugation (so if J 2 is a scalar operator then this scalar must be real, and R n T = { 1, -1) ). Let J 2 z = - Az with some z # 0 . If X # 0 then J z # 0 , and J 2 ( J e ) = J 3 z = J ( A z ) = AJz , i.e., J z is an eigen-vector for J 2 with the eigen-value 1. If a surjective morphism p : C e H C , an isomorphism 2 : T H Ker p and a morphisni U : GeH U'(7-I) are given as above (in particular, V ( z ( z ) )= r l for any I E T ) and A is defined by A o p = A O U then A is a projective representation and s ( p ( s ) ,U ( s ) ) is an isomorphism from Ge onto GA intertwining everything. -+
In the sequel we are going to study continuous projective representations of Lie groups. For this we have to choose a topology on P(7-I). We do not hide that the real aim is the situation when GA becomes a Lie group and A a continuous almcst linear representation, because these objects are tractable mathematically. We have indirect physical support: nobody has tried to consider models in quantum physics with non-continuous representations. Since the continuity of a unitary representation over a separable IIilbert space follows even from pointwise weak measurability, non-continuous representations are pathological indeed. Let G be a Lie group and A be a projective representation of G over 2 . Suppose that for any one-dimensional projection Q the mapping A ( . ) Q is continuous in the wo topology (this is the weakest continuity assumption among the natural candidates; the formal further weakening of considering just the one dimensional projections generated by vectors from a dense set yields the same notion). We want to show that better continuity properties then hold automatically. Consider the strong and weak topologies on U'(7-I) (these are defined by the families of pseudometrics F l , F2 ll(F1 - F2)ell and F1, F2 I ( ( F l - F 2 ) e , y) I , respectively). They coincide because -+
-+
The multiplication is continuous in the strong topology on any equibounded set of real-linear operators, in particular on the semigroup generated by U'(7-I) U P(7-I). Taking inverses is obviously a homeomorphism of U(7-I) in the weak topology and the same holds for U ' ( R ) \ U(X). Now these sets are clearly open in U'(7-I) in the strong topology, hence U*('Fi) is a topological group and U(7-I) is an open normal subgroup of it. Endow C A with the subspace topology of the product space G x U'(7-I) , then CA becomes a topological group, the unitary part r of C A is an open normal subgroup of index 2 or 1, and 21 is a continuous unitary r representation. Furthermore, if T is the factor topology on G then, by the openness of factorizations of topological groups, A defines a continuous action with respect to T and the so topology on P(7-I).We have not used the continuity assumption on A yet; it is needed to show that the canonical mapping p : GA H C is open (i.e., T equals the original topology of C , for p is, obviously, always continuous) and that
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176
GA is a Lie group. Since p is a morphism of groups, for the openness of p it is enough to consider a neighborhood base of the unit element in G A . Anticipating the proof of the fact GA is a Lie group, we choose neighborhoods of the form GAn (W x Z ) such that W is a compact neighborhood of 1 in G and
z= { F E U(R); I I F X j - "jll< 6 v j }
(9-3)
with a finite orthonormal system 2 1 , . . . ,x, and with E > 0 . Recall that the so and wo topologies coincide on P('H). Indeed, for projections we have II(P - P o ) z = ~ ~( P~ x , x ) (Pox , z ) - 2 R e ( P z , Pox) and if P -+ PO a t the functionals (. 2,x) and (. x,Pox) then this expression tends t o 0 . Thus if Q is a one dimensional projection and x is a vector then ( A (.)Q)x is a continuous function from G into 'H . Let y be a unit vector generating Q and let (9, F) E GA , then we obtain that F Q F - ' x = F ( ( F - l x , y) . y) = (2, F y ) . F y is close to ( I ,y) . y whenever g is close to 1 . Apply this with y = z / ~ and ~ x utilize ~ ~ that llFyll = 1 . Write X = ( F y , y ) . We obtain that for any x E 'H and 6 > 0 we can find a neighborhood V of 1 in G such that IIFx - ffx < 6 whenever (g, F ) E GA
+
l
and g E V . Now consider the finite set of xj's in (9-3) and apply the above in the cases x = x j and x = x j xk . We get that if F corresponds t o some g close enough to 1 then there exist numbers a j , bjk E T such that IIFxj - ajxjII < 6 and IIFxj Fxk - bjkxj - bjkxkll < 6 . Then Il(aj - b j k ) x j + ( a h - b j k ) x k l l < 36 and therefore laj - ah1 < 66 because xj and x k are orthogonal unit vectors. Consequently, IlCFzj - xjll < 76 for all j,and since ( g , F F ) € GA , we see that there exists a neighborhood V of 1 in G such that a-'{A(g)} intersects Z for all g E V . Hence ~ ( G n A(W x Z ) ) 3 W n V and p is an open mapping indeed. We assert that I<- = GAn ( W x Z ) is compact. Since W is compact, for an arbitrary net in K we find a subnet (g,, F,) such that gn + g in W . Choose F from ir-l({A(g)}) , then the openness of p implies that for any neighborhood S of (9, F ) in GA there exists an no such that for R 2 no we have (g,, XF,) E S with suitable X E T ( A depends on R and S ) . Consider the directed set N1 x N2 where N1 is the original and N2 consists of the neighborhoods of (g, F) in G A with the usual partial ordering ( S1 S2 in N2 if S1 3 S2 ). If ( n , S )E N1 x N2 then choose R O for S with the property above and choose nl E N1 such that nl 2 no and 721 2 n . Write cp(n,S) = nl and choose X ( n , S ) so that ( g n , , X(n,S)F,,) E S . We find a convergent subnet of X ( n , S ) because T is compact, i.e., we have a directed set N and a mapping z : N H N1 x N2 tending t o infinity such that limk X o z = s . Then limk F v ( t ( k ) ) . = s F and limk g?(,(k)) = g , so we have found a subnet of (g,, F,) converging in GA . Since Z IS closed in U * ( R ) ,we have S F E Z and so I( is compact. We show next that GA has no small subgroups. Since G is a Lie group, we can find a neighborhood V of 1 in G such that it contains no non-trivial subgroups. Hence if H C p - ' ( V ) is a subgroup of GA then p ( H ) = (1) , i.e., H c T . Thus if H is also contained in a neighborhood whose intersection with T is small then H = { 1) . Note this argument just used that p is continuous and the groups G and T have no small subgroups. Since we proved the local compactness of G A above, it must be a Lie group by Theorem F . 6 . Collect our results.
+
+
<
Projective Represen tations
177
Theorem 9.3. Let G be a Lie group and A be a projective represen tation of G over 3-1 such that A( .)Q is continuous in the wo topology for any one dimensional projection Q . Then A is, in fact, a continuous representation if P(3-1) is considered with the s o topology, and the wo topology coincides with this on P(3-1). The strong and weak topologies agree on U’(3-1) and make it a topological group, and it is the semi-direct product of the open normal subgroup U(3-1) by any subgroup {I,J } where J is an anti-unitary operator satisfying 5’ = I . The product topology of G x U’(3-1) turns GA into a Lie group and p is a continuous open mapping; A is a continuous almost linear representation and its restriction to the open normal subgroup r = A-l(U(3-1)) is a continuous unitary representation. We remark that the properties of being a locally compact group, a compact group or a Lie group go over to a topological group if they are possessed by normal subgroup and factor group. We include the first two statements as Exercise 9.1, the third one is a consequence by Theorem F.6. As a matter of fact, this property of Lie groups had been proved before the solution of IIilbert’s 5th problem w a s found, and was considered as a possible step towards that solution (but things happened differently).
Dc!finition 9.4. Let A l , A z be two projective representations of the group G over the Hilbert spaces 3-11 and 3-12. They are said to be equivalent (written A l 21 A z ) if a unitary or anti-unitary operator F from 3-11 onto 3-12 can be found such that A z ( g ) Q = F ( A 1 ( g ) ( F - ’ Q F ) ) F - ’ for all g E G and Q E P(3-1~) . Wigner’s theorem implies that if one considers projective representations whose IIilbert spaces have the same dimension 2 3 then this equivalence is the same as the one derivable from the catcgory of orthocomplemented lattices. There is another version of Wigner’s theorem which shows that we can omit the restriction about dimensions for they are also determined by the orthocomplemented lattice structure (but if both spaces are two dimensional then, of course, Wigner’s theorem does not apply). If A1 N Az by the operator F then the mapping @ ( g , T ) := ( 9 , F T F - ’ ) is an isomorphism from G A , onto G A such ~ that p z o @ = pl and Az(@(t)) = F A l ( z ) F - ’ . If F is unitary then Qi is identical on T , while if F is anti-unitary then @IT equals the conjugation. If G is a topological group then Qi is an isomorphism of topological groups, i.e., also homeomorphic between the suhspace topologies of the corresponding product topologies. Conversely, let (GI, p 1 ) and (G2, p z ) be two extensions of a group G by T , i.e., p , : Gj H G are surjective morphisms with kernels isomorphic t o T and these isomorphisms are considered fixed: “ T c G, ” . Suppose that an isomorphism @ exists from GI onto G2 such that p z o Qi = pl and Qi is either identical or the conjugation on the kernel T . For j = 1 , 2 let U, be a morptlism from G, into U’(3-1j) satisfying U ( z ) = z I for z E T and suppose that a unitary or antiunitary operator F from 3-11 onto 3-12 exists intertwining U1 and Uz o@ (of course,
178
Chapter 9
the unitarity of F is decided by the behaviour of @IIT). Then A1 N A2 by F if they are defined in the natural way, i.e., Aj(pj(s))Q = Uj(s)QUj(s)-’ for s E Gj and Q E ”(‘If,.). Let G be a Lie group. Consider the triples ( z l G e , p ) Definition 9.5. where G e is a Lie group, p : G“ H G is a surjective open morphism of Lie groups, z : T I+ Ker p is an isomorphism of Lie groups a n d
I ’ : = { s E G e ;s z = z s V z ~ K e r p } is an open normal subgroup of index 2 or 1 such that if G“ #
r
then sz(z)s-’
=
z(Z) for s 6 Ge\I‘. In thesequel wesirnplyuse the word ‘extension’for these, a n d
an ‘isomorphism’ bet ween two extensions is a n isomorphism of Lie groups between the two Ge ’s intertwining both p a n d 2 . By Cartan’s theorem if p is just a surjective continuous open morphism of groups and z is a n isornorplrism of topological groups then the above is satisfied. T h e slight abuse of language “2 = id” (in other words, “ T c G e ” )is common, a n d we shall apply this whenever it causes no confusion. Theorein 9 . 3 sliows that, any continuous projective representation A of t h e Lie group G defines an ext,ension ( G A , ~of) G and a continuous almost linear represent,at,ion A 1ly Definition 9 . 2 . Conversely, if a n extension (G“,p ) and a continuous representation are given as we considered aftjer Definition 9.2 then we obtain a cont,inuous projective representation A antl the canonical ident.ification K : C e H GA is certainly continuous, Iicncc analyt,ic by Cartan’s theorern. Rut dim C;‘ = 1 dim G = diin G‘A , t.liercfore K must be an isomorpliism of Lie groups and t I i m also of extmsions. By tlie remarks after Ilcfinit,ion 9 . 4 two equivalent continiloris projeci.ive representat,ions of a Lie group give rise to either unitarily eqirivalerit cont,iiriious “almost unit,ary” represcnt,a.t,ioiisof isomorphic extensions or ant,i-iinit,arilyeqriivalerit continilous almost unit.ary representations of extensions which are ‘anti-isomorphic’ (i.e., t,he corresponding isoniorpliisni of Lie groups conjugates the elenictnts of T ); in both cases t h e rcprcscntatkms are “identical” on T . ‘rlius the equivalence classes of continuous project,ive represcnthtions fall into disjoint objects corresponding to t>heclasses of extensions deteriirined by ‘isomorphism or airt,i-isoinori)liisin’. ‘ l h e composition of two sucli generalized isomorplrisms is an anti-isoiiiorpliisiii if and only if exactly one of tlieiri w a s sucli, Iicnce t,lie classes above consist of one or two isoniorphisiri classes. Let C be such a coiiibined class and fix ( G e , p )E C . If C is a single isomorphism class t.lrcn any continuous projective represent,atiori of G over ‘H connected wit,h C is irnplement~edby a continuous morpliisni U : Ge H U’(31) such that U ( z ) = z l , while in the other case we must also consider continuous morphisms with U ( z ) = FI . Denote by D ? O the union of the sets of continuous morphisins of these two types and let !D be the subset of morphisms of the first type. If F is any anti-unitary operator then U H FU(.)F-’ is a bijection between !XI and ! D o \ !D and morphisms corresponding by this bijection implemeiit equivalent projective representations. T h u s both !D antl ! D O can be used to study tlie equivalence classes of projective representations connected with C (no matter whether C
+
Projective Represen tations
179
consists of one or two isomorphism classes). Now let U1, Uz E !ZD , then the condition that F E U*('H) implements an equivalence of the corresponding projective representations means that Ul(s)QUl(s)-' = F U ~ ( S ) F - ' Q F U ~ ( S )F-' -' for all s and Q , i.e., a function c : Ge H T exists such that c(s)FU2(s)F-' = U l ( s ) for all s . Since FUz(.)F-l and U1 are representations, we have c(st) = c(s)c(t) if s E r and c(s2) = c(s)c(l) otherwise (see Definition 9.5). Since U1,Uz are continuous, c must be continuous and therefore it is analytic by Cartan's theorem. If F is anti-unitary then c(z) = t 2 for z E T , hence anti-unitary equivalence is possible in !ZD only if such a c exists. Fix a conjugation J of 'H , i.e., an anti-unitary operator whose square is the identity; it is well-known that this is equivalent to fixing a real Hilbert space whose complexification is 'H (in particular, any two conjugations are unitarily equivalent). Also fix a c with r ( z ) = z 2 if exists. Then U c := c J U ( . ) J is a bijection from !ZD onto itself and the condition that the projective representations implemented by 171, U2 E ZB are anti-unitarily equivalent is the same as to say that U1 N d . U,C where d : Ge H T is an analytic "almost morphism" as described above satisfying d ( z ) = 1 for t E T . Thus we have the following. Let G be a Lie group and consider the equivalence Proposition 9.6. classes of continuous projective representations of G . A class evidently determines the dimension of the underlying €filbert space; consider classes over a fixed [filbert space 'H. They form the disjoint union of certain objects corresponding to the combined isomorphism classes of extensions of G . lf C is such a class then the corresponding object is the following. Choose ( G e , p )E C and let ZB be the set of ContiflIJOuS morphisrns M : Ge H U*('H) satisfying U ( z ) = z I for z E T . Define
(9-4)
9' = { d : Ge H T ; d is analytic and d ( s l ) = d ( s ) . d(t)' }
where t s = szs-l through the identification ofT with a(T) c Ge ;in other words, zs = z if s E r and z s = F otherwise. Set 9= { d E 9* ; d ( z ) = 1 for z E T } . We say that C is of the first type if there exists a d E 9* such that d ( z ) = z 2 for z E T (it is easy to check that this property of the extension just depends on the class C). I n this case we fix such an element of 9' and denote it by c . Furthermore, we choose a conjugation J of the Hilbert space and define U c ( s ) := c ( s ) . J U ( s ) J for U E ZB . In the other case we say that C is of the second type. For U l , U 2 E !ZD write U1 U2 if they implement equivalent projective representations by the formula A ( p ( s ) ) Q := U ( s ) Q U ( s ) - ' . On the other hand, write U1 x Uz if a d E 9 can be found such that U1 e d . U2 where 2: means unitary equivalence. If C is of the second type then coincides with x , while if e is of the first type then U1 U2 if and only if either U1 x U2 or U1 x U i .
-
-
-
Suppose that G is such that the combined isomorphism classes of its extensions are known. Then the proposition above shows that the theory of continuous projective representations of G can be splitted into an analysis of continuous unitary representations of certain Lie groups and of their relation with the anti-unitary
180
Chapter 9
operators, cf. (9-1) and (9-2). In the remainder of this chapter we try t o obtain closer information about extensions of Lie groups. Let ( G " , p ) be an extension of the Lie group G , then d p is a surjective morphism of Lie algebras and Ker(dp) = iR where we identify the Lie algebra of Kerp = T with iR so that the exponential mapping of T coincide with the restriction of the ordinary exponential function. Denote by g and Be the Lie algebras of G and G " . Let u be a linear functional u : 8" I+ R such that u ( i t ) = t for it E iR , and write V = Ker u . Then dplv is bijective, denote its inverse by q . Since T is central in an open subgroup, iR is central in g" . On the other hand, d p is a morphism, thus for any x , y E g and 1, s E R we have
where ~ ( z , y )= u ( [ q ( x ) ,q ( y ) ] ) . So the structure of g" is determined by the bilinear mapping K . T h e K ' S arising from all possible u's are usually called the 'commutator cocycles' of the extension; we denote their set by Ii' . One checks immediately that Ii' just depends on the isomorphism class of the extension, and the set of commutator cocycles of an extension anti-isomorphic to (Gelp ) equals -A'. If u , u1 are two such functionals then q1(x) - q ( x ) E iR for any z because d p o q, = i d , so the fact iR is central implies n l ( z , y) = u I ( [ q ( x ) ,q(y)]) . This expression is nothing else than u1 applied to the left hand side of (9-5) if the latter is considered with u and with t = s = 0 thus we have
Conversely, if cp : g I+ R is an arbitrary linear functional then we define u 1 ( z ) = cp(dp(z)) for z E V and u1(it) = 2 for it E i n . Then u1 o q = cp , hence we obtain
If another extension ( G : , p l ) has the same cocycles then choose a functional u1 for this extension whose cocycle equals K and set T z = q l ( d p ( z ) ) for z E V and T ( i t )= it for it E iR . Then T is an isomorphism of Lie algebras, it is identical on iR and dpl o T = d p . Unfortunately, this does not imply the isomorphism of extensions (not even for connected Lie groups, see Exercises 9.2 and 9.7 for examples). Nonetheless, the set of commutator cocycles (equivalently, one cocycle by (9-6)) thoroughly restricts the possibilities. Quite often in practice we meet G's for which the cocycles do determine the isomorphism classes of extensions; try t o analyze this question. Let H and H1 be the simply connected covering groups of the unit components of G" and Gt , respectively, and denote by h and hl the covering morphisms. Then we have a unique isomorphism (of Lie groups) @ : H I-+ H I such t h a t dhl o d@ o dh-' = T . We have dpl o dhl o d@ = dpl o To d h = d p o d h and hence
181
Projective Represen tations
O n t h e other hand, if we identify the corresponding parts of the Lie algebras of H and H1 with iR through d h and dhl then @(expH i t ) = expH, it
(9-8)
for all it E iR because T is identical on i n . Let A = h-’(T) = Ker(p o h ) , it is a closed subgroup, its Lie algebra equals iR and A = e x p H ( i R ) . Ker h , hence A is central because both K e r h and e x p H ( i R ) lie in the center of H . We have @(A) = h;’(T) = e x p H , ( i R ) . Ker h l by (9-7). T h u s we can define a morphism from A into T by setting
(9-9)
Of course, f = 1 on the unit component A0 = e x p H ( i R ) of A by (9-8). From Ao.Ker h = A 3 @-‘(Ker h l ) we now obtain t h a t the condition @(Ker h ) = Ker hl is equivalent to the requirement t h a t f be trivial (i.e., f ( a ) = 1 for all Q E A ). In other words, @ yields an isomorphism of the unit components of t h e extensions if and only i f f is trivial. T h i s certainly holds if A = A0 . Observe t h a t t h e unit component Go of G is isomorphic (as a Lie group) to ( H / A o ) / ( A / A o ) ,hence if Go is simply connected then necessarily A = A0 . I f f is not trivial then we may still hope for an isomorphism. Suppose t h a t a continuous morphism ’p : H/Ao H R can be found such t h a t for a E A we have eiv(aAo) = f ( a ) (it is enough to know this for a E Ker h ). Then one vcrifies t h a t @1(s) = expII, (-i’p(sAo)) @(s) is an isomorphism satisfying t h e analogues of (9-7) and (9-8) a n d , in addition, its f is trivial. Such a ‘p does not always exist as Exercises 9.2 and 9.7 show. Another important aspect of this cocycle theory is the problem of finding all possible cocycles (in a number of cases the classification of extensions is reduced to this as we have observed). At the Lie algebra level we can say t h a t K is an ‘abstract’ cocycle if it defines a Lie algebra by (9-5). This means that K is a real bilinear form such t h a t (9-10) (9-11)
K(Z,
y) = -tc(y, x )
and
K ( [ 2 , Y l , Z ) + ~ ( [ Y , ~ I , ~ ) + ~ ( [ Z , ~ l ,= Y )0
‘
We call such objects the ‘cocycles of the Lie algebra g ’. If a bilinear form satisfies just (9-10) then it is called ‘symplcctic’; if g is commutative then (9-11) trivially holds and therefore each syrnplectic form is a cocycle of g . We see from (9-6) t h a t any two cocycles are inequivalent in this case. In the general case the LCvi-Malcev Theorem enables u s to prove the following result. Proposition 9.7. Let r be the radical of g a n d choose a complementing semi-simple subalgebra 5 . Then a n y class (as described by (9-6)) of cocycles of g contains a K satisfying (9-12)
K(rl
+ s1
1
r2
+
s2)
=4f-I
1.2)
Chapter 9
182
for r, E t , s, E 5 . On the other hand, a cocycle K of t defines a cocycle of g this way if and only if K is invariant for a d s , i.e., K((ad s ) ~ y) , = -K(x, (ads)y) for ~ € and 5 x , y t~.
PROOF: Let K be a cocycle of t. It defines a symplectic form k on g by (9-12). We have %(I, (ads)y) = ~ ( [ y , s ] , z ,) while i ( . , s ) = 0 for s E 5 , hence the ad 5 invariance of K is equivalent to the condition that ii: satisfy (9-1 1) for triples t,wo of which is taken from t and the third one from 5 . If a t least two of the triple is taken from 5 then each term in (9-11) becomes 0 for ii: , while for a triple from t (9-11) holds because K is a cocycle of t . Now consider an arbitrary cocycle of g and let (gelT) be the corresponding extension of g defined by (9-5) (the factorization a equals dp in the case of a cocycle of a group extension). ~ - ' ( 5 ) is a subalgebra of g" and A maps tliis Lie algebra onto the semi-simple 5 . By a consequence of the Ldvi-Malcev Theorem (see, e.g., Subsection 1.6.8 of [Boul]) we have a morphism q : 5 H ~ ~ ' ( such 5 ) that T o q = i d . Write 5 0 = q ( 5 ) , then g" = T-'(t) @ 5 0 (as a linear space). 50 acts on the ideal T - ~ ( c )by ad and this action is trivial on Ker A (for Ker T is central even in the whole g" ). Then, by the theorem of II. Weyl, we have an ad 50 invariant linear subspace V complementing Ker a in 7r-l (t) and so V @ 50 complements Ker T . Consider the cocycle arising from this subspace, it satisfies (9-12) because the corresponding selection q of A maps 5 to 50 and V @ 50 is invariant under ad 50 . Given a cocycle K of g we consider the corresponding Lie algebra g" together with the factorization, and choose a realization ( H ,PO) of the simply connected Lie group of ge and its corresponding morphism into the simply connected covering group G of the unit component Go of G . Let 29 : G H Go be the covering morphism, then d29 o dpo is the factorization connected with the class of K . If this class is not the trivial one then it also determines a scaling of Ker(dt9 o dpo) which we identify with iR as usual. Let A = Ker(29 opo) and A . be its unit component, then A0 = expH(iR) is central i n H because iR is central in ge . po implements a locally homeomorphic morphism from H / A o into the simply connected G , hence it is isomorphic, A0 = Kerpo and A/Ao N Ker 29 through po . If K was obtained from an extension (G",p) then we have a covering morphism 29, : H Gg such that
-
(9-13)
8, expH it = expG. it = eit E T
(9-14)
pOfle=d0~0
c Ge
and
1
where Gg = p-'(Go) is the unit component of G " . Thus Ao/(eXp~(2TiZ))21 T through 29, and A = 29i1(Kerp) = 29;'(T) = A0 . Ker29, . So in this case A is central in H and a discrete subgroup A1 exists such that A = A1 . A0 and expH it E A1 if and only if t E 2aZ . Conversely, if A is central and such an A1 can be found (for an abstract K ) then let 19, : H H H / A 1 be the factorization, define p by (9-14) on H / A l and
Projective Represen tations
183
set z(eit) = (expH i t ) A l . Then we get an extension of Go whose set of cocycles contains K . The fact that more than one A1 may exist and there might not be an automorphism taking one to the other is responsible for the existence of different classes of extensions with the same cocycle. We have proved the following.
Theorem 9.8. Let K be a cocycle of g and f : H H G be an open morphism of Lie groups where H is a simply connected Lie group with Lie algebra isomorphic to the one constructed from K by (9-5) and f is the mapping for which the q of (9-5) is a selection of df and Ker df = iR . Let A = Ker f and A0 be its unit component, then A0 = expH(iR) . Then K is a cocycle of some extension of the unit component GOof G if and only if a discrete central subgroup A l of H can be found with the properties that A = A0 . A1 and expH it E A1 if and only if t E 2sZ . In this case if we define z ( e i t ) = (expH it)Al then ( 2 , H / A 1 , f ) is an extension ofGo such that K is a cocycle of it. Let, M = { cp E (Be)* ; [g" , gel c Ker cp } , i.e., the set of Lie algebra morphisms from ge into R . Each cp E M defines a unique morphism (p : H I-+ T such that (p(expz) = eiv(s) for 2 E ge . If G is connected then the set D* defined i n (9-4) equals the set of (p's satisfying A l c Ker (p , while in general the operation h + hlGg implements a mapping from 9* into this set. T h e elements of 9 are singled out by the additional condition iR c Ker p , while the condition @ ( z ) = t 2 (for all z E T ) is equivalent to cp(it) = 2t (for all it E iR ). If G is not connected then it is difficult to say anything in general. We focus our attention to the case when G = Go x Gd and denote by @ the action of Gd over GO ( i t . , @ ( s ) t = s t s - ' ). The notation Gd is intended to suggest that Cd and @ determine the "discrete structure" of G . Of course, this circle includes all discrete groups, so one can not expect any important general results; in the back of our minds we think of applications when Gd is not only finite but rather small. Let ( G e , p ) be an extension of G and choose a selection Q : Gd I+ G" , i.e., a funct,ion satisfying p ( p ( s ) ) = s for all s E G d . Since p is a morphism, we have p ( e ( s ) e ( t ) )= p ( e ( s ) ) p ( e ( i ) = ) st = p ( e ( s t > ), i.e.,
with some function w : Gd x Gd H T , called 'the cocycle of the selection e'. Evidently, if el is another selection then el(s) = q ( s ) e ( s ) with some function 7 : Gd H T and any such function defines a selection this way. By immediate calculation we obtain that (9-16) where z S = e(s)te(s)-' , it is independent of e and just depends on the extension: we have zs = z if s E p ( r ) and zS = 2 otherwise. In the sequel we just consider selections with e(1) = 1 (such obviously exist for any extension) and by a 'cocycle'
Chapter 9
184
we shall mean a cocycle of such a selection. Then we must only consider 7's with q( 1) = 1 . Since Ge and G d are groups, these cocycles satisfy (9-17)
w(s,l)=l=w(l,s)
and
w ( a b , c ) . w ( a , b ) = w ( ~ , b c ) - w ( b , c .) ~
(9-18)
Isomorphic extensions have the same sets of discrete cocycles, while the discrete cocycles of an extension anti-isomorphic to our one are obtained by conjugation. The 'cocycle axioms' (9-17,18) can also be considered abstractly if we define z" . Suppose that a normal subgroup f d of index 2 or 1 is given in G d , then we set z" = z if a E f d and z" = 2 if a E G d \ f d (for all z E T ). If a function w : G d x G d H T satisfies (9-17) and (9-18) (the latter with respect to a fixed r d ) then on the set S = T x G d define ( ~ , u ) ~ ( w , b ) : = ( z ~ w " ~ w ( u , b ) - '. , a b )
(9-19)
This operation is associative by (9-18) and ( 1 , l ) is a unit element by (9-17). Any element ( z , u ) has a left inverse (P. w ( a - ' , u ) , u - ' ) and a right inverse ( F a . w ( a , a - ' ) " , u - ' ) with respect to this unit, so S is a group and (9-20)
w(u-',a)
= w(a,u-1)"
for any a (of course, this is also an immediate consequence of (9-18) and (9-17)). Let Z d ( z ) = ( z , 1) and p d ( % , u ) = a , then ( Z d , S , P d ) is an extension of G d (in the Lie group sense) and its f equals p , ' ( f d ) . Two such extensions (with common f d ) are isomorphic if and only if their defining abstract cocycles are connected by (9-16) with some function 7 ; extensions with different f d ' s can not be isomorphic or anti-isomorphic. If w was obtained from some Ge and e then ( z ,u ) + z . p ( a ) is an isomorphism of extensions. Since T is central in the unit component Gg , the automorphism P ( s ) z:= e(s)xe(s)-' of G; is independent of the selection e and P is a morphism from G d into the automorphism group of GE by (9-15). We have (9-21) (9-22)
p o Q i e ( s )= @ ( s ) o p @'(s)z
= tS for
tE
and
T .
Now suppose that the following are given: (a)
(b)
(c)
an extension ( (Go)e ,PO) of the unit component Go of G , r d of index 2 or 1 in G d and an abstract discrete cocycle w on G d x G d as described by (9-1 7,18) (with respect to r d ) , a morphism 6 from G d into the automorphism group of (Go)e satisfying the analogues of (9-21) and (9-22) (the latter with respect to r d ) . a normal subgroup
185
Projective Represen tations
s
with the morphism 8 o pd where (Zd, S , p d ) is the Let = (Go)e M extension of Gd corresponding t o r d and w as constructed above. Set p ~ ( z 2 := ) po(z)pd(t)
for z E (Go)e and t E S
.
Since 6 and po satisfy the analogue of (9-21), this p~ is a morphism (of Lie groups) into G . Obviously, K e r p K = { z . z d ( w ) ; z E T C ( G ~ ) ~W, E T }
.
Immediat,e calculations show that the coordinates 2 ,w above identify Ker p~ with T2 as a Lie group, it is central in I<, := (Go)" x p d l ( r d ) and z ( z . Z d ( W ) ) z - l = P . zd(w) for z E Ii'\ I<, (use (9-22) ). Define
N:={2.2d(2)-1;ZET}
,
then N is a closed normal subgroup of I< and z(z) := r N is an isomorphism from T onto (KerpK)/N such that the pair (2, A'/N) satisfies the condition in Definition 9.5 with r = Z<,,/N . Since N c KerpK , the m o r p h i s m p ~implements a morphism p : I
for 2 E (Go); . Thus all objects necessary to the construction are isomorphic in the way as they should; we have proved the following. Proposition 9.9. Let G = GOx Gd where Go is the unit component of the Lie group G and we denote by @ the corresponding morphism from Gd into the automorphism group of Go. Then the isomorphism classes of extensions of G can be labelled with triples ( r d , a,%) where r d Is a normal subgroup of index 2
186
Chapter 9
or 1 in Gd , % is an equivalence class of discrete cocycles compatible with r d (see (9-16)) and % is a class o f p a i r s ( ( ( G ~ ) ~ , p&) o ) consisting , of an extension of the unit component Go of G a n d a rnorphisrn 6 from Gd into the automorphisrn group of (Go)‘ satisfying (9-21) and (9-22) (the latter with respect to r d ) . Two such pairs are considered equivalent if a suitable isomorphism between the extensions also intertwines 6l(s) and &(s) for all s E Gd . We can see that the task of classifying the extensions of such a “semi-direct Lie group” splits into three disjoint problems: classify the extensions of Gd , those of Go and, finally, if an extension ((Go)‘,,) and a r d is fixed then classify the morphisms Oe satisfying (9-21) and (9-22). T h e first problem is hopeless in general but may be quite tractable for “tiny” discrete groups. We have already discussed the second problem and close this chapter by recording some easy observations about the third one. @“(s) is uniquely determined by its tangent F ( s ) := d @ “ ( s ) , and then (9-21) and (9-22) are equivalent t o the pair of conditions
d p o F ( s ) = d ( @ ( s ) )o d p F(s)liR = kid
(9-23)
and
with id exactly for s E r d here. Let % denote the group of those autoniorpliisms of 8‘ which can be lifted t o an automorphisrn of (Go)‘ (of course, one can identify % with the automorphism group of (Go)‘ but we want t o think of the elements of % as linear operators on g‘ ) . Set
(9-24)
% = { T E % ;d p o T = d p
and
TI,,=id}
Then the question can be posed as follows: consider those morpliisms F : Gd I+ % which satisfy (9-23), call Fl and F2 equivalent if an operator from ‘13 intertwines them, and try t o describe these morpliisrns up t o equivalence. Choosing a functional u on ge satisfying u(it) = t for it E iR we have a selection q of d p and a commutator cocycle rt describing the structure of g‘ by (9-5) . Let T be a linear operator on g‘ for which i R is invariant, then T ( z + i t ) = q ( d p ( T x ) ) + i ( u ( T t ) + c t ) for 1: E K e r u and it E iR with some real constant c . In other words, T(q(1:) i t ) = q(To1:) i(w(z) c t ) where To and v are a linear operator and a linear functional on g , respectively. Writing this into (9-5) one obtains that T is a rnorphism (of the Lie algebra) if and only if TOis a morphisrn and
+
+
+
and T is an automorphism if, in addition, TOis bijective and c # 0 . For T E 23 we have TO = I , c = 1 and Kerw 2 [g,g] ; if an operator possesses these three properties then it is a n automorphisrn of g‘ and if, in addition, T E % then T E ‘13 . (9-23) can be written with these notations as follows: F ( s ) o = d ( @ ( s ) ) and c : Gd H (1, -1) is a morphism with Kerc = r d . T h u s the only possible
187
Projective Represen tations freedom of F is described by the mapping v : Gd following axioms.
H
0'
which must satisfy the
The relation of Q, and K may be such that no v satisfies these axioms. In that case there can not be any extension of G whose unit component would yield ( (Go)",p ) . If v satisfies (9-26,27) and, in addition, U (q (. )
+ i t ) := q(d(Q,(s)).) + i ( ( v ( s ) ),. + c ( s ) t )
defines an element U E U for all s (i.e., the corresponding automorphism of the simply connected covering group maps the homotopy group of (GO)" onto itself) then we obtain an F . If two functions v l , v 2 satisfy these conditions then their equivalence implement,ed by the operator T ( q ( 2 ) it) = q ( x ) i ( w ( 2 ) + 1) is described by the formula
+
(9-28)
v2(s) -
q(s)=
( d ( @ ( S ) ) ) * W-
+
c(s)w .
The possible w's are those functionals which vanish on [g, g] and are such that the automorphism T of ge constructed above lifts to an automorphism of (Go)" .
EXERCISES 9.1. Let G be a topological group and N be a closed normal subgroup of G . Show that if N and GIN are locally compact groups then so is G . Furthermore, if N and GIN are compact then G is also compact. 9.2. On the set R2 x T consider the following multiplication:
Check that this defines a Lie group G and its simply connected covering is isomorphic to the Heisenberg group of freedom 1 (cf. Exercise 5.5). Show that the trivial class of commutator cocycles (i.e., the one containing 0 ) occurs for non-isomorphic extensions; moreover, one can find such extensions so different that even their respective Lie groups are not isomorphic. 9.3. Apply the theory of this chapter t o the Lie group R 2 . (Results: each symplectic form defines exactly one isomorphism class of extensions; the cocycle 0 yields a combined class of the first type, the other classes are of the second type; 9 "equals" R2 ; as regards classification of the unitary representations arising a t non-zero cocycles cf. the next chapter.) h
188
Chapter 9 9.4. Classify the extensions of T" . (Hints: if K is a non-zero cocycle of the Lie algebra then the group A in
Theorem 9.8 is not central; for the 0 cocycle the f of (9-9) is not necessarily trivial but our method is applicable.) 9.5. Classify the extensions of O(2,R) .
9.6. Show that a simply connected semi-simple Lie group has no non-trivial extensions.
9.7. Let G = S L ( n ,C)/Z, where Z, denotes the center { zZ; z E T , zn = 1 } of S L ( n ,C ) . Prove that the extensions of G form n isomorphism classes, while the number of combined isomorphism classes is [S] 1 (all with commutator cocycle 0 ) .
+
189 10. The Galilean and Poincark Groups
An important principle of classical (or Newtonian) mechanics is that if a so called inertial system is given then the following changes of coordinates yield inertial systems and must not affect the form of certain laws: translation of the origin (in space and time), rotation of the space, motion with a constant velocity vector. Essentially, this was discovered by Galileo, so it goes back to the Renaissance. Identifying the physical space and time with R3 and R , respectively, the rotations and motions above can be described by operators A ( z , t )= ( R z , t ) and B ( z , t )= (z - tv,t) with R E SO(3,R) and v E R3 , while the translations of the origin are of the form C(z,1 ) = (z - z0,t - t o ) with ( 2 0 ,t o ) E R3 x R . One calculates that A B A - ' ( z , t ) = (z - tRv,2 ) , A C A - ' ( z , t ) = (z - Rzo,t - t o ) and B C B - ' ( z , t ) = (z - ( z g - tov) , 2 - t o ) . This implies that the group Go generated by such transformations can be written in the form
Go = R4 x (R3 x SO(3,R))
(10-1)
where t,lie morpliisms in both semi-direct prodncts are identical (considering SO(3,R) as a set of operators over R3 and then R3 x SO(3,R) as a set of operators over R4 = R3 x R in the way described above). Some call 60 'the Galilean group', others call it tlie 'connected Galilean group'. Denote tlie space and time reflections by S and T , i.e., S ( z , t ) = ( - z , t ) and T ( z , t )= ( ~ ~ - 1. ) The greater groups . { I , T } = GO . { I ,S,T ,S T } are = GO . { I , S} and G = also relevant to physics; we shall call them and Go 'the Galilean groups'. In the other direction, if we exclude the translations of the origin of time thcn we obtain a subgroup Gs := Ker 2 4 x (R3x SO(3,R)) (we denoted by 2 4 the fourth coordinate function, i.e., tlie one assigning the time to the points of the space-time). Some call this subgroup tlie simple Galilean group. It is easy to check that
Gs E (R3 x
(10-2)
R3) x SO(3,R)
where the elements of SO(3, R) act on the two copies of R3 as the linear operators they are. The time translations form a subgroup isomorphic to R and they commute with space translations and rotations, while their action on motions is the following: if B ( z , t )= ( z - v t , t ) and C ( z , t ) = ( z , t - ~ )then C B C - ' ( z , t ) = ( z - v ~ - v t , 2 ) , i.e.,
(10-3)
Go
P
Gs x R
where T E R and ((: GL(4,R) , one can write
(10-4)
G = Go
L',
with R ) E 6,
X Gd
T ( ( , 11,
R)T-' = ((
+
TV
(cf. (10-2)). Sirice
= R4 X ((R3 X SO(3,R))X
, v,R) G d :=
{ I ,s , r , S T } C
Gd)
where the expression (R3xSO(3, R)) X G d is considered as a subgroup of GL(4,R) . Observe that ST is central in this group but not in G . We consider the Galilean
190
Chapter 10
groups as Lie groups by regarding the semi-direct products in (10-4) in the sense of Lie groups. In the special theory of relativity the Galilean group is no longer assumed to preserve inertial systems; instead, the subgroup (R3>a SO(3, R)) >a G d is replaced by 0 ( 3 , 1 ) = LO >a G d , where LO is the unit component of O ( 3 , l ) . We shall call &I ‘the connected Lorentz group’, while we call the three groups LO, C1 := Lo M{Z, S } and O ( 3 , l ) ‘the Lorentz groups’. Some call Lo “the” Lorentz group and the others ‘extended Lorentz groups’. We mention that the following different notation is widespread enough: C = O ( 3 , l ) , LT = C1 and C: = Lo . Furthermore, some prefer considering the time coordinate of a particular system as 10 (instead of 2 4 ) and, accordingly, the space-time is regarded as R x R3 instead of R3 x R . We call the Lie groups
(10-5)
P o : = R 4 ~ C o , P1:=R4>aC1=Po>a(I,S} and P := R4 >a O ( 3 , l ) = PO x G d
‘the Poincare groups’. Similar ambiguity of names and notations is found in the literature about PoincarC groups as Lorentz groups. It is not hard to check that the Lie algebra of O ( 3 , l ) is simple and isomorphic to that of S L ( 2 ,C ) , hence a comparison of the centers yields (10-6)
Lo 21 S L ( 2 , C ) / { I ,- I }
as real Lie groups. We note that S L ( 2 , C ) is simply connected. Of course, one can choose a particular isomorphism realizing (10-6) and calculate it explicitly. The present chapter is devoted to the classification of the continuous projective representations of the Galilean and Poincard groups. The first step is to find the extensions of them.
Proposition 10.1. The isoniorphism classes of the extensions of the groups G1 = 2 2 and G2 = Z2 x Z2 are the following. For G1 = ( 1 , s ) and r d = G1 we just have the trivial c h s (for which w 1 is a cocycle), while for r d = { I } we have one other class whose only cocycle is described by w ( s , s ) = - 1 . So we have 3 classes for G1 . For G2 we have 2 classes if r d = G2 and 4 classes for each proper r d , this yields 14 classes. Fix a normal subgroup r d of index 2 or 1 in G2 and introduce the notations x ,y , t for the non-unit elements of G2 so that x E r d ; let w ( z , x ) = w(x,y) = w ( x , z ) = 1 ,
Y) = W ( Y , Y) = r1 , w(y, z ) = w ( z , z ) = r2 w ( y l z ) = w(t,x)= rlr2 4 2 ,
(10-7) and
where r1,r2 E (1, -1) . The corresponding 4 cocycles are mutually inequivalent if = ( 1 , x ) and fall into two classes labelled by the product 1-11-2 if r d = G2 .
rd
The Galilean a n d PoincarC Groups
191
Each isomorphism class is invariant under an ti-isomorphism. P R O O F : Let 0 be an equivalence c ~ a s sof cocycles for some ( G d , f d ) as described by (9-16) (we mean by 'cocycle' a function satisfying (9-17) and (9-18)). It is clear from (9-16) t h a t an w E R exists such t h a t
On the other hand, if s 2 = 1 and s $ f d then w ( s , s ) E { I , -1} for any w by (9-20) , and this w ( s , s) is determined by R because of (9-16). To complete the proof about Zz we should just check t h a t w ( s , s ) = -1 defines a cocycle with respect to f d = { l} . Observe that if one of a , b , c in (9-18) is the unit, then (9-18) follows from (9-17) . It, remains to consider a = b = c = s , then t h e equation reduces to w ( s , s) = w ( s , s ) which holds for w(s,s) = -1 . Turn to G2 and i t s normal subgroup r d . If 7 : G2 I-+ T equals 1 at 1 , x ,z ldzy) = , so by (10-8) we can choose w E f2 from any class such 7( X 117 (Y)" t h a t w ( x , z ) = 1 = w ( z , y ) . Furt,hermore, if f d = G2 then we achieve w ( t , t ) = 1 at the same time. Fix such an w . Since s2 = 1 for any s E Gz , the special cases of (9-18) with a = b and b = c yield
then
w(a,a )
(10-9) (10-10)
= w ( a , a c ) . w ( a , c)" ,
w ( a b ,6 ) . w ( a , b ) = w ( b , b)"
.
Of conrsc, in our case eithrr w ( b , b ) E { 1 , - I } for each b or f d = G2 , so the right hand side of (10-10) equals w ( b , b ) . Writing a = z , b = y and c = y into these formulas we obtain and
1 =w(z,z)
because w ( z , y) = 1 b = z yicltls
.
w(z,y)=w(y,y)
Since w ( z , 2 ) = 1 , a similar applicat.ion with a = x and w( y,
2)
= w ( z ,2 ) .
Writing a = y and c = z into (10-9) we now obtain W(Y,
Y) = W ( Y l Z > 4 Y l .IY '
= w(z,2) .W(Y,
.IY
I
and r2 := W ( Z , Z ) then w ( y , z ) = r l F (for if y $! r d then r 1 , r 2E { l , - l } ). 'The choice a = y b = z in (10-10) yields w ( z , z) . w ( y , z) = 1 , hence w ( z , z) = -1'2 . T h u s if f d = { 1, x} then w equals one of the four functions we defined in (10-7). If f d = G2 then 1'2 = 1 for our w and and w ( z , z ) = 6. Apply (9-18) with a = z , b = y and c = 2 ; so w ( y , x ) = we obtain w ( z , x ) .w(x,y)= w ( x , z ) . w ( y , x ) , i.e., P I = r1 . T h i s means T I = 5 1 . Tlnts we have proved t h a t w satisfies (10-7). Conversely, assume t h a t a function w is defined by (10-7) and (9-17). Any value of w is k1, this simplifies the checking of (9-18). In particular, we need not care for f d . One can verify (9-18) so if
1'1
:= w ( y , y )
r, = { l , c } and
Chapter 10
192
by immediate calculation but we prefer a smarter approach. First of all, if we set ~ ( t=) ~
( 1= ) 1 and '(y)
= '(2) = i then d ( s , t ) :=
'(st) '(5)
..'(t)
equals the w of
(10-7) constructed with T I = r2 = -1 and if w1 is defined with ( r 1 , r z ) = (1,-1) then 9 . w1 equals the w derived from ( q ,r2) = ( - 1 , l ) . Hence it is enough t o prove that this last w turns {1,-1} x G2 into a group with the operation (9-19). We assert that this structure is isomorphic t o the dihedral group 0 4 . This is defined as the group generated by a reflection R in a line through the origin and by the rotation J around the origin by a/2. In a suitable basis we have
Set cp(1,l) = I , cp(l,+) = R , cp(1,y) = J , cp(1,t) = RJ and cp(-l,a) = -cp(l,a) for all a E G2 . Since -I = J 2 is central in 0 4 and J R = -RJ , we obtain that t h e w defined with r1 = -1 and r2 = 1 describes the group cp-'(D4). Since w ( s , s ) is determined by the class for s 4 f d (because s2 = 1 ) and r l = w ( y , y) , 7-2 = ~ ( zt) ,, we have that the four w ' s are mutually inequivalent if f d = {1,2}. Now let f d = G2 , w , w 1 be two such cocycles and suppose that 71implements an equivalence between them as described by (9-16). Then
i.e., "the rlr2 of w1 " equals that of w . We already showed an 71 establishing the asserted equivalences. The statement about anti-isomorphisms follows from the fact, that each cocycle class of these groups contains a real cocycle. Lemma 10.2. T h e Lie algebra o f the Poincark groups h a s no non-trivial cocycle classes. We note that a great part of [Wig] is devoted t o an elementary proof of this fact (and the smaller part has become the dawn of whole theories).
PROOF: By Proposition 9.7 we have to analyze those syiiiplectic forms K: on R4 which are invariant under the Lie algebra of the Loreiitz groups. Let n ( z , y ) = C4=1 C",=,ajkzjyk and A = ( a j k ) be the corresponding matrix. Then B'A = - A B for B E SO(3,l) (where := Bkj ), hence also for the complex combinations of these B's. This is equivalent to the condition that U'A = AU-' for all U E G where G is the corresponding connected subgroup of GL(4, C ) . Since the complexification of the form defining O ( 3 , l ) is symmetric and non-degenerate, G is spatially isomorphic to SO(4, C ) . Thus G is an irreducible set of invertible operators and so its each invariant complex bilinear form is a scalar multiple of its defining symmetric form (by a well-known version of Schur's lemma). Hence K is symmetric, but it is also symplectic, so K = 0 and the lemma is proved.
The Galilean and PoincarC Groups
193
Lemma 10.3. Let g be the Lie algebra of the Galilean groups and gs be the Lie algebra of the subgroup Gs.Denote by t, ,K,tD and o the subalgebras in g corresponding to the subgroups of time translations, space translations, motions with constant velocity and rotations, respectively. Define a bilinear form (., .) : ,K x t~ H R through the natural identification of r and t~ with R 3 : ((,v) = &Vk (physically the iinatural” identification above means the choice of a measure unit of time, for then a velocity vector corresponds to the space vector covered during a time unit). For each p E R define a symplectic form K,, on g by setting
xi=,
K/l((T,E>V,T),
( M , v l , q ) = p . ( ( E , v ’ ) - (E’ ,v)>
where the linear space g is considered as t @ r @ tV @ o . They and their restrictions to gs represent the equivalence classes of cocycles of the Lie algebras g and g, .
PROOF:t @ @ tV and r @ IU are the radicals of g and g, , respectively, o acts as S o ( 3 , R) on r and tV through their identification with R 3 , and o commutes with t . Fix a class of cocycles and choose a corresponding ad o invariant cocycle K on the radical by Proposition 9.7. In the same way as in the previous proof, the complexification of K is a symplectic form invariant under the operators (T,<,v ) -+ Ru) whenever R E S O ( 3 , C ) . Since for any ( one can find an R such = -( , we must have K ( T , < ) = 0 , and similarly K ( T , V ) = 0 , so that the t-component does not count in K . The method of the proof of the previous lemma is applicable to the restrictions of the complexified K to C @I r x C @ I[ , to C @ tV x C @I IU and to C @I ,K x C @ tV , and we obtain that these restrictions are scalar multiples of the symmetric form defining SO(3, C ) because SO(3,C ) is an irreducible set of operators. Since the first two restrictions must be symplectic and K ( v , [ ) = - K ( < , V ) , we obtain that each class of cocycles must contain some K,, (for both g and gs ). Obviously, each K,, is symplectic, satisfies (9-12) and its restriction to the radical is ad o invariant. This radical is commutative for g J , while for g we have [t, IU] = ,K and the other pairs formable from the decompositon t @ ,K @ tD are commuting. Since K,, also vanishes for most pairs, the checking of (9-11) is reduced to the case when two of z,y, z is taken from tV and the third one from t , and in this case it looks like K , , ( [ T , 2111 , v 2 ) = K ~ ( [ T 7121, , v 1 ) . It follows from (10-3) that [T,v ] = T . v through the identifications of t, K, and tV with R, R3 and R 3 , respectively, hence the above holds indeed. It remains t o check that K,, j. K , if p # v . We have K,, - K , = K,,-,, and it does not vanish on r x t~ though this is a commuting pair, so this difference can not be written in the form cp([.,.]) and the proof is complete.
(r,m,
On the base of Proposition 10.1 and Lemmas 10.2 and 10.3 it is not hard to classify the extensions of the Galilean and Poincard groups. Let and P be the simply connected coverings of G o , Gs and Po,respectively. They can be described as similar semi-direct products, just replace SO(3,R) by SU(2) in (10-1) and (10-2), and Lo by S L ( 2 , C ) in (10-5) and define the morphisms of the corresponding semi-direct structures through the covering mappings SU(2) H
c , <,
Chapter 10
194
SO(3,R) and S L ( 2 , C ) H Lo (these mappings are far from unique; we should choose one). For the PoincarC groups we can choose K = 0 and then the group H from Theorem 9.8 is a direct product R x P and f = 6 o e where e ( t , g ) = g and 6 : P H Po is the covering morphism. The scaling of the direct component R can be such that expH it = t for all it E iR with the notations of Theorem 9.8. We have Kerf = R x Z where Z has two elements, and so the possible A l ’ s are 2sZ x 2 and the cyclic group generated by ( s , - l ) if -1 denotes the nonunit element of Z . The corresponding two extensions are not isomorphic or antiisomorphic because of the following. T h e square of the Lie algebra of H equals the Lie algebra of the subgroup P and hence this connected subgroup is invariant under any automorphism; and the first A1 above contains a non-unit element of P while the second one does not. We included this argument mainly because it hints a t a method applicable quite often; in the present case we could have pointed to the fact that the first A1 is isomorphic to Z x 22 (in particular, it contains an element of second order) while the second one is isomorphic to Z , so even the homotopy groups of the Lie groups of these extensions are different. For the trivial cocycle class of the connected and the simple Galilean groups the above can be repeated verbatim (referring to Z x 22 Z in the end). Now consider a K = K , with ,u # 0 . Let g“ be the Lie algebra created from the Lie algebra g of 6 and from K , by (9-5). Consider g as a part of g“ (i.e., “ q = i d ” in (9-5)). It is clear that L = J @ tu @ iR is an ideal and t @ o is a complementing subalgebra (analogously, o is a complementing subalgebra in the K , extension of the Lie algebra of 6, ). The ideal L is such that [ L ,[ L ,L ] ]= 0 , hence the Hausdorff-Campbell formula takes a very simple form and the simply connected group corresponding to L can be described as follows: on the set R3 x R3 x R consider the multiplication (10-11)
, , t 1) . “2
(€1 V l
-
t
v2 112) =
(€1
+ (2
-
, 211 + 02
1
tl
+ 12 +
g ( ( F 1 , .2)
-
-
(12
1
.l))
)
,
this is a Lie group and its Lie algebra is isomorphic to L through the correspondence exp(lJ) (t1,0,O) , exp(tv) (O,tv,0) and exp(tis) (0, 0 , t s ) where on the left we mean the one-parameter subgroups corresponding to the elements [ E F , v E tu and is E iR , while on the right we use the canonical identifications of J and tu with R 3 . Of course, one can check that (10-11) defines a Lie group without any knowledge of the IIausdorff-Campbell formula and then the above isomorphism is easily established. We denote this Lie group by H , ; it can be called the ‘Heisenberg group of freedom 3 and with scaling p ’ . Then a semi-direct product H , x (R x S U ( 2 ) ) (or H , M S U ( 2 ) for 6, ) can serve as H in Theorem 9.8 and one can also choose and calculate explicitly a corresponding morphism from R x S U ( 2 ) (or from S U ( 2 ) ) into the automorphism group of H , . For us just the following facts are important. Choose a covering morphism 9 : S U ( 2 ) I--+ SO(3,R) , then the f of Theorem 9.8 can be such that flSu(2) = 1.9 , the restriction o f f to the subgroup R in R x S U ( 2 ) is the canonical identification of R with the subgroup of time translations in and flHp is the taking of the first 6 coordinates from 7 such that the first three establishes the canonical identification of R3 with the
T h e Galilean a n d Poincare' Groups
195
subgroup of space translations and the second three the canonical identification of R3 with the subgroup of motions with constant velocity. T h e scaling of i R c L is such t h a t expH i t = ( O , O , 2 ) where on the right we mean an element of H , . I t follows from the structure of ge that in this case for (7, R ) E R x S U ( 2 ) we have
for all ( < , u , i s ) E L . T h e center Z N 2 2 of SU(2) is thus central in H and K e r f = R x Z where R denotes the corresponding subgroup of H , . Similarly to the case of trivial cocycles we have the same two possibilities of A1 and t h e corresponding extensions are such that even their Lie groups are not isomorphic (by the same reasoning). On t h e other hand, one obtains t h a t the constructions above with - p yield a pair of extensions each of which is anti-isomorphic to one of the ext,ensions obtained from p (and the homotopy group decides which to which). So we can restrict our attention to p > 0 when interested in continuous projective representations of the Galilean groups n o t connected with the trivial class of commutator cocycles. We mention that even though the extensions with p > 0 may seeni more complicated than the ones with a trivial commutator cocycle, their representations are much simpler. Having disposed of t,he extensions of PO , and 6, turn to t h e analysis of the extensions of the non-connected groups. For the Poincar6 groups the situation is very simple. Then we should only consider K = 0 and t,he Lie algebra is the square of itself, so any u satisfying (9-26) must be 0 . Then (9-27) is also satisfied and the corresponding automorphisnis of the simply connected covering of (Po)' are of the form @ ( t , g ) = (c(s)t, @o(s)g) where c is the *l valued morpliisin with kernel I ' d and @(s) o 21 = 21 o @ o ( s ) where 21 : H PO is t h e covering niorphism. These @'s leave both possible A l ' s fixed, and so by Proposition 10.1 PI and P have 6 and 28 isomorphism classes of extensions, respectively. Each of these classes is invariant, under antJi-isoniorphism. For the Galilean groups observe t h a t , witah the notations of Lemma 10.3,
Since each K~ vanishes on o x g and o is invariant under Ad S and Ad T , we can see t h a t if u satisfies (9-26) with some K , then gs c Ker u ( u ) for all a . From now on we shall not discuss the case p = 0 in details because i t does not seem to have physical applications, but interested readers may try their hand at it (cf. the Exercises). Since [g,g] = g, , the sides of (9-26) must vanish. Hence if p # 0 then we obtain c ( S ) = 1 and c(T) = -1 from (9-26) because on @ tD we have Ad(S) = -I and Ad(T) = I @ - I . T h u s fd can only be { 1 , S } in this case (for groups with unit component,s Go or 6, alike). Observing t h a t g, is invariant under Ad G d , Ad SI, = I and Ad TI, = -I we get from (9-27) t h a t
+ +
2 u ( S ) = (Ad S ) * v ( S ) c ( S ) V ( S= ) ~ ( 1=) 0 , -2u(T) = (Ad T ) * v ( T ) c ( T ) v ( T )= ~ ( 1= )0 .
Chapter 10
196
Thus v must be trivial. Then (9-27) obviously holds and (9-26) is also satisfied because r d = (1,s). We can define automorphisms @ ( S )and @ ( T ) of the simply connected extension group such that for ([, v , is) E L and (7, R ) E t @ o we have (10-13)
d ( P ( S ) )( t ,v , is,r , R ) = (-€, - v , is, r, R ) and d(@(T)) ((,v, is,r , R ) = ([, - v , -is, - T , R ) .
P(S) and @ ( T )are identical on the subgroup SU(2) corresponding to o , hence on Z and therefore map both possible Al's onto themselves. Thus we obtain extensions this way. Using Proposition 10.1 we infer that for each K,, # 0 there correspond 2 isomorphism classes of extensions of the Lie groups G1 and Gs x { 1,S } , and 8 isomorphism classes of the extensions of 6 and Gs >a G d . Anti-isomorphism takes these classes to the classes with commutator cocycle class containing K - ~ .
Calculate the sets 9*and 9 from Proposition 9.6 for the extensions of the Galilean and Poincard groups. The square of the Lie algebra of the extension equals the Lie algebra of the subgroup P in H for the PoincarPl groups, and iR@gs for the extensions of the Galilean groups with K,, # 0 (we refer to Exercise 10.2 concerning the case p = 0 ). So with a non-zero K,, we can only have extensions of the second type (in the sense of Proposition 9.6) and 9 = 9*. For these extensions of Gs 9 is trivial, while for Go the elements of 9 can be labelled by real numbers s so that d, = 1 on the normal subgroup p - l ( G 3 ) and d,(~.Al)= e i s T , where on the left T is the element of H corresponding to a time translation in the distinguished subgroup R x S U ( 2 ) , and on the right T is the corresponding real number. In the case of Po the two extensions can be described as follows. If 9.1 : P H Po is the covering and we denote by -1 the non-unit element of Ker 29 then consider the Lie group T x P and its central subgroups 21 = { ( 1 , l ) , (1, -1) } and Zz = { ( 1 , l ) , (-1, -1) } . The Lie groups in the extensions will be ( T x P ) / Z j ( j = 1 , 2 ), ~ ( z=) ( z , l ) Z j for z E T , and p ( ( z , g ) Z j ) = d ( g ) . We observed above that any element of 9' has a tangent vanishing on the Lie algebra of P Iso one easily checks now that the elements of 9' are exactly the morphisnis dn((z,g)Zj) = z" where n E Z in the first case and n E 2 2 in the second case. So both extensions are of the first type, we must take c = dz , and 9 is trivial. Consider the non-connected Poincard groups. Choose an extension S of Zz or Zz x Zz by Proposition 10.1, then the construction described in the previous chapter on p . 185 can be written as (10-14)
K = ((T x P ) / Z j ) M S
with
6(a)((z,g)Zj) = ( z " , aga-l)Zj
where aga-' can be computed from A d a . Let d E 9* and denote the corresponding mapping on K by d , too. The restriction of d to the unit component must equal some d, , so d ( ((z,g)Zj). (w, 1 ) ) = (zw)". (9-4) takes the form
The Galilean a n d Poincare‘ Groups where z,yE (T x ? ) / Z j
, so t h e function
(10-16)
6(ab) = W ( U , b)” . 6 ( ~ .6(b)” ) .
197
& ( a )= d(1 . ( 1 , a ) ) must satisfy
Conversely, if this holds for some 6 then we can construct a d E 9* by setting d ( ( ( . z , g ) Z j ) . ( w ,a ) ) := (rw)“ . 6 ( a ) . Since we are interested just in determining the type and finding 9 , we have only to consider the cases n = 2 , O , a n d then w ( a , b)” = 1 for any w described in Proposition 10.1. If we consider t h e “unitary extensions” of PI and P (i.e., for which T is central) then we have two and four solutions of (10-16), respectively, because for 2 2 we may take S(S) = f l , while for Z2 x 22 = { 1,z, y, z } the non-trivial 6’s are characterized by the property that they equal -1 a t two of z , y , z and equal 1 otherwise. For the non-unitary extensions of PI 6(S) can be arbitrary. Consider a non-unitary extension of P and denote by z the element of {S,TIS T } which belongs to r d . Then the solutions of (10-16) are the following: 6(z) = f l and if { y , r } = { S , T , S T }\ { z } then 6(z) = S(z)S(y) is the only stipulation on 6(y) and 6 ( r ) . Observe t h a t (10-15) is t h e general description of (9-4) in t h e context of the construction of p. 1 8 5 . One obtains 9 for the (/I # 0)-extensions of G, M { I , S } and Gs x G d as special cases of the above (with n = 0 and r d = (1,s)). For as described earlier, and (10-13) determines and we have to consider 6.Let d, be the morphism on Go described above (i.e., d , ( ~ A. l ) = e i s r and (Gs)e c Ker d, ), and 6 be a solution of (10-16) with n = 0 and r d = { I , . T h e n d ( z . ( w , a ) ) := d,(z) & ( a ) defines an element of 9 because d,(i(a)z) = d,(x)” by (10-13). T h u s we have described 9 for the ( p # 0)-extensions of G1 a n d G .
s}
If we have a non-unitary extension of a (necessarily non-connected) Galilean or Poincarl group and z E G d \ r d then y2 = f l E T for y = ( 1 , z ) N E GO)^ >a S ) / N (from the construction on p . 185), and the sign is determined by w from Proposition 10.1. So (9-2) takes the form J 2 = &Z (the sign is determined by the extension). In some cases one can find another y “lying askew” in ((Go)“x S ) / N with y2 = 1 even if y2 = -1 for the y’s considered above. This is not very important because anti-unitary operators J with J 2 = - I are tractable enough. We remark that a complex Hilbert space with a distinguished such J is t h e same as a ‘quaternian IIilbert space’ with a distinguished choice of the subset C and of the element j in the skew-field of quaternions (in particular, such a J can not exist on a IIilbert space of finite odd dimension). When seeking unitary representations of the extensions of the connected Galilean and Poincarl groups t h e role of the two possible Al’s is exactly t h e following. T h e corresponding unitary representation U of the simply connected covering must be such that (besides U(expH it) = e i t I ) we have U(-1) = I in the first case and U(-1) = - I in the second case where -1 denotes the non-unit central element of S U ( 2 ) or S L ( 2 , C ) when these groups are canonically imbedded into t h e corresponding H’s. So the continuous projective representations of the connected Galilean and Poincard groups are the same as t h e ones implemented by those continuous unitary representations U of the corresponding simply connected
Chapter 10
198 extension groups H for which (10-17)
U(exp, it) = e"l
U(-1) = f Z
and
.
Theorem 10.4 (Stone-von Neumann Theorem). Let p > 0 and consider the Heisenberg group H , defined in (10-11). If U is a continuous unitary representation of H , such that U(O,O,t) = e"1 for all t E R then U is a multiple of the Schrodinger representation of mass p . This is defined on L2(R3) by setting
W is an irreducible continuous unitary representation.
<
PROOF:Let A = { ( [ , O , O ) ; E R 3 } and N = { (0, v , t ) ; ( v , t ) E R 4 } , then H , = N x A and N = R4 also as a Lie group. For E A and (v,t) E N we have <(v,t)<-' = ( v ,t p ( v , < ) ) . Hence the orbits of the action of A over fi are the following: the one point sets in 0 := { C E fi ; C(0,t)= 1 V t E R } (in the usual identification of fi with R4 we have 0 = Kerx4 ), and the sets 0 where C 6 0 . Let E be the spectral resolution of U I , . T h e requirement U(O,O,t) = e i t Z then can be written as JR4eitZ4 d E ( x ) = e " 1 . Let E' be the spectral measure on R defined by E ' ( B ) = E(R3 x B ) for any Bore1 set B , then JR4e i t 2 4 d E ( z ) = J"eit2 d E ' ( x ) by the rule about substitutions in integrals. Applying the unicity part of the SNAG Theorem to R we can see that E'({l}) = Z and E is concentrated on C 0 where is the character C(v,t) = e i t . Observe that what we really need in Proposition 7.9 is not the assumption ' U is a factor representation' but it,s consequence that the central disintegration of U l , is concentrated on a single orbit. Thus Proposition 8.5 applies and U can be described by (8-14a). We mention t,hat this could have been inferred more immediately from the SNAG Theorem applied on R3 and from the Imprimitivity Theorem (and, as a matter of fact, a major step towards the Imprimitivity Theorem was when G. W . Mackey proved it in this special case). We did otherwise to emphasize the connection with the more general theory. Of course, the same argument works with Heisenberg groups of freedom n instead of just 3 . Since the stabilizer of C is trivial, we can see from (8-14a) that W is nothing else than the representation obtained by inducing from the one dimensional representation of the trivial subgroup {0} of the semi-direct component A and with our C above; in particular, W is an irreducible continuous unitary representation. Furthermore, U is almost the same, just we should induce from an arbitrary representation of the trivial group. Let K be the space of this representation and consider a maximal orthogonal family of one dimensional projections on K: . They commute with the trivial representation and therefore induce a maximal orthogonal family of projections on the induced space. On the other hand, it is obvious that the restriction of U to the space of a projection induced from a one dimensional projection is equivalent to W .
<
+
<+
+
<
The Galilean a n d Poincare' Groups
199
T h e theorem above severely restricts t h e p # 0 possibilities of continuous projective representations of the Galilean groups. First we construct a "minimal" representation of the simply connected group H = H , x (R x S U ( 2 ) ) as described around (10-12). T h i s representation acts on L2(R3) and continues t h e Schrodinger representation W of the normal subgroup H , , we shall also denote it by W . Set (10- 18) (10-19)
W ( R ) f := f o (d(R))-' and W ( T ):= e x p ( - i ~ ( 2 ~ ) - ' A )
where 19 : S U ( 2 ) I-+ SO(3,R) is the covering distinguished by our construction of t h e extensions, t h e subgroup of time translations is identified with R similarly, f E L2(R3) , A is the Laplacean operator on L2(R3) (it corresponds to the multiplier of the function - 1 1 ~ 1 1through ~ Plancherel transform) and (10-19) means t h a t t h e generator of the one-parameter group T -+ W ( T )equals the skew adjoint operator $ A . Here (10-18) may seem natural enough, while onc may derive motivation for (10-19) partly from some calculations in the enveloping algebra of the Lie algebra of cj and partly from physics. One easily checks t h a t (10-18) defines a continuous unitary representation of S U ( 2 ) (with W(-1) = I ), while (10-19) defines a continuous unitary representation of R by definition o r , if one considers its right hand side through spectral theory, this follows from the easy direction of Stone's theorem. A simple calculation yields t h a t each W ( R ) from (10-18) conimut,es with the Plancherel transform. Since they also commute with the unitary inultipliers exp(is 1 1 ~ 1 1 ~ for ) any s E R , we can see t h a t (10-18,19) defines a continuous unitary representation of the Lie group R x SU(2) . One may ask why do not we pass altogether to the other image through Plancherel transform. T h e reasons are: tradition and the accepted language of physics (the latter has something to d o with visualizing quantum mechanics). An easy calculation and (10-12) yields that W ( R ) W ( z ) W ( R ) - '= W ( R x R - ' ) for R E S U ( 2 ) and z E H , . To get the same thing for time translations we apply the Plancherel transform V :
V W (
P
-
(<, x) - 2 ( u , ~ ) .)f ( T
+ /Lu)
,
comparing this with (10-12) we obtain the asserted statement. So we have a continuous unitary representation W of H over L2(R3) which is, of course, irreducible because WIHp is already irreducible. Now let U be a continuous unitary representation of H satisfying (10-17). T h e n UIHp N 1, @ WIHr with some cardinal cr by Theorem 10.4. Changing the notation we may assume equality here (and t h e new U is equivalent t o t h e original one). Let W, = 1, @I W and set Vo(s) = U(s)W,(s>-' for s E R x S U ( 2 ) . Since both U and W , are representations and they agree on H , , we have Vo(s)E W,(H,)" for all s . Since WIHr is irreducible, each Vo(s) is of the form V ( s ) @ I I with some unitary operator V ( s ) over ['(a) (cf. Proposition 6.6). Consequently, Vo(s)W,(t) = Wa(t)V0(s) for all s , t E R x S U ( 2 ) . But U ( s ) = Vo(s)W,(s) and W , are representations, so we obtain t h a t V is a representation. It is evidently
200
Chapter 10
continuous, and V(-1) = f I because U satisfies (10-17) and Wa(-l) = I . T h e argument above also works for 6, instead of 60 , then we get a representation V of the group S U ( 2 ) with these properties. The continuous unitary representations of S U ( 2 ) are very well known (see Theorem 6.22 for the theory and Exercise 6.6 for the particular result). T h e condition V(-1) = fl is satisfied if and only if V does not contain irreducible representations of odd and even dimensions a t the same time. For T E R we have V(T) E V(SU(2))‘ , so by resorting t o Proposition 6.6 again we can neatly separate VI, and we obtain the following theorem.
Theorem 10.5. Those continuous projective representations of the connected Galilean group Go and the simple Galilean group G8 for which 0 is not a commutator cocycle are the following (up to equivalence). Choose a representation V, from the equivalence class of irreducible n dimensional continuous unitary representations of S U ( 2 ) . Let a l , a’, a g , . . . be a sequence of cardinals and consider the Hilbert spaces w
31, = @(!’(a,)
!’((an)
L’(R~)>) ,
n=l 03
31, =@(!’(an)@!’(2n-
1)@L2(R3)) .
n=l I E H, and s E SU(2) set U ( z s ) = $, I @I Vz,,(s) @ W ( z s ) and U ( I S )= $, I @ V~,-I(S) @ W ( I S ) over these Ifilbert spaces, respectively. Here W depends on p > 0 as was described in Theorem 10.4. These represent the
For
equivalence classes of continuous projective representations of G8 with non-trivial cocycle class. Setting U ( T ) = $, X,,(T) @ I @ W ( T ) where X , is an arbitrary we obtain the continuous continuous unitary representation of R over !’(an) projective representations of 60 and two such representations are equivalent if and only if both of them are even (odd) and d X , N C . I dXA for all n where C is an imaginary number independent of n and N means unitary equivalence (in particular, a,, = a; and only those n’s must be considered for which this cardinal is not 0).
+
We leave the detailed proof about equivalence to the readers but mention the main tool is Proposition 6.6. If just one a , differs from 0 and that equals 1 then we are dealing with an “ergodic” projective representation: no non-trivial projection is fixed under the action. These representations correspond to the ‘quantum mechanical elementary particles’, p is the mass of the particle and if N is the multiplicity of the Schrodinger representation (i.e., N = 2n or N = 2n - 1 with our notations) then is called the ‘spin’ of the particle. Try t o apply our method to the non-connected groups. Set
9
(10-20)
W ( S ) f ( z ) = f(-z)
and
W(T)f= 7 .
We have W(S)’ = I = W(T)’ and one easily checks that W ( S ) W ( g ) W ( S= ) W ( S ( S ) g ) and W ( T ) W ( g ) W ( T= ) W ( S ( T ) g ) where !P is described by (10-13) and g E H ( H and WI, are the objects considered in the proof of Theorem 10.5).
The Galilean and Poincare‘ Groups
20 1
Consider the construction on p. 185 and write e : G d H K / N , e(a) := 1 . (1,u) . N . Then the continuous morphisms U : K / N H U’(7i) are determined by the two objects UO = U o 6 and u d = U o e where d : H H GG is the covering morphism. If U is a morphism connected with a continuous projective representation then UO satisfies (10-17) and so it must be a unitary representation of structure described in Theorem 10.5. Furthermore, we have
’ = UO (*(a)g)
U d ( a)uO (S)ud (a)-
and
Ud(ab)= u(a,b)Ud(a)Ud(b). Conversely, if UO is a representation from Theorem 10.5 and Ud is a mapping such that U d ( I ) = I , Ud(S)E U(X),U d ( T ) E U’(7-l) \U(7i) and the above is satisfied then Uo and Ud define a continuous projective representation of G . T h e same reasoning with (1,s) instead of Gd yields “the restriction” of these conditions when studying G I , and one can deal with the non-connected extensions of Gs similarly. Now let W,(S) = Z@W(S) where the I refers to the identical operator of the Hilbert space K = $, 1 2 ( a , ) @ 1 2 ( 2 n )or Ic = $, !?(a,) @ 12(2n- 1) , and define W a ( T )by W,(T)(z@y) = T @ W ( T ) y where - denotes the canonical conjugation of K . Then W,(S)2 = I = W,(T)’ , W,(S)W,(g)W,(S)= W,(P(S)g) and Wa(T)W,(g)W,(T) = W,(P(T)g)for g E H . Setting Vo(S):= W,(S)Ud(S) and Vo(T):= W,(T)Ud(T) we obtain unitary operators satisfying
Vo(S)(V(s)8 W(gs))Vo(S)-’= V(P(S)s)@ W ( g s ) and Vo(T)(V(s)8 W(gS))Vo(T)-’= V(S(T)s)@ W ( g s ) for g E H , and s E R x S U ( 2 ) , where V is the representation of R. x S U ( 2 ) over K connected with U o . In particular, Vo(S),Vo(T)E W,(H,)c and therefore Vo(S)= V(S)@ I and Vo(T)= V(T)@ I with some V(S),V(T)E U(K). Thus the above can be reformulated as
(10-21)
R) . V(S)E V ( R x S U ( 2 ) ) c , V(T)V(r,R)V(T)-’ = V(--T,
We have U d ( S T ) ( z@ y) = u(S,T)Ud(S)Ud(T)(z @ y) = w(S,T)V(S)V(T)z 8 W(ST)y and similarly Ud(ST)(z@ y) = u(T,S)Ud(T)Ud(S)(z 8 y) = u(T,S)V(T)V(S)x @ W(ST)y , which implies V(T)V(S)= rlr2V(S)V(T) where r 1 , r 2 are the signs from Proposition 10.1 (writing z = S and y = T into Proposition - 10.1). From S 2 = I = T 2 we obtain in the same way that V(S)’ = I and V(T)V(T)= r l I . So one should seek unitary operators V(S)and V(T)satisfying these three equations and also (10-21) in order to get all continuous projective representations of G having non-trivial commutator cocycle class. We leave the question at that. It is clear from (10-17) that finding the continuous projective representations of Po is equivalent to finding those continuous unitary representations U of
202
Chapter 10
P which satisfy U(-1) = 4 d . The situation is similar with the projective representations of the connected Galilean groups having trivial cocycle class. One tries to apply Theorem 8.7 successively and, in fact, this program works with complete triumph. Moreover, the method extends to the non-connected groups (with some extra work about anti-unitary operators). We relegate the case of Galilean groups to Exercises 10.4 and 10.5. The group O ( 3 , l ) is given by the equation X’QX = Q where
&=
iR : i p,) 0 0 1
0
In other words, X-’ = QX‘Q . Hence X -+ (X‘)-’ is an automorphism of O ( 3 , 1 ) , and since it leaves the space reflection S fixed, it maps Lo and L1 onto themselves. T h u s the orbits of these groups in R4 coincide with their orbits in R4. Determine these orbits. Since the bilinear form (z,y)= z’Qy is invariant under O ( 3 , l ) , its quadratic form (.,I) must be constant on any orbit. For T E R set O ( r ) = { I E R4;( I , I )= T } . For r > 0 O ( r ) is a one-sheet hyperboloid, for r < 0 it is a two-sheet hyperboloid, and O ( 0 ) is a (double) cone. For r 5 0 introduce the notations Ot(r) = { I E O ( r ); 14 > 0 } and O-(r) = { I E O ( r ) ; 14 < 0 } . We assert that the orbits of Lo are O ( T ) with r > 0 , Ot(r) and O - ( P ) with r 5 0 and the trivial orbit (0). The orbits can not be greater because they are continuous images of a connected group. T h u s it is enough to show the following. Let m > 0 . If I = (O,O,m,O) or I = (0,0,O,m) or I = ( O , O , 0,- m ) or I = (O,O,1, 1) or I = ( O , O , 1,-1) then the orbit Lo 2 contains o(?n2) or Ot(-m2) or o - ( - m 2 ) or @(o) or o-(o),respectively. Since LO contains SO(3, R) (acting in the first three variables), we can forget the first two variables and may assume the third is non-negative, i.e., it is enough t o show that LOI contains the corresponding half or quarter of a hyperbola or the corresponding half-line in the 1 3 ~ 1 4 plane. For this the “one dimensional Lorentz group” will be enough. This is isomorphic to R and acts on the plane by the matrices cosht sinht sinht coshl ’ h
(
)
Our statements now reduce to elementary problems of analytic geometry. Since each of the LO orbits is invariant under S , they also constitute the orbits of L 1 . Then the orbits of O ( 3 , l ) must be O ( r ) with r # 0 , O(0) \ (0) and ( 0 ) . We can see that the Lorentz groups act on R4 separably. Let h
a
= ( O , O , 1,0) ,
b = (O,O,l,l) ,
c = (0,0,0,1)
and denote by G,, Gb, G, the stabilizers of these points in LO. In fact, the images of these groups under the mapping X + (X‘)-’ are the stabilizers of the dual action, but this mapping is an automorphism of all three Lorentz groups, so we need not bother about this. Setting J1 = (0) , Jz = ( O , + o o ) . a , J 3 = { b , - b } and
The Galilean and Poincark Groups
203
54 = ( R \ (0)) . c we have a family with the properties described in Theorem 8.7 (except possibly for the tameness of the stabilizers LO, G , , Gb and G , , though we shall see this holds, too). The same J’s serve for the action of L1 but the stabilizers are twice as big; for O(3, 1) the “positive halves” of 53 and J4 should be used instead, the first two stabilizers are greater than those for L1 while the second two coincide with the corresponding stabilizers in L 1 . It is not hard to determine these stabilizers. For 3: = a and 2 = c the subspace {y ; (y,.) = 0 } is a direct complement of the line through 2 and the elements of the stabilizer are decomposed by this direct sum. For c this yields O(3, R) for the non-connected Lorentz groups and G , N SO(3, R) ; for a we obtain O ( 2 , l ) and its corresponding parts, “the Lorentz groups of space-time with two dimensional space”. Finally, one calculates that a matrix in O(3, 1 ) stabilizes b if and only if it is of the form cost sint
-rsint rcost rv rv
-I
-y 1-w -w
w
l+w
+
where t , I ,y E R , r = *l , u = 3: . cost y . sin t , v = y . cos 2 - 2: . sin t and w = { ( I ? y2) . The subset A defined by I = y = 0 is a closed subgroup isomorphic to O ( 2 , R ) (as a Lie group), while t,he subset N defined by t = 0 and r = 1 is a connected closed subgroup such that the coordinates I, y establish an isomorphism with the Lie group R 2 . N A equals the st,abilizer by an obvious calculation, and since the Lie algebra of N consists of the matrices
+
one obtains that this stabilizer is isomorphic to R2 M O ( 2 , R ) with the natural action of O(2, R) over R 2 . This stabilizer is contained in L1 , while Gb is its connected component and Gb 2 R2 M s o ( 2 , R) . If we want to analyze the continuous unitary representations of then the stabilizers L o , G, , Gb and G , must be replaced by their inverse images in SL(2, C) . Therefore we have to examine the covering morphism d : SL(2, C ) H LO a little bit closer. What we need is the fact that d can be chosen so that d - l ( S O ( 3 , R ) ) = SU(2) where by S O ( 3 , R ) we mean the set of these matrices acting in the first three variables of R 4 . This follows from the possibility of finding an isomorphism from 5[(2, C) onto S o ( 3 , l ) which maps m ( 2 ) onto So(3, R) . If one proves 51(2, C) N 50(3,1) by showing a particular isomorphism then it is customary to do this by providing the above property, too; if not then the theorem asserting that Cartan decompositions are equivalent by inner automorphisms helps. Now the Lie algebra 5 4 2 ) N 5 0 ( 3 , R ) 2 (R3,x ) is such that any line through 0 can be mapped to any other one by an inner automorphism, hence if v E SU(2)\(0} and T = min(t > 0 ; exptv = 1 } then exp !j~v = -1 because this evidently
Chapter 10
204
holds for
The SO(2, R) of the first two variables is a subgroup of G, as well as of Gb . where exptw is the rotation by angle t in this S O ( 2 , R ) . Let w = dd-'(w) Then T = 47r for this w by the above. This implies that the unit components of d-'(G,) and d-l(Gb) contain -1. Therefore these unit components cover the connected groups G, and Gb twofold and they equal the whole d-'(C,) and d-l(Gb), respectively. This implies d-'(G,) N S L ( 2 , R) because SL(2, R) is the only connected Lie group with center 2 2 and Lie algebra 5 [ ( 2 , R ) . Let L be the Lie algebra of d-l(Gb) and M be its subalgebra corresponding to N . Then j := 29 o exp = exp odd is an isomorphism of Lie groups from M onto N . Consider the Lie group N x R where R acts over N by
tzt-l = (exptw) . z . (exptw)-l
-
-
.
This is a simply connected group and its Lie algebra can be identified with L by setting exptv ( 0 , t ) and e x p t z ( j ( t z ) , O ) for z E M (considering the Lie algebras as sets of one-parameter subgroups). Hence we obtain a covering ?I, : N x R H d-l(Gb) whose tangent is this identification. We have Ker ?I,nR= 4aZ and Ker II, C Ker(d o ?I,) = 2aZ c R , i.e., d-l(Gb) 2: R2 x T where T acts on R2 by a twofold covering T H SO(2, R) . So we have to consider the following four stabilizers when trying to apply Theorem 8.7 to G = p : SL(2, C) , SL(2, R) , R2 x T and S U ( 2 ) . It seems just the last (and simplest) of these has found physical interpretation. The third one can be disposed of by another application of Theorem 8.7. The tameness of the first two stabilizers follows from the general theorem asserting that connected semisimple matrix Lie groups are even CCR (see, e.g., Theorem 15.5.6 of [Dixa]). T h e same also holds for non-linear connected semi-simple groups by a famous theorem of Harish-Chandra but the proof is much harder (cf. Theorem 9.5.1 of [Dix3]). Note these theorems extend to Lie groups having finitely many components whose unit component is semi-simple by our Theorem 7.12. Observe that the extensions S of 2 2 and Z2 x Z2 contain finite subgroups F such that T . F = S (using the cocycles from Proposition 10.1 we certainly obtain such an F when the T coordinate is restricted to (1, -1}, see (9-19)). Now the group A' of (10-14) is the factor of ( R x 3 ) x S (by the normal subgroup A1 ), and since the representations to be considered are prescribed on R x T I their restriction to the closed subgroup 3 x F determines them. Here any f E F acts on @ ' in the same way as an element of G d , so it is completely feasible to analyze the spectrum of 9 >a F in the same way as we did for 3 (in this case we may have non-connected stabilizers). The spectra of the CCR Lie groups SL(2, C ) and SL(2, R) are rather thoroughly known (for an exposition see, e.g., [Kna]). It seems that the now enormous semi-simple representation theory derived its main starting impetus from Wigner's question: he asked in [Wig] what are the irreducible continuous unitary representations of these two groups (he was interested in this because of the continuous
The Galilean and Poincare' Groups
205
projective representations of the connected Poincar6 group). These questions were answered after about eight years (by I. M . Gelfand and M . A . Naimark for S L ( 2 , C ) and by V . Bargmann for SL(2,R)).
EXERCISES 10.1. Determine the extensions of the non-connected Galilean groups corresponding to the trivial commutator cocycle class. (Result: one can choose Y = 0 for any r d because of (9-28) and we have 6 and 28 classes for the groups of two and four components, respectively.) 10.2. Prove that the extensions in the previous exercise are of the first type (in the sense of Proposition 9.6), and determine D for them.
10.3. Show that the Heisenberg group H , (defined in (10-11)) is tame and describe its spectrum. 10.4. Show that all Galilean groups are tame.
10.5. Describe the spectrum of the connected Galilean group 10.6. Check that the Lorentz groups and the stabilizers of- a , b , c are unimodular, as well as the analogous groups arising at the analysis of P (this simplifies a bit the description of the corresponding induced representations).
206
APPENDIX In this Appendix we want to establish notation (whose usage is facilitated by the Index) and to collect rather standard material for the reader's convenience and also for the sake of easy citation in the main text. Sometimes we shall sketch proofs. It is outlined in the preface what is the level a t which this book is about to start. This Appendix should clarify this question further. We shall mention books which, more or less, cover what we need; but there are many others and our practice is not meant to restrict the reader's choice. A part of Appendix E concerns with delicate results about partial differential equations, and should go logically after Chapter 2 of the main text, but would have been too much of a digression there. We try to sort the material by mathematical branches. Some notations are so widespread in mathematics that we should fix them now. So we use the notations R , C, Z, N t o denote the sets of real numbers, complex numbers, integers and nonnegative integers, respectively, together with the usual algebraic operations and, for R and C , the usual distance on them. In this book the letter F will always mean a field which is either R or C . We use the letter T t o denote the group R/Z as realized by the multiplicative group of complex numbers having absolute value 1 (so T is t,he 'unit circle'). Z, denotes the cyclic group of order R . We denote the first partial derivatives on Rn by 8 1 , .. . and means the derivative a t the direction v , so i f f is a differentiable function then
,a,
a,,
j=1
Partial derivatives of higher order are abbreviated with the help of multi-indexes when applied to smooth functions. A multi-index is an element of N" and if cy is a multi-index then 8" = 8:' . The altitude of a multi-index cy is Icy( = cyj , so it equals the order of 8.. Sometimes we also use the symbols of partial derivatives to refer to the components of some direct product, the factors of which may be Banach spaces in general. We use the symbol I" to denote the homogeneous polynomial cQ = 2': . . . I"" on F". If X is a set and H c X then we always denote by 1~ the characteristic function of H , i.e., l ~ ( c=) 1 if c E H and 1 ~ ( c=) 0 if 2 6 H . The domain and range of a mapping are usually denoted by V and R .
cj
207 A. Topology General reference: [CsA], [Kel], [Kur] . We shall meet topological questions in connection with topological vector spaces and locally compact groups. These also carry uniform structures, and it is sometimes more convenient t o think of them as such, although the algebraic structure enables us to handle them while not knowing anything about a ‘uniform structure’. The notion of a ‘topological space’ can be defined in several equivalent ways. One of the quickest is to say that we have a set X and a collection 7 of its subsets, called the open sets, satisfying the axioms that 8 and X are open, the intersection of finite number of open sets and the union of any collection of open sets are also open. Then the complements of the open sets are called closed sets, the interior i n t ( A ) of an arbitrary set A c X is defined to be the union of open sets contained in A , and we say that A is a neighborhood of x E X if I E i n t ( A ) . The closure 2 of A is defined by 2 = X \ i n t ( X \ A ) . Other definitions start with the concept of ‘closed set’, ‘closure’ or ‘neighborhood’, respectively, impose corresponding axioms on it, and then define the other concepts to get the same thing eventually. If two subsets A c B are such that they have the same closure then we say that A is dense in B ; if B = X then we simply say ‘ A is dense’. We call a topological space separable if an a t most denumerable dense subset can be found. A base of a topology is a collection B of open sets such that for any open set G and any point x E G we can find a B E B between them, i.e., satisfying x E B c G . Topological spaces possessing an a t most denumerable base are called ‘Mz’. For metric spaces this is equivalent to separability. A ‘neighborhood base’ at a point 3: is a family n/ of neighborhoods of x such that for any neighborhood V we have a W E N satisfying W c V . A topological space is called M I if a t most denumerable neighborhood bases exist a t all points. This is weaker than the A42 property. We may say that the central notion of topology is convergence. Unfortunately, it is not enough to consider just sequences for a general topological space. The two twin concepts generalizing sequences are those of a ‘net’ and a ‘filterbase’. A net is a function from some “index set” I into X (we get a sequence if Z is the set of positive integers), where Z is partially ordered and ‘directed’, i.e., for any pair i , j E Z there is a k E Z with k 2 i , k 2 j . A filterbase is a non-empty family of non-empty subsets of X having the property that the intersection of any two members contains a third one. A filterbase is a directed set if i 5 j means i 2 j , and if we choose an element xi E i for all i then we obtain a net. All these nets are called associated to the filterbase. In the other direction, if we have a net 2, then consider the sets {x, ; n 2 no} for all no . This family of sets is a filterbase, called the filterbase of the net. We say that a filterbase converges to a point x if any neighborhood of x contains a member of it. A net I,, is said to converge to I if its filterbase converges to x, i.e., if for any neighborhood U of x we can find an no such that x, E U for n 2 no . It is easy to see that a filterbase converges to a point if and only if all nets
208
Appendix A
associated t o it converge t o that point. The closure of a set A equals the set of the limits of all convergent nets taking values from A . Thus a topological space can be defined by telling which nets converge to which points. A topological space is called ‘TZ’or ‘Hausdorff’ if any two points have disjoint neighborhoods in it. The subscript in T2 refers to the place of this axiom in the hierarchy of separation axioms. The Hausdorff spaces are exactly the spaces where a net can not converge to more than one point. An important concept is that of the subnet. Let t, be a net with index set I . If J is another directed set then a function f : J H I is said t o ‘tend t o 00’ if for any i E I there is a j E J such that k 2 j implies f(k) 2 i . In this case z ~ ( as ~ a) net with index set J is called a subnet of t, . A net t, does not converge to a point t if and only if there is a subnet of t, outside of some neighborhood of 2 .
When we work with nets, it is not bad to visualize them as sequences in some familiar topological space, e.g., the plane. But sometimes we must be more careful. Do not forget that for a net there may be infinitely many indexes which are not beyond a fixed one (a nuisance in some cases). If X , Y are topological spaces then a function f : X I-+ Y is called continuous if it preserves the convergence of nets; equivalently, if f-’(U) is open whenever U is open. f is called an open mapping if f ( V ) is open for any open set V . A uniform space differs from a topological space in the following way: in a topological space it is given for any fixed point which other points are close t o it, while in a uniform space it is known for any pair of points how close they are to each other, i.e., the concept of closeness is “uniform”, does not depend on the place. We do not burden the reader with the precise definition (it can be found, e.g., in [Cszi] ) but mention that one way to think of a uniform space is to consider it as a set endowed with a family of pseudemetrics (a pseudemetric is a nonnegative, symmetric function on the set, which satisfies the triangle inequality but may differ from a metric in allowing for different points t o have distance 0). Then uniform continuity can be defined analogously as in metric spaces, and two families of pseudo-metrics are said to define the same structure if the identical mapping is uniformly continuous between them in both directions. Uniform structure is something “richer” then topological one, i.e., it is natural to assign a topology t o a uniform space but different uniform structures may define the same topology (just think of the interval ( 0 , l ) and of R). It is interesting that the topological spaces admitting a uniform structure are exactly the spaces where any closed set can be ‘separated by a function’ from any fixed point outside of it (to ‘separate by a function’ means to find a continuous real function which equals different constants on the two sets). This theorem is intimately related to the generalized Uryson Lemma, see [CsG] . A net in a uniform space is called ‘Cauchy’ if for any pseudemetric of the structure the distance of elements with “large enough” indexes in the net is arbitrarily small. A uniform space is called complete if any Cauchy-net is convergent. If the space can be defined by a denumerable family of pseudemetrics, then it is enough to consider the Cauchy-sequences to get the same notion. In general a uni-
209
Topology
form space is called sequentially complete if at least the Cauchy-sequences converge in it. Any uniform space is a dense subspace of a complete one. A uniform space is called totally bounded if it can be covered by a finite number of subsets of diameter < E with respect to any pseudc-metric of the structure and for any E > 0 . A uniform space is called separated if the distance of different points is positive for at least one pseudc-metric (which may depend on the points). Each uniform space can evidently be “factorized out into a separated one”. A uniform space is separated if and only if its topology is Hausdorff. A topological notion of outstanding importance is compactness. The archetype of a compact space is a bounded closed subset of R” . A compact space is a topological space in which any net has a convergent subnet. We refer the reader to the literature for the various equivalent definitions and elementary properties. The slightly different notions of sequential and countable compactness are also of some importance. A topological space is called sequentially compact if any sequence has a convergent subsequence, while it is called countably compact if any sequence has a cluster point. Compactness or sequential compactness implies countable compactness, no other implication holds in general between them. A compact Hausdorff space admits a unique uniform structure, and these uniform spaces can be characterized as the complete, totally bounded and separated spaces. A continuous function from a compact uniform space into an arbitrary uniform space is necessarily uniformly continuous. A very important. property of compact spaces that the product of any (may be infinite) number of them is compact, too (product topology is defined by saying that a net converges if and only if it converges in each coordinate).
a
Definition A . l . In this book the expression ‘locally compact space’ means Hausdorff topological space in which any point h a s a compact neighborhood.
Some authors call a space locally compact if any point has a base of neighborhoods consisting of compacts. It is easy to see that our notion then should be called ‘locally compact Hausdorff’, but since we shall be concerned only with Hausdorff spaces, Definition A . l is more convenient for us. Let X be a locally compact space. We can “augment” it with one point denoted by ‘00’ in the following way: let the open sets in X U ( 0 0 ) be the open sets of X and the complements of the compacts of X . Thus we obtain a compact Hausdorff space, the so called one-point compactifcation of X , which contains X as a subspace. As an example, think of the sphere as the one-point compactification of the plane. On the other hand, a compact Hausdorff space “minus” one point is always a locally compact space. A crucial property of locally compact spaces is that if ‘I\ is a compact subset, U is a n open subset and K c U then there is a continuous function f from the space into [0,1] such that = 1 , supp f c lJ and suppf is compact (the support of a real or complex function is the closure of the set where the function does not equal 0).
fl,
Definition A.2.
Let X be a topological space. A family of continuous
210
Appendix A
complex valued functions on X is called locally finite if any poin t h a s a neighborhood intersecting the supports of only a finite number of the functions of the family. Such a family is called a ‘partition of unity’ if the members are non-negative, and their sum equals 1 at every point. If ( U i ) i E l is a collection of open sets whose union is X and (f a ) o E A is some family of continuous complex valued functions on X then we say that the family (fa) is ‘subordinate’ to the covering ( U i ) if for any (Y E A we can find an i E I such that supp fm c U, . In this book a topological space is called ‘paracompact’ifit is Hausdorff and for any open covering we can find a partition of unity which is subordinate to t h a t covering. The above definition is equivalent to the notion of a ‘paracompact T2 space’ in the sense of [Csii], where one can find a comprehensive treatment of the versions of paracompactness. Any metrizable space is paracompact, though it is not so easy t o prove t h i s fact. The following statement is more important for us.
Proposition A.3. A locally compact space is paracompact if and only if it is the disjoint union of closed-open sets, each of which is a denumerable union of compact sets.
A subset of a topological space is called nowhere dense if the interior of its closure is empty. Countable unions of nowhere dense sets are called ‘meager’ (or ‘of first category’). A topological space is called a ‘Baire space’ if any non-empty open subset is non-meager in it; complete metric spaces and locally compact spaces are Baire spaces. We close this chapter with a special result related to Baire spaces which we use in Chapter 5 . We also give the easy proof here. Proposition A.4. Let X be an arbitrary topological space, Y be a metric space and f i , f 2 , f 3 , . .. be continuous functions from X into Y such t h a t f ( t ):= f n ( z ) exists for all t E X . Then the set of the points at which f is not continuous is meager.
PROOF:For t E X write ~ ( t= )inf{ S ( f ( U ) ) ;t E i n t ( U ) } where 6 means diameter, i.e., 6 ( H ) = sup{ d ( y , z ) ; y , r E H } for any non-empty H if d is the metric. Of course, ~ ( is t ) a non-negative number or $00 and f is continuous at t if and only if ~ ( t =) 0 . It is also obvious that T(c) = w W 1([c, +a]) is closed for any c . Let A ( n , k , s ) = { t E X ;d ( f , ( t ) , f k ( t ) ) 5 s } with s > 0 and B ( j ,s) = n,,k>jA(n, k,s) . Then B ( j ,s) is closed because the fn’s are continuous. Taking limits we obtain d ( f j ( t ) , f ( t ) ) 5 s for 3: E B ( j ,s) . Now suppose 2 E in2(B(j,s)) . Then the continuity of f j implies that w ( t ) 5 2s . Consequently, if c > 2s then the closed set T(c) n E ( j , s) is nowhere dense. Using the pointwise convergence of fn once more we see UjOO,lB(j, s) = X . Thus T(c) is meager for any positive c ; applying this to c = 1/n the proof is done.
211
B. Measure and Integration General reference: [BouS], [Hall]. We suppose that the reader is familiar with the Riemann and Lebesgue integration on R” . Let X be a set, and S be a collection of subsets of X . S is called a ‘ring’ if it contains the union and the difference of any pair of its members (then it must contain the empty set and the intersection of any two members). It is called a ‘semiring’ if the difference of any two members can be expressed as the finite disjoint union of some members and it contains the intersection of any two members. It is called a ‘lattice’ if it contains the union and the intersection of any two members. It is called an ‘algebra’ if it is a ring and X is a member of it. A subfamily of S is called an ‘ideal’ of it if it contains the intersection of its any member with any member of S. S is a ‘a-ring’ if it is a ring which contains the union of any denumerable family of its members (then it must contain denumerable intersections, too). It is called a ‘u-algebra’ if, in addition, it contains the set X . These notations come from abstract algebra when applied to sub-structures of the Boolean algebra of all subsets of X , but are only important here when one wants to construct a measure from a set-function defined just on some family S of “nice” subsets. The details can be found, e.g., in [Hall]. If S is a u-ring then the set X endowed with the “structure” S is called a measurable space and the members of S are called measurable sets. A ‘measure’ on a measurable space is a function p assigning non-negative numbers or +m to the measurable sets such that
i= 1
where U stands for disjoint union. Somewhat distractedly, a complex measure is a complex valued set-function which may be defined just on some ideal but there satisfies (B-l)l and therefore a complex measure is not a measure in general, while a measure is not a complex measure if its range contains +m. Some authors call positive measure what we call measure, but we keep that expression for “nobler use”, see Notation B . l below. A measurable space endowed with a measure is called a measure space. Assume that a complex valued set-function X is given on a semi-ring P and satisfies (B-1). First we define the ‘absolute value’ of X by setting
(B-2)
IXl(A) = sup
IX(Ai)l ; A is a finite disjoint union of the Ai’s
Then 1x1 also satisfies (B-1) . We call X a pre-measure if this supremum is finite for any A E P . Assuming this consider the so called Carathkodory extension of 1x1,
Appendix B
212
which is defined by
IXl(A;) ; H c U,eO,,A; , A; E P
03-3) for any H
(B-4)
cX
. Finally, let
S = { A c X ; p * ( H ) = p * ( H nA )
+p*(H \ A)
for all H C X }
,
and let p = p*Is . Then S is a a-algebra, p is a measure on it, P c S and 1x1 = p i p . Let S1 be the a-ring generated by P and p1 = plsl , then we call ( X , S1,p1) the measure space of A . We mention that p1 is the unique measure on S1 which is an extension of 1x1. We also note the following facts. With notation SO= { A E S ; p ( A ) < + 0 0 } we have that the completion of the measure space of X contains So (i.e., if A E So then A1,Az E S1 can be found such that A1 c A c A2 and pl(A2 \ A l ) = 0 ); and S can also be described as the family of those sets B for which B n A E SO for any A E SO. There is a unique complex measure A 1 on S1 nSo which extends A . The absolute value of this extension equals the restriction of p . Sometimes a set-function is given on a lattice, which is possibly the restriction of some complex measure. Therefore it is useful to note the simple fact that if C is a lattice then the differences of pairs belonging t o C U (0) form a semi-ring containing L . Let a measure space ( X ,S , p ) be given. Then we want t o integrate with respect to it. Denote by the set R U ( - 0 0 , +00} endowed with the usual ordering, convergence and operations, in particular, +00 . 0 = 0 = -00 . 0 . First we integrate the non-negative step-functions, i.e., the functions f : X I+ it whose range is finite, non-negative, and f - ’ ( { t } ) E S if t > 0 . In this case let
n
(B-5) Now i f f is a pointwise limit of an increasing sequence of non-negativestep-functions, then let its integral be the limit of the integrals of the step-functions. The crucial observation in integration theory is that this limit is independent of the choice of the sequence of step-functions. Thus we can integrate the so called non-negative measurable functions (a function is called measurable if the inverse images of intervals not containing 0 are measurable sets). A set N is called p-negligible if there is a measurable set M of p-measure 0 such that N c M . A set A is called p-measurable if there is a measurable set B such that A A B is p-negligible, where ‘A’means symmetric difference, i.e., A A B = (A\B)U(B\A) . Analogously, a function is called p-measurable if it equals a measurable one ‘almast everywhere’, i.e., the equality holds on the complement of a p-negligible set. Then we integrate the non-negative p-measurable functions by assigning the integral of a corresponding measurable function to them (it is trivial to check that
213
Measure and Integration
two measurable functions which are the same almost everywhere have the same integral). An arbitrary p-measurable function f : X H is said to have an
n
has sense, i.e., a t least one of the integrals on the right is finite. If both of them are finite then we say that f is integrable. A complex valued function is called (,+)measurable if its real and imaginary parts are (p-)measurable. Functions with values in R” are treated similarly. They are called integrable if the coordinate functions are integrable, and the integral is the corresponding vector (or number in C which is identified with R2).Functions with values in infinite dimensional vector spaces are also subject to integration, but there the situation is more complicated. We return to this in Appendix C. A measurable set in a measure space is called ‘a-finite’ if it is the union of a denumerable number of sets of finite measure. The measure space itself is called a-finite if each measurable set is a-finite. It is called ‘totally a-finite’ if, in addition, the whole space X is measurable. Assume that (X IS,p ) is a a-finite measure space and v is a complex measure defined on some ideal of S. Then the Radon-Nikodym Theorem asserts that if we fix a set Y in this ideal then we can find a set A C Y of p-measure 0 and a complex valued integrable function f such that
(B-6)
v ( B )=
J , f d p := J f . I B d p
for any measurable B c Y
\A .
The function f is unique on Y up to a change on a p-negligible set. This function is called the Radon-Nikodym derivative of Y with respect to p on Y . The set A can be chosen empty if and only if p ( N ) = 0 implies v ( H ) = 0 for any measurable subset H of Y . If, independent of a-finiteness, an integrable f can be found satisfying (B6) with an empty A (of course, it must be unique up to a p-negligible change) then we say that v is absolutely continuous with respect to p on Y . If Y = X then it is omitted from the expression. If a p-measurable f defined on some 2 E S can be found such that satisfies the condition above whenever Y c 2 and v is defined on Y then we also say that v is absolutely continuous with respect to p on 2 (in this case we do not have uniqueness for f in general, but this pathology usually does not occur in practice). If X is a pre-measure then the extended X is absolutely continuous with respect to the corresponding p1 on any Y E S1 nSo (observe that S1 is a-finite). It can be shown that in this case i f f is a Radon-Nikodym derivative on Y then If1 = 1 almost everywhere on Y . If h is any complex valued p1-integrable function then we say that h is integrable with respect to X and set
fly
(B-7) where is ‘a Radon-Nikodym derivative’, i.e., any measurable function which satisfies (13-6) for those Y’s of S1 nSo on which h never vanishes (such a derivative
Appendix B
214
can be constructed because h , being integrable, vanishes outside of some a-finite set). Observe that ‘pl-integrable’ is the same as ‘p-integrable’ (cf. the relevant remark to the construction of p ) . We have the important inequality
with equality only trivially, i.e., when h& = clhl with some constant c almmt dl4 everywhere. One can define versions of absolute continuity between pre-measures (differences of a-rings and non-a-finiteness may require extra care) and the ‘Rule of Substitution’ (i.e., a formula similar to (B-7) ) holds under fairly general conditions. We shall go into the details for the special case of Radon measures after Notation B . l . Lebesgue’s theorem asserts that if a sequence h, of integrable functions converges to the function h almost everywhere and the function supn lhnl is integrable then h is integrable and J” h, + h . Yegorov’s theorem states that if a sequence f, of p-measurable functions converges (in C) alrnmt everywhere on a set A of finite measure, then for any E > 0 we can find a B c A such that p ( A \ B ) < E and fn converges uniformly on B. I f ( X I,S1,P I ) , ( X 2 ,S2, p2) are two measure spaces then the ‘rectangles’ A1 x A 2 , where Aj E Sj , form a semi-ring in XI x X 2 . Write v ( A 1 x A 2 ) = p 1 ( A l ) . p z ( A 2 ) . Then the set of rectangles for which v is finite form a smaller semi-ring. It can be shown that we obtain a pre-measure v this way. Take the measure space corresponding to this pre-measure. It is called the product space, its measure is denoted by p1 x p 2 . If the measure spaces were not totally a-finite then it is customary to extend the notion of (p1 x p2)-measurability by using the corresponding S of ( B - 4 ) . Then Fubini’s theorem asserts that i f f vanishes outside of a a-finite set and it is (PIxps)-measurable (equivalently, it is (PI x,u2)-measurable in the stronger sense) then
in any case; and if this integral is finite then we have
Of course, the roles of
p1
and pz can be interchanged.
Now we turn to the special properties of integration on locally compact spaces. Let X be a locally compact space. We shall denote by L? the a-algebra generated by the open sets, and call its elements ‘Bore1 sets’. Note that in [Hall]
215
Measure and Integration
a slightly smaller family, the a-ring generated by the compacts, is called the family of Borel sets. We denote by B1 the smallest a-ring with respect to which every compactly supported continuous real function is measurable. Its elements are called ‘Baire sets’. It is not hard to show that B1, as a a-ring, is generated by those compacts which are countable intersections of open sets. Thus if X is metrizable and separable then each Borel set is also a Baire set. A measure p on B1 (on B ) is called a Baire (Borel) measure if p ( K ) < +m for any compact Baire (for any compact) I<. Such a measure is called ‘regular’ if the measure of any Baire (Borel) set is the supremum of the measures of the compact Baire sets (compact sets) contained in it. In fact, any Baire measure is regular and can uniquely be extended into a regular Borel measure. Moreover, this measure is uniquely determined by the integrals of the compactly supported continuous real functions, and any linear functional on this space of functions which takes non-negative values at non-negative functions is the restriction of such an integral. These facts are usually referred to as one of the representation theorems of Riesz, while a more general result is the Daniell-Stone Theorem. Now consider the complex linear functionals on the space of compactly supported continuous complex functions. It is not hard to show that such a functional 9 is the linear combination of positive ones (i.e., of integrals with respect t o Baire measures by the theorem above) if and only if
(B-9)
VA’ compact 3C : lp(f)I
5 C . max [).(fI X
if
supp f
cK
.
These and only these functionals are integrations with respect to complex measures defined on the ideal of B, generated by the compacts.
Notation B . l . We use the term ‘Radon measure’ t o mean the following. First, if a linear functional p is given such that (B-9) holds, then we say that a Radon mcasure is given. Then consider the corresponding linear combination of regular Borel measures on the ideal of B generated by the compacts. This complex measure X could be called the Radon measure itself but if X is not a countable union of compacts then it is natural to expand slightly the set of integrable functions by the following twist. Consider the so called ‘essential outer measure’ ji of 1x1 :
p(H) =
sup
K compact
min{ IXl(A);
N n A’ c A c A’ } = supp*(H n I<) K
where H is any set, the A’s are Borel measurable and p* is the CarathCodory extension of 1x1 (cf. (B-3)). If X is u-compact then, of course, ji = p* . Now let S be the family of ji-measurable sets (i.e., it is defined by writing instead of p* in (B-4)). We shall call these sets measurable with respect to the Radon . It turns out that a /Imeasure in question. Denote by p the measure measurable function f exists (and it is unique up to p-negligible change) such that f d p = X(A) for any Borel measurable A whose closure is compact. We have If1 = 1 p-almost everywhere. Now we write X(A) := f d p for any A E So , the ideal of sets of finite measure and call the resulting complex measure X the Radon measure associated t o p .
sA
sA
Appendix B
216
Some remarks on the above will clarify the picture. S contains the p*measurable sets. Any p-integrable function equals a Baire measurable one p-almost everywhere. S also contains B (in fact, any Borel measurable set is even p*measurable) and the restriction of p to B is the regular Borel measure corresponding to that linear functional I‘pI for which (B-10)
A set A belongs to S if and only if A n Ii‘ is measurable with respect to the above mentioned regular Borel measure for any compact Ii‘ (see o u r definition of measurability with respect to a measure; some call that ‘almost measurability’).
p also equals the CarathCodory extension of the absolute value of the extended A . If ’p = Cjcj’pj is a finite linear combination of positive ’pj’s and p, is the
regular Borel measure corresponding t o p, then X(A) = Cjc,p,(A) whenever A E B is such that pj(A) < +oo for each j . If we use here the so called Jordan decomposition of p then this finiteness will hold for any A E BnSo . Furthermore, if we consider the terms of the Jordan decomposition as Radon measures then the intersection of their measurability domains yields exactly S and on So the decomposition formula persists. In the general case of linear combinations the corresponding intersection is a part of S and there the combination formula remains valid. We say that a Radon measure is non-negative if ‘p = IpI , i.e., if it is the corresponding extension of a regular Borel measure. We say that a Radon measure is positive if it is non-negative and the non-empty open sets are not negligible. If X is a Radon measure then supp X denotes the smallest closed set whose complement is ]XI-negligible. Its existence follows from the regularity. Thus a positive Radon measure is a non-negative Radon measure whose support is the whole space. Now examine the Rule of Substitution for Radon measures. If X is a Radon measure and f is a ‘locally X-integrable function’ (this means that f . 1~ is 1x1integrable for any compact I< ) then we denote by f X the Radon measure for which u d ( f X ) = uf dX for any compactly supported continuous function u . Then the same holds for any IfXI-integrable u,we have lfXl = Ifl.lXl , and if u is any function satisfying the condition ‘ uf is X-integrable’ then necessarily u is (fX)-integrable. It is not surprising after all this that we have a very neat version of the RadonNikodym Theorem for Radon measures. Namely, if v and X are arbitrary Radon measures on X then we have a set A and a complex function f which are both uand X-measurable such that f is locally X-integrable, f X = 1 ~ uand X \ A is IXI-negligible. Such objects are unique (in the strongest possible sense, i.e., A can be changed (1x1 Ivl)-negligibly and f can IXI-negligibly). The singular set X \ A can be empty if and only if u ( B ) = 0 for any IXI-negligible B and it is enough to know this for Baire sets with compact closure. For Radon measures we have a stronger version of Fubini’s theorem. Namely, if XI,X 2 are locally compact spaces and p1, p2 are Radon measures on them, then consider the functional ’p(f) = f(z,y) d p l ( c ) ) d p 2 ( y ) on the product space. It defines a Radon measure. If a function is just measurable with respect to this
s
s
+
s(s
Measure and Integration
217
Radon measure but vanishes outside a denumerable union of compacts then the statements in Fubini's theorem hold for it even though it may not be measurable with respect to the product measure defined in the general way. Of course, for separable metrizable spaces there is no difference. Luzin's theorem asserts that if p is a non-negative Radon measure on a locally compact space, f is a complex valued p-measurable function and A is a measurable set of finite measure then for any E > 0 we can find a compact B c A such that p ( A \ B ) < E and f l B is continuous. Finally, we mention Sard's theorem, though it concerns with the ordinary Lebesgue integral. Theorem B.2 (Sard's t h e o r e m ) . Let F be a once continuously differentiable function from an open subset of R" into R" . Let S be the set of points at which det F' vanishes. Then the set F ( S ) has Lebesgue measure 0 .
An easy consequence of this theorem is that if f is a once continuously differentiable function from an open subset of Rk into Rn with k < n then the range o f f is negligible.
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219
C . F u n c t i o n a l Analysis General reference: [RudS], [Yos]; [Arv], [B-D], [Day], [Scha], [S-Zs]
.
N o t a t i o n C . l . If X is a locally compact space then C ( X ) denotes the complex vector space of the complex-valued continuous functions on X . C c ( X ) and Co(X) denote its subspaces consisting of the functions which are compactly supported and vanishing at infinity, respectively; the latter means that the function can be extended continuously to the one-point compactification of X such that it takes 0 at 00. In other words, f vanishes at infinity if I f 1 < E outside of some compact for any E > 0 . Of course, if X is actually compact, then these three spaces coincide. The notation ‘ C o ( X ) ’shall always mean this space endowed with the usual C*-algebra structure, i.e., pointwise multiplication, complex conjugation as involution. and Ihe maximum norm
We recall that ‘C*-algebra’ means a structure isomorphic to some normclosed star-subalgebra of the ring of operators on some Hilbert space, and it is known that any commutative C*-algebra is isomorphic to some Co(X) described above. It is evident that C,(X) is dense i n C o ( X ) . The following family of statements provides smaller dense subspaces.
Theorem C.2 (Versions of the Stone-Weierstrass T h e o r e m ) . Let A c C o ( S ) be a linear subspace which contains the complex conjugates of its elements. Assume that A is separating, i.e., if z
#y
for a n y
then z
3f EA : 3gE A :
f(z) # f(y)
and
g(x)# 0
(the second line says that A separates X from 00). Then if A is such that the real-valued functions in it form a latlice (i.e., maximums and minimums of pairs belong to it) and, in addition, 1 A f belongs to it for any real-valued f E A , then A is dense in C o ( X ) .On the other hand, if A is an algebra then the closure of A satisfies these properties, and hence A is dense in C o ( X ) . If A c C , ( X ) and it is an algebra then for any f E C c ( X ) we can find a compact I< and a sequence fn E A such t h a t fn + f uniformly and supp f,, c A’ for all n.
PROOF:We just sketch how one can reduce the last assertion to the other statement. Let L = suppf and g E C o ( X ) be such that g l L = 1 . Then there
220
Appendix C
is a real-valued function h E A which is close to g, and hence its range on L is close to 1. Let T be this range, then we can find polynomials pn on R such that pn(0) = 0 , pn + 1 uniformly on T and the pn’s are under a common bound on the whole range of h. Now if u, is a sequence from A tending to f in C o ( X ) then f n = Un . (pn o h ) is good with K = SUPP h .
(s lflP
If ( X , S , p ) is a measure space and 1 5 p < $00 then write Ilfllp for any p-measurable function f . Holder’s inequality says that dp)
llfsll1 I Ilfllp . 11gIlq
if
1
=
1
-+-=1, P Q
and if the left hand side is finite then we have equality only trivially, i.e., if one of the functions l f l P and 1914 is constant times the other one (almost everywhere). Holder’s inequality also holds when p = 1 and q is infinite, where the ‘essential supremum’ ll.llm is defined by Ilfll, = min(2 2 0 ; p( If[-’( (1,+00])) = 0 } . I n this case we have equality for a finite left hand side only if 191 = Ilgll, almost everywhere on the set where f does not vanish. Notation C . 3 . P ( X , S , p ) denotes the linear space of those p-measurable functions for which II.Ilp is finite, where 1 5 p I +co (the fact that it is a linear space follows from Minkowski’s inequality, which is a consequence of Holder’s inequality). Denote by N the subspace of functions which vanish almmt everywhere. Then LP(X, S , p ) denotes the Banach space ( C P ( X , S , p ) / N , 11.11,) . If X is a locally compact space and A is a Radon measure on it, then we use the notation LP(X, A) = LP(X,S, 1x1) with S from Notation B . l , and similarly with L P . If there is a distinguished measure on X then it is usually also omitted from the notation; in particular, we always do this when X is some open subset of R” and the distinguished measure is the Lebesgue measure.
C , ( X ) / ( N n C , ( X ) )is dense in LP(X, A ) for finite p for any Radon measure A , and if X is positive (in our sense, see Notation B . l ) then we need not factorize by N since it intersects C , ( X ) in (0) then. On the other hand, if p = +00 then the closure of C e ( X )is just C o ( X ) . A subalgebra A c C c ( X ) which satisfies the assumption of Theorem C.2 must be dense in these LP-spaces (for finite p ) , too. Another representation theorem of Riesz asserts that “(Lp)* = L q ” . More precisely, the theorem is the following. Let + = 1 ( p or q can also be infinite), denote by LP and L9 the corresponding Banach spaces with respect t o the same measure space, and let j : L9 H (LP)* be defined by ( j ( F ) , G )= S f g if f E F and g E G . Then
(C-3)
j is an isomorphism of Banach spaces if 1 < p < +oo , or if p = 1 and the measure is localizable; j is an imbedding of Banach spaces except if p = 1 and the measure has an infinite atom,
Functional Analysis
22 1
where an ‘infinite atom’ is a measurable set with measure $00 whose any subset of finite measure is negligible. We refer the reader to [Seg] for the various equivalent definitions of ‘localizability’, but we mention that the p in Notation B . l is localizable, and the totally a-finite measures are trivially localizable.
Proposition C.4.
Let f, E P ( X , S,p ) be such that M
Then C, fn(z) exists (in C ) for almost every I , and the finite sums tend to it in LP . A s a consequence, any Cauchy-sequence in LP(X, S,p ) has a subsequence con verging almost everywhere.
Notation C.5. For Radon measures we define the ‘local LP-spaces’ as follows. Let A be a Radon measure on the locally compact space X . Assume that f is a complex function such that f . 1~ E P ( X , A ) for any compact I<. T h e linear space of these functions divided by n/ is denoted by Lyoc(X,A ) . This is a typical example of a ‘locally convex space’ (see below), if we use the family of seminorms f + I l f . l K l l p . L y o c ( X A, ) will always mean this locally convex space.
A ‘topological vector space’ is a vector space X: over the field F = R or C which is, a t the same time, a topological space in such a way that the operations are continuous, i.e., the addition from X x X into X and the multiplication by scalar from F x X into X are continuous. Then the closure of the set (0) is a closed subspace, and the corresponding factor space is a Ilausdorff topological vector space, which also carries a natural separated uniform structure. In this book we shall only be concerned with topological vector spaces which are ‘locally convex’. One definition of this is t o require that any point have a base of neighborhoods consisting of convex sets. An equivalent definition, which does not use the concept of topological vector spaces, is the following. Let X be a vector space over F, which is endowed with a family of seminorms (a seminorm is a non-negative function p on the space satisfying the a x i o m
Then we say that a net I, converges to a point I if p(1,--2) -+ 0 for any seminorm p in the family. This defines a topology, and 1 as endowed with this topology is called a locally convex space. Thus if another family defines the same topology then we have the same locally convex space. The union of all equivalent families coincides with the set of continuous seminorms and is also a family defining the locally convex space. There is a unique translation invariant uniform structure consistent with this topology, and it can be given by the pseudo-metrics d(t, y) = p(z-y) where p runs through a family of seminorms defining the locally convex space. Thus a filterbase is called Cauchy if for any E > 0 and any seminorm p in the family it has a member
Appendix C
222
A such that p(x - y) 5 E for z,y E A . A net is called Cauchy if its filterbase is Cauchy and the space is called complete if all Cauchy-nets converge. Any locally convex space is the dense subspace of some complete locally convex space. A locally convex space is called sequentially complete if all Cauchy-sequences in it converge. Notation C.6. In this book we shall use the expression ‘locally convex space’ as an abbreviation of ‘locally convex Hausdorff space’. Of course, such one can be constructed from a non-Hausdorff space if we divide it by the subspace where all seminorms of the family vanish. The following method of constructing locally convex spaces is useful in the theory of distributions.
Definition C.7. The ‘inductive limit of locally convex spaces’ is defined be a family of locally convex as follows. Let X be a vector space over F, spaces over F , and let a linear operator Ai : Xi I+ X be given for all i E I . Then there is a finest locally convex topology on X so that each Ai be continuous. X with this topology is called the inductive limit of the system ( X i , A i ) . In the special case when the Xi’s are subsets of X and the A, ’s are inclusions we say that X is the inductive limit of the Xi ’s. Proposition C.8. Let X be the inductive limit of the system (xi, , be another locally convex space, and F : 1 I+ !2J be a linear mapping. Then F is continuous if and only if F o A; is continuous for every i E I .
!2J
Notation C.9.
Let X be a locally convex space defined with the family c X is called bounded if p,(A) is a bounded set
( p a ) , of seminorms. A subset A of numbers for any a .
Let X’ be the linear space of the continuous linear functionals, i.e., mappings into F, on X . We consider two locally convex topologies on it. The weak* topology (abbreviated by w*) is defined by the seminorms q z ( f ) = If(x)l where t runs through X (this is the topology of pointwise converger,ce). The strong topology (usually abbreviated by 0)is defined by the seminorms q A ( f ) = supzEAI f ( t ) l where A runs through the bounded subsets of X . On the other hand, the seminorms pj(x) = If(x)l define a locally convex topology on 3 , which is called the weak topology and abbreviated by w . We frequently use the notation ( f ,x) instead of f(t); especially, when the object f can also be thought of otherwise than a functional, e.g., in the case of (LP)’ = Lq . Similarly, we sometimes call the weak* topology ‘weak’, because the elements of X are sometimes more naturally thought of as functionals on x’ . Recall some basic results and notions concerning locally convex spaces. The Hahn-Banach Theorem says that if A, B are disjoint convex sets in X such that A is compact and B is closed then there is an f E 3E’ which strictly separates them.
Fun c tional Analysis
223
The Krein-Milman Theorem asserts that a convex compact set is the closed convex hull of its extremal points (a point of a convex set is called extremal in the set if it is not an interior point of any segment belonging to the set). Milman's Converse Theorem states that if K is a convex compact, D is a closed subset of it and the closed convex hull of D equals I<, then D contains all extremal points of I<. If A is a linear operator from a dense linear subspace of a locally convex space X into another locally convex space g then it is closable (i.e., the closure of its graph is the graph of some linear operator) if and only if the adjoint A* is defined o n a w*-dense subspace of 13' ( A*'p is the continuous linear extension of ' p o A when the latter is continuous). In this case the closure of A equals the second dual A** , where the dual spaces are endowed with their respective w* topologies. We use the notation A#'p := cp o A if A is an everywhere defined linear operator from some linear space into another one and 'p is a linear functional on the latter ('algebraic dual'). If Cl is a neighborhood of 0 in X then the set
{ f E X' ; If(z)l 5 1 for z E U } is w*-conipact. In particular, the closed unit ball of the dual of a normed space is w*-compact. A complete metrizable locally convex space is called FrCchet space. T h e Closed Graph Theorem is valid for Frkchet spaces, i.e., an everywhere defined closed operator is necessarily continuous between them. Tlie Banach-Steinhaus Theorem asserts that if A is a family of continuous linear opcrators from 1 into 9 such that the set
{ ~ E X ; { A ~ ; A Ei sAb o}u n d e d i n Z J } is not meager, then the family A is equicontinuous. FrCchet spaces are not meager in themselves. The Banach-Steinhaus Theorem is often applied to pointwise convergent or pointwise Cauchy sequences of operators. Unfortunately, a pointwise convergent net of operators does not necessarily satisfy the corresponding boundedness condition. Now turn t o the problem of vector-valued integration. Let (TIS,p ) be a measure space, 3E be a locally convex space, and f : T H X be some function. If we want to integrate it, we should a t least require that 'p o f be integrable for all 'p E x' . In this case we call f 'weakly integrable' and the integral is a linear functional v on X' defined by
An important question is: when does this integral actually belong to X ? Frequently in practice, one can check that v is continuous with respect t o the strong topology, and in this case if X is reflexive (e.g., LP-space with 1 < p < +m ), then it
Appendix C
224
contains v . If the space X is the dual of some other space (‘w* integrals’) then it is enough to check that v is continuous with respect to that topology, which is often easy. For more general spaces we need nicer ‘integrable functions’. A useful theorem in connection with this is the following (cf. [Scha], [Day]).
Theorem C.10 (Generalization of Krein’s Theorem). Let Ji’ c X be compact in the w topology, and let L be the closed convex hull of (I (then L is also w-closed by the Hahn-Banach Theorem). Then L is w-compact if and only if it is complete with respect to the original locally convex structure of X . Thus, e.g., if p is finite, f is weakly integrable and the closure of the range o f f is w-compact and, on the other hand, X is quasi-complete (i.e., each bounded closed set is complete in it) then the integral belongs to X (use the Hahn-Banach Theorem on the space !J of all linear functionals on X’ , where !J is endowed with the corresponding w* topology). An important consequence of the Hahn-Banach Theorem is that if the weak integral v o f f exists and belongs to X then for any continuous seminorm p we can find a cp E X’ such that IcpI 5 p and p(v) = o f)dp ; consequently, we have
s(cp
p(v) L
l(P
0
f ) dP
s,
where denotes the lower integral, i.e., the supremum of the integrals of not greater measurable functions (the function p o f is not necessarily measurable). ‘Strong integrals’ can be defined by requiring that the integral be the limit of some suitable net of integrals of step-functions. Instead of exploring this idea, we turn to an ‘‘even stronger” integral, the ‘Riemann integral’. Let T be a locally compact space, and let p be a finite Radon measure on it, i.e., for which lpl(T) < $00 . Let X be a locally convex space, f : T H X be some function, and let w denote a finite collection of disjoint Bore1 sets whose union is T. Then the ‘Riemann approximation’ with respect t o w is the set
Heuristically, the statement ‘the Riemann integral of f equals v’ should mean that the set R, is in a prescribed neighborhood of v whenever w is “fine enough”. For compact metric T one says that w is &-fineif the diameters of its members are 5 E . So it is natural to use the unique uniform structure of the one-point compactification of T in general. Thus if D is a family of pseudo-metrics defining this uniform structure then for any finite H c D and E > 0 we say that w is (H,&)-fine if the diameters of its members are 5 E with respect to the pseudo-metrics in H . Let R ( H , c ) be the union of all R,’s such that w is ( H ,&)-fine(the existence of such an w follows from the compactness). Thus we constructed a filterbase, and v is said t o be the Riemann integral o f f if this filterbase converges to v .
f i n ctional Analysis
225
We mention that one can circumvent the uniform structure by using the compact-open method. Namely, let K 1 , . . . , I<, be compacts in T , let K j c U, with open U, , and call w fine with respect to them if A n K j # 0 implies A c Uj for any member A of w . This yields an equivalent notion of fineness. Let Y be the set of those points in T a t which f is continuous, and suppose that a certain compact Ii' is contained in Y . Now if 2 1 , 2 2 E R ( H , E ) then 2 1 - 2 2 is of the form i ,j
with s, E A , , I, E B, and small Ai, B, . Split the sum into two groups according to whether I( intersects A, n B, or not. Then the first sum is small for suitable ( H , E ) . The second sum is always in lpl(T\ I<) times the absolute convex hull of the set R ( f ) - R ( f ), where R ( f ) is the range of f. Thus we see that the filterbase R(H,E ) is Cauchy if R ( f ) is bounded and T \ Y is IpI-negligible. If X is quasi-complete than this implies the existence of the Riemann integral in X . If T U {m} is metrizable then sequential completeness is enough. The Riemann integral has great advantages. One is that we can choose a net of ( A i , t i ) 's to compute the integral from a big collection. Another advantage is that it is trivial t o check the following properties. Proposition C.ll. Assume that v is the Riemann integral o f f . Let A be a linear mapping from some linear subspace of 3E containing the range off into a locally convex space '2). If A is continuous and v E D ( A ) then the Riemann integral of A o f exists and equals Av. On the other hand, if A is j u s t closed (i.e., its graph is closed in X x '2)) but the Riemann integral of A o f is known to exist, then it equals A v .
So we can see that a Riemann integral is also a weak integral. For weak integrals we have the following. Let 2 1 be the closure of the domain of A and f be a function whose range is contained in this domain. If the weak integral u o f f exists then it vanishes on the polar of El , so it can be considered as a functional on the factor which is naturally identified with X; because of the Hahn-Banach Theorem. Thus replacing X by 2 1 we may assume that A is densely defined. Then we have ( u , A * p ) = cp o A o f for any cp E D ( A * )c '2)* . Therefore if A is continuous then u o A* is the weak integral of A o f , while if we know the existence of the weak integral v of A o f then v 3 u o A* . If A is closed then we know that A** = A (the second dual is taken with respect to the w* topologies), hence if u and v belong to X and '2) then v = Au . Thus the statements of Proposition C . l l are also valid for "semi-weak" integrals (weak integrals lying in the corresponding ground spaces). The notion of barycenters is closely connected with integration. Let Ii' be a convex compact set in some locally convex space, and p be a non-negative Radon measure on Ii' such that p ( K ) = 1 . Then the barycenter of p is the Riemann integral of the identical mapping of It' with respect to p . I t is the only point E
226
Appendix
C
in the locally convex space X for which cp(z) = piK dp for all cp E x’ . Of course, if p is a finite convex combination of point-masses then its barycenter is simply the corresponding convex combination of those vectors. Choquet ’s theorem asserts that any point in Ii‘ is the barycenter of some ‘extremal measure’ p , i.e., for which p ( A ) = 0 whenever A is a Baire set containing no extremal points of I<. This is a much stronger result than the Krein-Milman Theorem. Unfortunately, the set of extremal points is not a Baire set in general; but it is if 11‘ is metrizable. Choquet’s theorem is applied in representation theory when I( is the set of the so called states on a unital C*-algebra, since then the extremal points are exactly the so called pure states, i.e., which generate irreducible representations by the GNS-construction (see, e.g., [Arv]). Enormous numbers of proofs in analysis depend on the interchangeability of integrations with respect to two variables (which is provided by Fubini’s theorem) and on t,he interchangeability of integration with respect to one variable and differentiation with respect to the other one. Unfortunately, the latter “rule” seems to elude a precise formulation, partly because we have several different versions of vector-valued integration and differentiation. Of course, we know from calculus that if we have a continuously differentiable function in two (say, real) variables and integrate it with respect to one variable on an interval then the result can be differentiated “under the integral sign” with respect to the other variable. Obvious generalizations of this statement require no more explanation. We mention a few more delicate versions below. We use the notions of ‘derivative’ and ‘weak derivative’ as follows. If X is a normed space, ZJ is a locally convex space and f is a function from an open subset of X into ZJ then we say that f is differentiable a t the point z i n the domain of f and its derivative is the continuous linear operator A : X H ZJ if (C-5) In this case we write f’(z) = A . On the other hand, the weak derivative can be defined even if X is just a locally convex space, too. It assigns a scalar to pairs (v,cp) E X x ZJ* by setting
One may define derivatives requiring something between the extremities of (C-5) and (C-6) but let these ones suffice for us. Weak derivatives of higher order can simply be defined by considering more 21’s and the corresponding partial derivative in F” on the right hand side. To get (strong) derivatives of higher order, we must fix a locally convex topology on the space of continuous operators. Now if ( p a ) , is a family of seminorms defining the topology of ZJ and B is the unit ball of X then let qo(T)= supB p , o T , and consider the locally convex topology defined by this family of seminorms. Iterating this process we obtain the locally convex spaces
227
Functional Analysis
‘ d n ) ( X ,9)’, and the derivatives f ( n t l ) = (f‘”))’ can be defined. It is not hard to check that the elementary properties of differentiation are satisfied (we call attention to the rule concerning composite functions and to the ‘Leibniz Rule’, which tells us how to differentiate a “product”, i.e., continuously multilinear combination, of differentiable functions). Proposition C.12. Let ( T , S , p ) be a totally a-finite measure space, U be an open set in the field F, and f : U x T H F be a function such that h(x) = J f ( x , t ) d t exists on U , and f satisfies the following conditions: (a) f ( . , t ) is the integral of its derivative for almost every 2 (i.e., it is absolutely continuous if F = R and it is holomorphic if F = C ); (b) z f ( x , t ) is measurable (with respect to ‘Rorel(x)S’); (c) ,[I$f(x,t)l dt is a locally bounded function on U ; (d) v(x) = $f(z,t)dt is a continuous function on U . Then h is differentiable, h’ = v .
s
P R O O F : If z is a number such that the segment [ x ,21 is contained i n U then
we have
h ( Z )-- h ( x ) = z -- 2
J
(I’a,!(
[x
+ s ( z - x ) ] , t )ds) dt =
1
1 U(X
+ S(Z - x ) ) d s
(we used condition (a) in the first equality, and conditions (1)) and (c) enable us to use Fubini’s theorem in the second one). Now condition (d) completes the proof. Proposition C.13. Let ( T , S , p ) be a measure space, X be a normed be a locally convex space, and f : U x T H 9 space, U be an open subset of it, be a function differen tiable on U for almost every t E ?’ and satisfying the following conditions: Vx E U : h ( z ) = J f ( x , . ) d p exists weakly and belongs to 9; the analogous statement holds for V ( x ) = f ’ ( x , . ) d p ; for any x E U and seminorm p from some family defining the topology of 9 we can find a p-integrable function b such that
s
for small 11v11, where this smallness may depend on x and p . Then h is differentiable in the sense of (C-5) and h’(z) = V ( x ) . -+ Av is continuous from Therefore in the weak sense we have for small v
PROOF:Observe that the mapping A
into
9 for any fixed v.
B(X,9)
Appendix C
228
Now suppose the contrary of our statement, i.e., that we have an x E U , a sequence vn -+ 0 in X and a seminorm p such that
+ vn) - h ( z ) - V(z)vn )
P( h(x
llvnll
f.0
.
Then using the Hahn-Banach Theorem we can find cpn E ‘2). such that Ipnl 5 p and
But this contradicts to Lebesgue’s theorem because the integrand converges to 0 almost everywhere and under an integrable bound (observe that the assumptions of the proposition imply that
5 b a l m a t everywhere).
p ( f ’ ( x l ‘)’)
11v11
Proposition C.14. Let ( T , S , p ) be a totally u-finite measure space, X be a normed space and ‘2) be a Banach space. Let U be an open subset of X and 3 c ?)* be a “norm-showing” set, i.e.,
llYll = SUP{ I 9 ( Y ) l l IIVII ; 9 E 3- \ (01 1 for any y. Then consider the locally convex topology on ?) defined by the seminorms IpI where p runs through 3. Let the sign when applied to a vector-valued function, mean the weak integral with respect to this locally convex topology. Assume that a function f : U x T H ‘2) is given satisfying the following conditions: (a) f (., 2 ) is twice continuously differentiable for almost every t ; (b) h ( t ) = J f ( t , t ) d t and g ( x , v ) = f ’ ( t , t ) v d t exist in ‘2) for any x E U , v E X and the linear operators g(x,.) are bounded; (c) (p, f ” ( z X z l t ) ( v lv)) is measurable in ( X , t ) (with respect to ‘Borel(x)S’) for any p ~ 3 X, E U , % , V E X ; (d) I (” f ” ( x l t)(vd dt 5 b ( x ) where b is locally bounded on U .
s,
s
1
+
IIPII . llvll
’
Then h is differentiable in the sense of (C-51, h’(x)v = g ( t , v)
PROOF:We have t o prove that the function
tends to 0 uniformly in cp E 3 \ (0) for any fixed x when v imp1ies that
-t
0 . Condition (a)
(observe that to obtain this formula it is enough that f be twice continuously weakly differentiable with respect to the coarser topology defined by 3 ) .We can now apply
Functional Analysis Fubini’s theorem because of conditions (c) and (d), and infer that Irl small v with C depending only on 2.
229
5 C 11v(( for
Remark C.15. The conditions of Propositions C.12-14 may seem disgustingly complicated. But this is a consequence of seeking very weak assumptions, and so in practice it is often trivial to check them. It is sometimes useful to formulate stronger assumptions, e.g., observe that C.14.(d) follows if we assume that Ilf”(z, .)[I dp is locally bounded. Note that we even refrained from stating the weakest assumptions the proofs would allow (think of the weak F-differentiability instead of C.14(a) or of ‘almast measurability’ in the suitable sense instead of C.l2(b) and C.l4(c)). The using of the 3-topology in C.14 is motivated by the important special case when is the dual of some Banach space and 3 is a pre-dual of ?I. We discussed the topic of differentiating under the integral sign with the aim of helping the reader to “feel the method”. It is advisable to choose a “simplest way” for yourself as a particular problem arises.
A n important branch of functional analysis is the study of linear operators in a Hilbert space. Since we have a great variety of good books on this topic, it is not necessary to dwell on it very long here. Let it suffice to establish notations connected with the extraordinarily useful notion of spectral measures. be a measurable space such that R E S . Let Notation C.16. Let (0,s) E be a projection valued set-function on the measurable sets (i.e., E : S H B(3-1) with some complex Hilbert space 3-1 such that E ( A ) 2 = E ( A ) = E(A)’ for any A ) . E is called a ‘spectral measure’ or ‘projection valued measure’ if ( E ( . ) z , z ) is a measure for any 3: E 3-1 and E( R ) = I , the identity operator. We denote these measures by E, , while Ez,y:= (E(.)3:,y) , which is a finite complex measure. We use the convention of writing the scalar product so that it be conjugate linear in the second variable. I f f is any complex-valued measurable function on R then its integral with respect to E is a normal operator F (in general unbounded) which is defined on 3:’s for which f E C2(L?,S,E,) by setting
and we write J f ( w ) d E ( w ) = F . In the special case when ( R , S ) = ( C , B ) the spectral measure E is determined by the integral of the identical function f ( z ) = z . In this case E is called the spectral measure (or spectral resolution) of this normal operator, and any normal operator can be ‘resolved’ in this way. If R is a locally compact space and S is the algebra generated by the Baire sets then we talk about a ‘spectral Radon measure’ and we extend such a measure to the Bore1 sets so that E, be regular for any 3:.
Appendix C
230
The theory of abstract Banach algebras meets the theme of this book when we use group algebras. Suppose that the reader knows such notions as the spectrum of an element and the Gelfand map of a commutative Banach algebra. Also assume that the reader is familiar with the Fourier transform on the group R” , though it is discussed to some degree in the main text, being closely connected with the representation theory of commutative locally compact groups. In the remainder of this chapter we mention a couple of miscellaneous results of functional analytic nature. First recall that the Inverse Function Theorem is valid for Banach spaces, i.e., if X , ?) are Banach spaces over F, U c X is an open set, f : U H g is continuously differentiable and f’(x) is bijective from X onto for some x E U then f is diffeomorphic on a neighborhood of I. This implies the usual results about implicit functions, regular and coregular mappings. E.g., if g : U H ZJ is regular at x (i.e., g is continuously differentiable and g’(z) is bijective onto some ZJl where ZJ = !ZJ1 @ ZJ2 as Banach spaces) then we have a cont.inuously differentiable function h on a neighborhood of g(x) such that h o g is identical, and therefore i f f is some function from a third Banach space into X with f(z) = I then the differentiability of g o f and the continuity o f f at z imply the differentiability o f f a t z . Proposition C.l7.(a) Let X , ZJ be locally convex spaces, N be a directed set, and ( A , ) , € N and (x,),€N be a net of operators and vectors, respectively, sucli that the An’s are equicontinuous from X to g , x, + x in X and for any fixed j the net (Anxj), is Cauchy in 9.Denote by yj the limit of this net in the completion of ZJ . Then in this completion we have a limit y = limn yn , and it is also the limit of Anxn as wellasof A,x. (b) Let X , ‘2) be as above, U be an open set in a normed space 2 and F : U H B (X, 9), f : U H X be functions with the following properties. The range of F is equicontinuous; F ( . ) x is differentiable at a certain point z E U (in the sense of (C-5)) if 3: = f ( z ) ; F ( . ) x is continuous at z for any 1: and this continuity is uniform on the set { x = f ’ ( z ) v ; llvll = 1 ) ; the function f is differentiable at z . Then F ( . ) f ( . ) is differentiable a t z and its derivative is calculated by the Leibniz Rule, i.e., it equals [ F ( . ) f ( z ) ] ‘ ( z ) F ( z ) f ’ ( z ) .
+
PROOF:Obviously we have
for any z , and we consider the cases when z = x, or z = x . Let V be a convex symmetric neighborhood of 0 in the completion of !J , then by the equicontinuity the first term on the right belongs to V if n , j 2 no with a suitable no. Now for a fixed j 2 no the second term belongs to V if n 2 n1 . This can be applied to another large j and we obtain another threshold n2 for n . Now choose an n satisfying n 2 no1n1,n2 and infer that the difference of these yj’s belongs t o 4 V . Hence we obtain that the net yn is Cauchy.
23 1
Function a1 Analysis
>
no which is large enough to have y - yj E V . Now Then choose a j if n is large enough to have small second term with this j a n d also n 2 no then A,% - gj will be in 2V with both choices of t . T h u s A,z - y E 3V a n d (a) is proved. For (b) consider the net of the small non-zero vectors in 2 with t h e ordering t h a t u v means llull 11011 and set A , = F ( z v ) and
>
<
2,
= -
- f(z) - ”(*)’
llull
+
. Then by the continuity of F ( . ) x at z we can
see that, the third’assumption of (a) is also satisfied and y,, = F ( . z ) x , , from which we get y = 0 for F(t) is continuous. So (a) implies that A,x, + 0 . Add to this convergence the one arising from the differentiability of F ( . ) z a t t with x = f ( r ) . v v Finally we can replace F ( z v)f’(t.)by F ( z ) f ’ ( z ) because of t h e third
+
llvll
llvll
assumption in (b) and the proof is complete. T h e reader may righteously think t h a t the proof above is trivial and the statement itself has disgusting conditions. Rut observe that quite often in practice we have just, slightly stronger condit,ions satisfied and, on the other h a n d , t h e trivial proof above becomes rather time-eating when performed in the head as is usually done. It is worth mentioning here that if the normed space 2 in (b) above is finite dimensional then the condition concerning S := { x = f ‘ ( z ) v ; llvll = 1 ) is a consequence of the others (it follows from (a) that F(C)[ is jointly continuous a t ( t , ~for ) any x and S is compact in this case). Theorem C.18. Let X be a sequentially coniplete locally convex space over F a n d U be a n open subset of Fn . Let F c x’ be a “boundedness-showing” set of functionals, i.e., if for some set A c X we have ‘ p ( A ) is a bounded set of scalars for a n y ‘p E 3 then A must be bounded. Suppose that a function f : U H X is given such that ‘p o f is k times continuously differentiable for any ‘p E 3 , where k 2 1 . Then f is k - 1 times continuously differentiable (in the sense of (C-5)).
PROOF: We prove this by induction on k. First consider the case when k = 1 . For the sake of the subsequent induction, we prove the following more precise version: if I< c U is convex and such that supK II(’p o f)’ll < +m for ‘p E 3 then f is uniformly continuous on I<. Set
If x E A and cp E 3 then
‘ p ( x ) is
of the form
which is evidently bounded by supK II(‘p o f)‘” . T h u s sumption, and hence f is uniformly continuous on Z<.
A is bounded by the
as-
Appendix C
232
Now let k = 2 . Fix some u E U , v E F“ and consider the function f(u -I- *’) where z is scalar. Then for any ‘p E 3 we have g(z) = z
:.l
( (‘p 0 f)’(u
+ zv) -
(‘p 0 f)‘(u
+ s z v ) , v) ds =
Here the right hand side has sense for z = 0 , too, and defines a continuous function on some neighborhood of 0, which neighborhood is independent of ‘p. Thus we can apply the already proved modified case k = 1 to g on suitable convex sets (half-discs or intervals) to obtain that g is uniformly continuous on a neighborhood of 0. Then the sequential completeness of X implies the existence of 8,,f(u) := lim,,og(z) . For ‘p E 3 we have ’p[d,,f(u)] = ((’p o f)’(u) , v ) , and the right hand side is continuously differentiable in u (for k = 2 ). Applying the case k = 1 t o 8,f we see that it is continuous. This is true for any v , in particular for a basis of F” , and that implies the existence and continuity of f’ . Now suppose that k > 2 and the statement is proved for smaller k’s. Then f is continuously differentiable and ‘p o ajf = 8j (’p o f ) is k - 1 times continuously differentiable for any ‘p E 3 . Thus by induction 8jf is Ck-2 for j = 1,. . . , n and f itself is Ck-’ .
Remark C.19. The whole dual X’ always satisfies the assumption imposed on 3 above, i.e., ‘w-bounded’ is the same as ‘bounded’ (though it is not so easy to prove it, see, e.g., Corollary 2 of Theorem lV/3.2 in [Scha]). For some spaces one can find much smaller F’s, as the followirig proposition shows. Proposition C.20. Let X be the space of continuous linear functionals on a sequentially complete locally convex space g , endowed with the strong topology (see Notation (3.9). If a set A c X is pointwise bounded (i.e., ( A ,y) is a bounded set of scalars for any y) then A is also strongly bounded.
PROOF:Assume the contrary, i.e., that we have a pointwise bounded set A which is unbounded in the strong topology. The latter condition means that we can find a bounded set B in 2!J such that ( A ,B ) is an unbounded set of scalars. Denote by q the seminorm on X defined by q(z) = supyEBI (I, y) I , and similarly, let p(y) = supzEAI ( z , y ) I . Then we recursively define I, E A , yn E B and a, > 0 in the following way. First, let c and C be fixed positive numbers such that c < 1 and C is large (what we shall need is c2(cC - 1) > 2 ). If I , y, a are defined up to n - 1 then let
233
F h c tional Analysis
(for n = 1 write M1 = 0 ). Now choose z, E A such that q(zn) > M,” (we can do that because ( A ,B ) is unbounded) and then choose y, E B such that I (z,,y,) I > c 2 q ( z , ) , which is possible because c < 1 . Finally let u , =
I
(zn,Yn)
1-1’2
.
an-1
Then we can see that a, < - for n 2 2 , and hence the series cc a l + a2 + . . . is summable. Since yn E B which is bounded and since 9 is sequentially complete, we obtain a vector y = alyl a2y2 . . . . We assert that p(y) is infinite, which will be a contradiction. Estimate ( z n , y ) . If n > j then we shall use I ( z n l y j ) I 5 p(yj) , while for n < j we shall use I (z,, yj) I 5 q(z,) . Thus
+
+
Therefore f C is large then
with some positive constant K , and since an -, 0 , we obtain p(y) = +m . Note that if 9 is a Banach space then the proposition above reduces to a special case of the Banach-Steinhaus Theorem.
Proposition C.21. Let ( X ,S,p ) be a totally a-finite measure space and ( Y , 7 ,u ) be a measure space such that its L2-space is separable (equivalently, it has an a t most denumerable orthonormal basis). Let F : X H L2(Y ,7 ,v) be a weakly p-measurable mapping, i.e., ( F ( . ) ,v ) is p-measurable for any v E L2(Y,7 , v) . Then there is an (S(x)7)-measurable function f : A‘ x Y H C which “realizes” F , i.e., f ( ~.),E F ( z ) for almost every z.
PROOF:Let v1, v 2 , . . , be an orthonormal basis of L 2 ( Y ,7,v ) , and let aj be a measurable function on X which equals ( F ( . ) ,v j ) almost everywhere. Choose a measurable representative g, from each v, and set f,(z,y) = C:=, aj(z)gj(y) . Then fn(z, .) is a square integrable function on Y for each z, and the non-negative measurable function
-
equals [IF(.) - fn(z, .)I1 almost everywhere (from now on we use the customary 0 if 2: is such abuse of notation which confuses LP with P). We have b,(z)
234
Appendix C
that a j ( z ) = (F(z),vj) for all j , i.e., almost everywhere. Since ( X , S , p ) is totally o-finite, we can infer by Yegorov’s theorem that there exists a sequence of measurable sets Ak E S such that the complement of their union is p-negligible and b, + 0 uniformly on each Ak . Then apply Cantor’s diagonal process to obtain a subsequence b,,, such that its sum is finite almost everywhere. If it is finite for a certain z then we can use Proposition C.4 to infer that f,,(z,.) tends to some function almost everywhere and in L 2 ( Y , 7 v) , a t the same time. Let f be the function which equals limi+m f,,,(z,y) whenever the latter exists (in C ) and 0 otherwise. Then f clearly satisfies our statement. It is interesting that the above proof does not use Fubini’s theorem. There are slightly different versions of Proposition C.21 which are proved on the base of Fubini’s theorem and the Radon-Nikodym Theorem. I n any case, some kind of separability of v is crucial. We mention that if v is a Radon measure and Y is metrizable and separable thcn L2(Y,v ) is separable.
235
D. Analytic Mappings It seems t h a t the usual approach to analytic mappings is t o define some 'weakly holomorphic' property first, and then invoke complex function theory (most notably, Cauchy's formula) and the Banach-Steinhaus Theorem to prove t h a t analytic mappings are subject to formal manipulations with their Taylor-series in a very nice way. T h i s is good and useful but hides an interesting aspect. We wish to show t h a t if we define analytic mappings to be the sunis of convergent power series (see precise definitions below) then t h e whole theory is very simple, while in practice it is often easy to check that a mapping is analytic, using rather elementary stages of the theory. Throughout, this chapter gothic capitals iiieaii locally convex spaces over F (recall that F always denotes a field which is either R or C), and italic capitals from the end of the alphabet mean normed spaces over F . A family of seminorms defining the locally convex t,opology of ZJ is usually denoted by ( p o ) .
ZJ) the set of those n-linear mappings Notation D . l . We denote by B" (S, f from X" into ZJ for which
is finite for any cy. A 'homogeneous polynomial of order n ' is any function h of the form h ( z ) = f(x,.. . , z ) with some f E R" (A', ZJ) . Let
then we have a natural 'symmetrization' S from B" onto S" defined by Sf (21,. . . , 2,) = C , f ( z ~ ! ~. ). ., , z ~ ( ~ )and ) Sf defines the same polynomial as f ; moreover, for a symmetric f the corresponding polynomial h determines f as we shall see.
5
T h e following inequality can be checked immediately. Let f E B" ( X , 9), gk E A') , p = j i . . . j , and f 0 g E BP (2,ZJ) be defined, naturally, by setting
@'(z,
+ +
Then we have
(D-3) where the norms on the right are the 4's corresponding to t h e norm of X .
236
Appendix D
Definition D.2. A 'power series' from X into 9 is a sequence fo, f i , such that fn E S" (X, 9 ) and there is a number t > 0 satisfying
.. .
Q3
C tn
(D-4)
qa(fn)
< $00
n=O
The supremum of such t 's is called the radius of the power series. The 'sum of the power series' is defined if 9 is sequentially complete; in thir case let h, be the homogeneous polynomial corresponding to f n , r be the radius. and the sum h be defined on the open ball around 0 in X with radius r by setting h(x)= h n ( t ) . If 9 is not sequentially complete but this limit exists by accident then we also call it the sum of the power series. Now let U c X be open and F : U H 9 be some function. F is callea 'analytic' if for any x E U we can find a power series summable around 0 such that F ( x v) = h ( v ) for small v (here h is the sum of the series, and the series itself is called the Taylor-series of F at x , since it is determined by F as we shal, see). On the other hand, a function F : X H 9 which is the sum of a power series with infinite radius is called 'entire'.
cyz0
+
We list a couple of elementary facts which are almost immediate consequences of the definition. Let be sequentially complete in the sequel. If h is the sum ol a power series fn with radius r then h is analytic with Taylor-series
at x (the radius of (f,")n is at least r - 11x11 ). In particular, the translation F ( x .) of a n entire function F is entire and any entire function is analytic. If F is an analytic function then it is 'differentiable term-by-term', i.e., F is differentiable, F' is analytic and if fn is a Taylor-series for F at x then
+
gn(vll...,Vn)u=
( n + 1).fn+l(u,vlr...,vn)
yields a Taylor-series of F' at x. As a consequence, we see that F is differentiable infinitely many times and the Taylor-series at 2: is determined by the formula
in short, fn = $ F ( " ) ( x ) , Applying this to a homogeneous polynomial we see that it determines the corresponding multilinear functional. If we have an analytic mapping f : U H 9 such that the spaces are real, then we can find an extension F of f t o some open subset of the complexification of X , mapping it (complex) analytically into the complexification of . If we fix a possible domain whose connected components intersect the domain o f f then F is unique. This extension enables us to invoke Cauchy's formula even if F = R .
Analytic Mappings
237
If F is analytic from an open subset U of X into 9 and G is analytic from an open subset V of 2 into U then F o G is analytic and its Taylor-series are computed formally, i.e., by manipulating with the infinite sums as though they were fini1,e sums. T h e proof is based on (D-3), and is reduced to the case when all three spa.ces equal C by the “method of positive coefficients”. The proof of the following theorem, which is important in the theory of analytic manifolds, provides a fine example of the method of positive coefficients. We use the following abbreviations. Let A 1 , . . . , A , be any objects for which an associative multiplication has sense. If k is a positive integer and a E pk (i.e., a is a sequence of length k taking values from { 1,2,. . . , p } ) then we write P
(D-6)
A” := A,, . . . A , ,
and
a! := n ( a = j)! j=1
where ( a = j ) denotes the number of i ’ s for which ai = j . Such a’s are sometimes called non-commutative multi-indexes. The lengt,h k of a is denoted by la1 , and the number p is called its dimension. We also use the conventions that “the non-commutative multi-index of length 0” is denoted by 0 and A e is interpreted as identity element.
Theorem D.3. Let U be an open subset of Fn and analytic vector fields on U , i.e.,
A l , .. . , A p
be
n j= 1
with scalar-valued analytic functions ai, for any scalar-valued differentiable function f 011 U . Then suppose that the vector fields B 1 , . . . , B , are analytic combinations of the Ai ’s, i.e., D
with analytic functions b k , . Then for any compact I<
cU
we can find a constant C such that
where f is any infinitely differentiable function, /3 is arbitrary m-dimensional noncommutative multi-index, and the a’s are p-dimensional.
PROOF:One can check by induction on the length of 0 that for any infinitely differentiable f we have
238
Appendix D
where the variable
'*I
runs through a set of piP[ . IpI! elements and
Ia(*)I+
c IPI
14*ls)l
= IPI
for any
*
.
S=l
N o w analyze the most important special case when n = p
A , = 8, bki(z) = and K = (0) , where
1
1
I -.ex, M , R , p , Q are positive constants. Then each factor in (D-7) is positive; namely,
(D-8)
A ~ ~ (=oM) . ~ 1 . 1 . a ! , A " ~ ( o=) p . elal . a ! .
Therefore if some other fixed collect,ion of d a t a is such t h a t the absolute values of the corresponding fact,ors i n (D-7) are bounded by the right hand sides of (D-8), then this other 1Bofl is bounded by the number c = c ( M ,Rip, e , P , p ) = B ' f ( 0 ) of o u r special case. An idea of this fashion is usually referred t o as the method of positive coefficients; in the present proof we shall refer to this particular instance of it as 'the Lemma'. Estimate c . Writing 1 = we have BPf = ( b D ) ' f with D = 8, . T h i s function is of the form
cj
P
+
+
with 2'-'p' terms i n the s u m , and with u ( * , j ) v ( * , j ) 5 21 1 for any factor, as one can easily check by induction on I . T h e numbers N will be multiplied by Ru or ~ Z when J 1 is increased by 1. Writing c(1) = ( b D ) ' f ( O ) we can see t h a t c(1 1) 5 M p ( R V ~ ) ( 2 1 1) . c ( f ) and hence
+
(D-9)
+
c(l)< p . [ 2 p M ( R V e ) ] ' . f !
+
.
+
+
We mention that for p = R we have c ( f 1) = M R . [ ( I l)p 1 1 . c(1) which indicates that (D-9) is not too bad. It is well-known t h a t t,he s u m of the m-dimensional polynomial coefficients of order 1 equals mi' and hence 5 mlPl . T h u s we see t h a t
(D-10) It follows from the term-by-term differentiability of Taylor-series t h a t if g is the s u m of a convergent power series with radius r (over F" ) then & P g ( z ) is the corresponding coefficient of the multi-linear functional in the Taylor-series of g at z,and the latter can be obtained by (D-5). T h u s it follows t h a t if s , t > 0 with s t < r then we can find a n M such t h a t
+
239
Analytic Mappings
(a stronger inequality, involving r! instead of IyI!,can be obtained from Cauchy’s formula; but this is not important for us because 5 nlYl ). Hence if a function h is analytic on U and Ii‘ is a compact in U then we can find positive constants M , R such that.
(D-11) for any n-dimensional multi-index y. Apply (D-11) to the analytic functions a i j , b k i of t h e original statement. Then we can use the Lemma with f = b k , a n d writing the A’s instead of the B’s in it (and writing 8, instead of the A’s i n it). T h u s we obtain from (D-10) that the AQbk,’sfrom t,lre original problem can be estimated by expressions like in (D-8), and writing
we get exactly the statement of the theorem by a sccond application of the Lemma.
Remark D.4.
Using the ‘remaining integral term’ k
1
F ( t ) - ~ ~ F ( ~ ) ( z ) ( . z. . -, z-.) z ,
(D-12)
pg
j=O
=
3.
F ( k t l ) [ z+ t ( z - x)](z - 2,. . . , z - x) dt
of t h e Tlylor-series (which is valid for any sequcn1ially complete g ,normed X a n d function F which is C ( ‘ t + ’on ) a neighborhood of the segment [x,z]) we can see that a C” function satisfying (D-1 I) is necessarily analytic (even if it is infinite diinerisional vector-valued, though in that case 1.1 must be replaced by seminorms p , a n d M may depend on p but R must not). Comparing this fact with Theorem 0 . 3 a n d using the Hahn-Banach Theorem we can see that the analyticity of a smooth (maybe vector-valued) function f can be decided by estiniating the A a f’s on compacts with some vector fields A l , . . . , A,, such that the coefficients az3 are analytic a n d their determinant never vanishes. By other applications of the method of positive coefficients one can prove the Analytic Inverse Function Theorem (for Banach spaces) and the possibility of solving an ordinary differential equation with analytic right hand side by formal manipulations with the Taylor-series a t least locally (again for Banach spaces). For these, it is even enough to use suitable ‘positive power series’ in one variable. By ordinary differential equation we meant a n equation of the form
240
Appendix D
thus including, e.g., systems of ‘explicit’ differential equations of any order in one variable (by the usual trick). Finally we mention that we have statements of the following type: h ( z ) = f(z,t ) dt is analytic in z if f is suitably analytic. A simplest example of this requires f to be analytic in (z,2) which may vary in Banach spaces but dt should be a Radon measure on a compact subset of the corresponding Banach space. More refined statements can be proved by using the existence of complexification, the methods of differentiation under the integral sign (see (3.12-15) and Cauchy’s formula.
24 1
E. Manifolds, Distributions, Differential Operators General reference: [Hell], [Hor]. We adopt the “global point of view”, i.e., we think of the objects of the theory as sets, functions, etc. having local representations in several systems of coordinates (the “local point of view” is to consider the objects as collections of data with prescribed rules for the change of coordinates). We shall be concerned with smooth manifolds modelled on R” and without a boundary. This is defined as follows. Let M be a non-empty set and suppose that a collection (‘pa) of bijective mappings, the so called charts, from subsets of M ont.0 non-empty open subsets of R” is given with the following properties: the union of the domains U , of ‘pa equals M , and the “change of coordinates” is smooth, i.e., if U , n Up # 0 then V = ‘p,(U, n U p ) is open and ‘pp o ’pi’ is infinitely differentiable on V . Then we say that this collection of charts defines a manifold-structure on M , and we call the collection itself an atlas of this structure. Another atlas is said to define the same structure, if the change of coordinates is smooth between them, i.e., if the union of the two atlases still satisfies the axioms above. It is then clear that the union of all atlases defining the same structure is an atlas of this structure, too. Therefore it is customary to define a manifold as a pair (M,d) where A is this largest atlas. Anyway, if a manifold is given then we use the word ‘chart’ to mean any element of A. Of course, the expressions ‘chart’ and ‘atlas’ came from the archetype of a manifold used in modelling the surface of the Earth. The simplest way to obtain new manifolds from old ones is to take the ‘product’: if Mil.. . , Mp are manifolds, then it is easy to see that the mappings (21,.
. . ,.P)
-
(‘pl(4,
’ ’ ’ I
‘pp(.p))
form an atlas on the set M1 x . . . x Mp if P I , . . . ,‘pp run through atlases. T h e corresponding manifold is called the product manifold. Submanifolds and factor manifolds are more tricky; we define them a little later. The domains of all charts form a base of open subsets for a topology on M , and a manifold is always considered as endowed with this topology. Another (equivalent) version of the definition requires M to be a topological space from the outset and the charts must have open domains and be homeomorphic. A slightly more general notion is obtained if the dimension n is allowed t o depend on the chart. But even in such manifolds the pathwise connected components are closed-open, and on a component the dimension is constant. Unfortunately, a manifold is not necessarily a T2 topological space ( a wellknown counter-example is, roughly, the letter ‘Y’with double point a t the junction). If we require the change of coordinates be analytic, we obtain the notion of an analytic manifold. It is also possible to replace R” by C“ and require the change of coordinates be holomorphic, thereby defining a complex manifold. Of course, a complex manifold carries a structure of a 2n-dimensional analytic manifold, and an analytic manifold carries the structure of a smooth manifold, but different “richer” structures may define the same “poorer” one.
242
Appendix E
Definition E.l. In this book the word 'manifold' is used to denote a smooth, analytic or complex manifold with constant dimension, which is a Tz topological space. If the manifold is just smooth, we shall omit the word 'smooth '.
We denote by C k ( M )the set of k times continuously Notation E.2. differentiable complex functions on M , i.e., f E C k ( M )means that f is complex valued and f o ( o i l is k times continuously differentiable (in the real sense) for any chart 'pa . C " ( M ) means the intersection of these sets of functions (the functions i n this space are briefly called 'smooth'). If M is an analytic manifold then we denote by C W ( M )the set of complex valued analytic functions, i.e., for which f oq;' is analytic (in the real sense) for analytic charts p a . If k = 0 then it is omitted, i.e., C ( M ) = C o ( M ) . We use the subscripts 'c' and '0' to denote the subspaces of functions which are compactly supported and vanishing a t infinity, respectively. Thus, e.g., C,"(M) denotes the set of complex valued, compactly supported smooth functions on M . We define locally convex topologies on the spaces C " ( M ) and C , " ( M ) . In C " ( M ) we consider the seminorms P ~ , ~ ,f K) = ( maxK lP(fo cp-')l for any multi-index Q , chart (o and compact subset Ii in the range of (o. These seminorms define a locally convex topology and C " ( M ) will always mean the corresponding locally convex space. The topology of C,"(M) is a bit more complicated. If A' is any compact subset of M then let DK be the subspace of C,"(M) consisting of those functions which vanish outside A'. Let DK be endowed with the topology of C " ( M ) . Then C , " ( M ) will always denote the locally convex space which is the inductive limit of these D K ' S (see Definiton C.7).
A manifold (in the sense of Definition E . l ) is not necessarily paracompact. Its paracompactness is equivalent to the requirement that its components be coverable by a denumerable number of charts (cf. Proposition A . 3 ) . In that case it is also metrizable, moreover, the components can be imbedded into some Rk (Whitney's hheorem). On paracompact, manifolds for any open covering we have locally finite partitions of unity, consisting of compactly supported smooth functions, subordinate to the covering. This is usually used when the covering consists of the domains of some atlas. Proposition E.3. If Ii c U are subsets of a manifold M such that A' is compact and U is open then we can find an f E C,"(M) such that 0 5 f 5 1 , f = 1 on Ii and supp f c U .
PROOF:We can find an open subset V containing Ii' which is paracompact even if M is not paracompact. Consider a locally finite partition of unity on the manifold V (consisting of compactly supported smooth functions) subordinate t o the covering (V n U , V \ A') . Let f be the sum of those members of this covering which do not vanish identically on A'. There can only be a finite number of them, hence f E C,"(V) and therefore, using M is T2, f E C,"(M) if it is defined to be 0 outside V . The other properties are trivial to check now.
Manifolds, Distributions, Differen tial Opera tors
243
This proposition and Theorem C.2 imply t h a t C y ( M ) is dense in C o ( M ) and in LP(M, p ) for any Radon measure p a n d for 1 5 p < +m . Another important density theorem (Theorem E.5 below) concerns with product manifolds. First we prove a lemma. Lemma E.4. T h e set of polynomials is dense in C"(Rk).
PROOF:It follows from the term-by-term differentiability of power series (see Appendix D) t h a t i f f is an entire analytic function then its Taylor polynomials approximate it in t h e Cm-topology. So it is enough t o prove the density of t h e set of entire functions. Let f E Cm(Rk) be arbitrary. Let gn be a compactly supported smooth function which equals 1 on the ball around 0 with radius n. T h e n clearly gnf -+ f . T h u s we may assume that f is compactly supported. Denote by pt the Gauss-kernel:
15
> 0 and 2 E Rk. Since 1 1 1 1 1 ~ = Cf=l , the above formula extends to C k and defines a holomorphic function. Now let
where t I
E
Since f is a compactly supported smooth function, it follows from Proposition C.12 that ft is holomorphic on C k . T h u s it is enough to check t h a t the restriction of ft to Rk tends to f when t 0 . Use the substit.ution 1: y - I and "differentiate under the integral sign" to infer t h a t -+
-+
for any multi-index a . Now the integral of the Gauss-kernel equals 1 for any t , thus is of the form s p t ( z ) . [g(y - I ) - g(y)] dx with some compactly supported smooth g . Then the expression [...I is uniformly bounded and converges to 0 uniformly in y when I 0 . T h u s the proof is completed by observing t h a t 1imt-o p i = 0 for any U whose closure does not contain 0.
a"(ft - f ) ( y )
s,
-+
Theorem E.5. Let M I , . . . , M p be manifolds, a n d T = realized by functions on MI x . . . x M p , i.e.,
fl (23 . .
(23 fp ( E l ,
Then T is dense in CF(M1 x
. . . ,EP) =
+ . .x M p ) .
I-J
fj ( Ij)
BjC,"(Mj)
as
Appendix E
244
PROOF: Since T is a linear space of functions, it is enough t o approximate the elements of a subset linearly generating Cp(M1 x . . . x M p ) . Now Proposition E.3 (or, indeed, a part of its proof) implies that, for an arbitrary open covering, any compactly supported smooth function can be expressed as the sum of smooth functions subordinate to the covering. Therefore we shall approximate a function f whose support is covered by a product chart 'p1 x . . . x 'pp with domain Ul x ' ' . x U p . Let K be this support, K, be the projection of it in U,, and g, be a function for K j c Uj by Proposition E.3 . Let k be the sum of the dimensions of M I , . . . , Mp and h E CF(Rk) be the function which corresponds t o f in the range of this chart and vanishes outside of it. By Lemma E.4, we find a sequence h, of polynomials tending to h in C W ( R k )Thus . h, is the sum of functions of the form uj(xj), and therefore the function
nPZl
(defined to be 0 outside the domain of the chart) belongs to T . Now if the support of ( g 1 @ . . . 8 g p ) is L then, clearly, f,, + f in D L , and hence in C y ( M 1 x . . . x Mp). Let M , N be manifolds. Then a function f : M I+ N is called C k if it is 'Ckon charts', i.e., y5 o f o 'p-' is C k if and 'p are charts on N and M , respectively. The notions of analytic and holomorphic functions (on analytic and complex manifolds) are defined in the same way. If I E M then the tangent space at x (denoted by M,) is the set of "derivations of first order" a t t ,i.e., $J
M , = { v : C"O(M)I--+ C ; 3u E R" ,
'p
E A : v(f)
= d,(fo'p-')('p(x))
}
where '8"' means the derivative at the direction u , i.e., d,h = (h',u ) . T h e elements of this tangent space are called tangent vectors a t 2. Using Proposition E.3 it is easy to see that if in the above formula we replace the domain of v by C k ( U ) with any 1 5 k 5 00 and open neighborhood U of x , we obtain the same thing essentially. Now if we choose U to be the domain of some chart 'p then we see that a linear bijection exists from M , onto R" which is sometimes denoted by d,'p : v(f)= a d z , ( v ) ( f p-')(y(x)) . Clearly d z q p ( d z p ~ ) - '= (YP P , ' ) ' ( ~ C Y (.~ ) ) The tangent brindle T ( M ) is another manifold which is, as a set, the disjoint union of the tangent spaces, and for which an atlas consists of the following charts: If 'p is a chart on M with domain U then consider the function on U{M, ; t E U } which takes v E M , to (p(z),d,p(v)). This function is called the chart of T ( M ) corresponding to 'p. Then we see that the tangent bundle is a In-dimensional manifold, and it can be considered as an analytic manifold if M is such. It is also natural to consider the tangent bundle as a complex manifold if M is complex. If F is a C' function from M into N then its tangent mapping d F is a function from T ( M ) into T ( N ) defined by dF(v)(f)= u ( f o F ) for any f E C ' ( N ) . If we consider charts of M and N and the corresponding charts of the tangent bundles then this tangent mapping is computed simply by taking the derivative of
245
Manifolds, Distributions, Differential Operators
the function. d F maps M , into N q , ) and this restriction is usually denoted by
d,F. Now we can define submanifolds and factor manifolds. Suppose t h a t a subset
S c M has a manifold-structure of its own such t h a t , denoting the inclusion by i : S H M , this mapping is 'regular', i.e., i t is C" and its tangents d,i are injective for all x E S . Then this manifold S is called a submanifold of M . Analytic and complex submanifolds are defined by requiring, in addition, i be analytic and holomorphic, respectively. It follows from the Inverse Function Theorem t h a t if S is a submanifold of M then for any x E S we can find a chart cp of M around x such that cplv is a chart of S with V = cp-'(cp(U)nRk)where U is the domain of cp a n d k is t h e dimension of S (here Rk means the set of vectors in R" with last n - k coordinates 0). If we have the stronger property t h a t we can find charts satisfying the above and V = U n S then we say t h a t S is a strong submanifold of M . In this case it is not necessary that S have a manifold-structure of its own, because it is determined by the set. Clearly, the topology of a submanifold is at least as fine as the subspace topology, and they are the same if and only if we consider a strong submanifold. Any submanifold is determined by its set and topology. Let M be a manifold, '-' be a n equivalence relation on it, N be t h e set of equivalence classes and p : M H N be the factorization. If N can be endowed with a manifold-structure such t h a t p is 'coregular', i.e., i t is C" and d,p is surjective for all z, then N with this structure is called the factor manifold of the equivalence . Of course, one should check t h a t this structure, if exists, is unique. But this follows from the Inverse Function Theorem (cf. Appendix C). We mention t h a t a theorem of Godement gives a nice characterization of equivalences defining factor manifolds; b u t we d o not need it became we shall only be concerned with equivalences defined by closed subgroups on Lie groups in which case it is much simpler to construct the factor structure. A smooth vector field V on M is a C" function from M into T ( M ) satisfying the axiom V(z) E M , Vx . It is clearly determined if we know the functions V(z)(f)for every f E C " ( M ) . So V can be identified with this function-valued mapping on C " ( M ) , whose value is denoted by V f . In this book the expression 'vector field' shall always mean a smooth vector field. It is trivial t h a t for 2, E M , , f , g E C"(U) we have v(fg) = g(z)u(f) f(x)v(g) . It is also clear that the tangent vectors are real, i.e., = .(f). On the other hand, it is not too hard to show t h a t the tangent vectors at z are the only linear functionals with these two properties, so this is an equivalent definition. Then it is easy t o see t h a t N
v(7)
{V :C"(M)
H
+
C "(M) linear ; V is real and V(fg) = g . V f + f . Vg}
is an equivalent definition of the family of vector fields. Unfortunately, this nice characterization does not pass to more general "manifolds" modelled on infinite dimensional spaces. In this book we d o not need the usual machinery of tensor-fields, exterior differentiation, etc. Even Riemannian manifolds are not quite necessary, though invariant distances on Lie groups are important for us, and these are intimately connected with the corresponding Riemannian structure.
Appendix E
246
A (smooth) Riemannian manifold is a manifold on which a “curved scalar product” is given, i.e., a function g on M whose values are scalar products on the corresponding tangent spaces and which is ‘smooth’; the latter property can , be smooth for any be characterized by the requirement that g(z) ( V l ( z ) V2(z)) pair V1,V - of vector fields. Assume that y is a piecewise C’ function from a real interval into a Riemannian manifold M , then the length of this ‘curve’ y is the J g ( y ( t ) ) ( + ( t ) ,+ ( t ) )dt where g is the scalar product of M and i. is the integral ‘tangent at y’,i.e., + ( t ) = d t y . Of course, this arc-length is modelled after the length of a path in Euclidean space. Now the Riemannian distance of two points in M is the infimum of the lengths of piecewise C’ curves connecting them. It is not hard to show that the infimum taken over the smooth curves yields the same, and it is also possible to consider only ‘regular) curves (these are smooth with never vanishing tangent). The Riemannian distance is in accordance with the topology of the manifold. ‘Rectifiable’ curves can be defined. If this metric space is complete then the distance can be realized by a ‘geodetic’ ( a regular curve satisfying a certain variational equation), and any rectifiable curve connecting the points with the same length is a reparametrization of some geodetic. We refer the reader to [Hell] for a nice exposition of Riemannian manifolds.
(2)
Since a manifold is a locally compact TZ space, we can consider Radon measures on it (cf. Notation B.l). If ‘p is a chart with domain U and p is a Radon measure on M then (’p*P If) =
b?f O ‘p)
defines a Radon measure ‘p*p on ’ p ( U ) . We say that p is absolutely continuous if ‘p*p is absolutely continuous with respect to the Lebesgue measure for any chart ‘p (equivalently, for the charts of an at.las). In this case denote by pv,p the corresponding Radon-Nikodym derivatives, and omit the measure from the subscript if this causes no confusion. It, follows from the int,egral transformation theorem t,hat if ‘p, I,!J are charts with domains U , V then
(E-1) pv = (QG
o
41 o ‘p-’) . I det[(I,!Jo ’p-’)’I I almost everywhere on ’p(U n V )
.
On the other hand, if locally integrable functions pv are given for a n atlas of ‘p’s such that (E-1) holds for each pair of charts, then these functions correspond to a well-determined Radon measure on M . We say that a subset A of M is negligible if for any chart ‘p the set ’ p ( A n U ) has Lebesgue measure 0, where U is the domain of ‘p. Equivalently, A is negligible if p(A) = 0 for each absolutely continuous measure p. A function f on M is called locally integrable if f o ’p-’ is locally integrable with respect to the Lebesgue measure for any chart ‘p. Clearly, if p is absolutely continuous with locally bounded pV’s and f is locally integrable then f p is absolutely continuous and pv,f,, = (f o ’p-’) . pV . Notation E.6. We call an absolutely continuous measure p C k if the corresponding Radon-Nikodym derivatives are C k l where k = 0 , 1 , 2 , . . . 00, w . Note that ‘Colmeans more than ‘absolutely continuous’.
247
Manifolds, Distributions, Differential Operators If p is Co then denote by p * ( z ) the measure on M , which is Lebesgue measure”, i.e.,
“e times
the
where 011 the right hand side we integrate with respect to the Lebesgue measure on R” , and e,,, is continuous. We remark that this measure-valued function p C ( z ) can also be considered as a mapping which assigns continuous functions t o n-tuples of vector fields, namely the function f ( z ) = p t ( x ) ( A , ) to the n-tuple (Vl, . . . , Vn) where A , is the parallelepiped in M , spanned by the vectors Vl(z), . . . , Vn(z) . Note the analogy with differential n-forms. It is not hard to see that if M is paracompact then there exists a C“ measure p on M which is ‘st,rictly positive’, i.e., p * ( z ) is positive for all z or equivalently eV,p is positive for all c p . In this case a set A is negligible if and only if it is p-negligible.
If M is a Riemannian manifold then on each tangent space M , consider that Haar measure which corresponds to the scalar product g(;c), i.e., which assigns measure 1 to the cube spanned by a n orthonormal basis. Then these measures form p + ( ~ for ) some strictly positive C” measure p . This measure is called the volume in the Riemannian manifold M . If F is a C’ diffeomorphism from M onto N and p , v are corresponding Co measures on them, i.e., ( v , f )= ( p , f o F ) , then d,F takes the measure p.(z) to v,(F(x)). We can generalize Sard’s theorem as follows. Theorem E.7. Let M , N be n-dimensional manifolds, M be covered by a denurnrrable atlas, and F : M H N he C’. Let v be a Co measure on N and m(y) be the number ofpoints in the set F-’({y}) (we write m(y) = +m if this set is infinite). Then there is a unique Co measure p OJI hf such that p.(z) ( A ) = ~ , ( F ( z )()d , F ( A ) )
for Bore1 measurable A C M ,
the function m is v-measurable, and F takes p to m u , i.e., p ( F - ’ ( B ) ) = whenever B is such that one side of the equality has sense.
;
s,
m dv
SKETCHOF PROOF:Let S be the ‘singular set’ of F , i.e., S= { 2 E M ; d,F is not injective } . This is a closed set, and pC(z.)= 0 for z E S (the existence and uniqueness of p is trivial). Thus Ipl(S) = 0 , for 1pIC= 1p.l . Since M is covered by a denumerable atlas, Sard’s theorem (see B.2) implies that F ( S ) is covered by a denumerable union of sets each of which has Lebesgue measure 0 through some chart. But IvI is absolutely continuous, hence Ivl(F(S))= 0 . So it is enough t o prove the theorem for S = 0 because otherwise we can replace M by M \ S and m will change only on a negligible set.
Appendix E
248
Then assume that d,F is always injective. Then F is locally diffeomorphic by the Inverse Function Theorem. Now define r n ~ ( y )to be the number of points in K n F-’({y}) for any compact I< c M . Fix such a I ( , then K is covered by a finite number of open subsets U 1 , . . . , Up such that F is diffeomorphic on each Uj . Assume that B is a Bore1 subset of N . Let A j = F - l ( B ) n l l ‘ nuj \ ( 4 : : U i ) and for any 01-sequence a of length p let
B, = njV,,,
where
V,,,=
B\F(Aj)
if a, = 1 , if aj = 0 .
Clearly, if y E B, then r n ~ ( y )= (a1 , the number of 1’s in a. Since B is the disjoint union of the B,’s, F-’(B) n K is the disjoint union of the Aj’s and F is diffeomorphic on each Uj , we infer that
It is possible to find an increasing sequence of such K ’ s whose union is the whole M . This finishes the proof for non-negative v. Finally use that 1 ~ corresponds to lvl, for lp*l = 1p1* and Iv+I = lvll . In the theory of linear differential equations it is frequently useful to consider ‘weak solutions’. These objects are differentiable in some weak sense, satisfy the equation, and the weak derivatives should coincide with the “normal” ones if they are actually continuous functions. The ‘distributions’ (introduced by L. Schwartz on Rn originally) do exactly this job. First of all observe that a continuous function f on M is perfectly determined if we know all the numbers
(f o c p - ’ )
where
cp is chart with domain U
and
e E C~(cp(U))
(cf. the remarks after Proposition E.3). Moreover, i f f is merely locally integrable then it is determined up t o a change on a negligible set. Now if f E C ’ ( M ) and V is a vector field then
where t = cp(z) and d,cp[V(z)] = ( a l ( t ) ,..., an(t)) . Since supported, integrating by parts we obtain
e
is compactly
1
Manifolds, Distributions, Differential Operators
249
Now we are in a position to define distributions and their differentiation on a manifold. Observe that what is used to ‘test’ f is really not e but the measure ~ ( tdt) . Thus we shall use to test functions and distributions the set of compactly supported C” measures on M I and denote it by D ( M ) (this is linearly generated by its subsd whose elements are supported by charts). We make D(M ) a locally convex space and the distributions will be the continuous linear functionals on it. If there is a strictly positive smooth measure p on M (which is true for a ny paracompact manifold) then it induces a linear bijection between C,DO(M) and D ( M ) by (o -+ ’pp . Through this bijection we obtain a locally convex topology on D ( M ) . It is independent of p because it coincides with the following one which is defined on any manifold. If K is a compact in M then let V K denote the subspace of V ( M ) consisting of the measures whose support is contained in Ii, and let D K be endowed with a linear metrizable topology such that p,, 0 if and only if -+
ev,pn- 0
in
CO3(4u))
for all charts (p, where we denoted by U the domain of ‘p. Then let V ( M ) be the inductive limit of the V K ’ S . Now let f be a locally integrable function on M . Then it is easy to check that the linear functional
(E-3) is continuous on D ( M ) . Definition E.8. The continuous linear functionals on D(M ) are called distributions. Their set is denoted by V ‘ ( M ) and is endowed with the weak topology, i.e., u,, u means (u,,, p ) --+ (u,p ) for all p E D ( M ) . The locally integrable functions are identified with the corresponding distributions by (E-3). The complex conjugate ii of a distribution u is the only distribution for which ( 2 1 , ~ = ) (ulp) for any real (or equivalently, for any non-negative) p E D ( M ) . If ii = u then u is called ‘real’. The support of a distribution is the complement of the greatest open set on which it vanishes, i.e., -+
If u is a distribution and ‘p is a chart with domain U then it is natural to denote by u o (p-’ the distribution (uop-’, edt)=(u,v)
where
eEC,OO(’p(U)) and v - e l
i.e., e = ev,” and IvI(M \ U ) = 0 . An equivalent approach to distributions is to define them as objects u having coordinates ‘uo (p-’ ’such that the change of coordinates works in the suitable way (thereby reducing the problem of their definition to the case when M is an open subset of Rn).
250
Appendix
E
Using Theorem E.5 we can extend the "tensor product" f1 x . . . x f p of functions t o distributions in the following natural way. Let M 1 , . . . , M p be manifolds. If uj E V ' ( M j ) then write
where p1 x
. . . x p p stands for the product measure.
Theorem E.9. We have a unique distribution u = u1 x . . . x up on the product manifold M = M1 x . . . x Mp satisfying (E-4). For any p E V(M ) we can calculate ( u ,p ) by 'successive application' of the uj 's. This means the following. Let I<, be the projection of s u p p p in M , , choose smooth measures A j on M, such that ( A j ) * do not vanish on I<, , and denote by f the Radon-Nikodym derivative j* . Note that j can be computed on product charts by
Then choose an arbitrary permutation u of 1,. . . , p and define u' be a function on M,(i+l) x . . . x M u ( p ) by setting
This has sense because the functions ui are compactly supported and smooth; and the number u" equals ( u ,p ) .
PROOF:If h is a smooth function on R" then it is easy to check that its corresponding difference-quotients tend to its partial derivatives 8, h in C"(R") , while the difference functions hy(z) = h(x y) - h ( z ) tend to 0 in C"(R") when y + 0 . Therefore if K = s u p p p is covered by one product chart then we infer by induction with respect to i that ui is smooth and compactly supported and its derivatives are constructed by similar recursions. This implies the same for general A' by the usual 'partition of unity' argument. This proves the existence of the number u" and, since the A's should only depend on Ii',thereby defines a linear functional on V K . It is easy to check now that the recursion ui-' -+ ui is continuous in the Cm-topology (for fixed u,(,)). Thus with fixed A's and u (E-5)defines a continuous linear functional p + u" on V K . We infer by Fubini's theorem that this functional satisfies (E-4). Observe that the A's can be constructed so that they are positive on open neighborhoods of the Ii'j's. Apply Theorem E.5 t o these neighborhoods. Then we see that the functional obtained above is independent of the X's and of u. Thus for
+
Manifolds, Distributions, Differential Operators
25 1
any It‘ we have a continuous functional and they coincide on V K nV,t for any K , L . The theorem is proved. We remark that this theorem reduces to Fubini’s theorem if the distributions u l , . . . , u p are locally integrable functions. So it can be considered as a generalization of Fubini’s theorem. It is evident from Theorem E.9 that this tensor product operation is partially continuous (in the weak topology). We know that vector fields can be considered as linear operators in the space C ” ( M ) (they are also continuous). If f E C ” ( M ) then the multiplication by f is also a continuous linear operator in C ” ( M ) . An operator which is built by addition and multiplication of these two types of operators is called a differential operator. Any such operator can be written in the form
where ‘[f]’means the multiplication by f , the V’s are vector fields, the number maxid, is called the order of the expression, and the order of the differential operator is defined to be the least order with which it can be expressed. T h u s the multiplications by a smooth function are exact,ly t,he differential operators of order
0. We mention that there is a slightly different version of the concept of differential operators and their orders which is defined as follows. Let D be a linear operator i n C ” ( M ) which is ‘local’, i.e., if f = 0 i n a neighborhood of z then D f = 0 in a neighborhood of z. Then using Proposition E.3 we can naturally define the ‘restriction’of D to a n y open subset. In this case let D v h = [DI, ( h o p ) ] o p - ’ , where, of course, CI is the domain of cp and cp is a n y chart. Then D is called “differential operator” i n this other sense if there is an atlas such that for 9’s in it we have that D v is a differential operator in our sense on p ( U ) c R ” , i.e., in the usual sense of calculus. Then the order of such a differential operator is defined to be the supremum of the orders of these Dv’s. This may be infinite. We shall call the operators satisfying this other definition locally differential operators because they can be characterized by the property that their restrictions to small enough neighborhoods of each point are differential operators in our sense. If D is a locally differential operator then we define the support of D t o be the smallest closed subset outside of which D vanishes, i.e., the restriction of D t o the complement of it equals 0 (of course, this notion can also be defined for any operator which is local). Now we want to extend the (locally) differential operators to distributions. The idea is very simple: take the second dual. To make this idea work we need the following simple generalization of (E-2). Lemma E.lO. Let N be an open subset of M and p be a smooth measure on M such that 11. never vanishes on N . Let V be a vector field.
252
Appendix E Then there is a smooth function h on N such that
(E-7) where ( a l ( t ) ,. . . , a , ( t ) ) = d,cp[V(t)]
with t = cp(t)
.
PROOF:If the support of g is in U n N where U is the domain of some ~ to obtain (E-6) for any h chart 'p then substitute p = ( g o cp-') . Q ~ in, (E-2) which satisfies (E-7) on U n N with this particular chart. Now observe that (E-6), if holds, determines h (cf. the remark after Proposition E.3), and this can also be said if N is replaced by a smaller open subset. Therefore if (PI, 'p2 are charts with domains 1 7 1 , UZ such that H = Ulnu2nN is not empty, then (E-7) must yield the same h on H using the two charts (of course, this can also be proved by immediate, but rather long, calculation). Thus (E-7) defines a smooth function h on N and (E-6) holds for g's whose support is covered by some chart. But any g is the sum of such ones because there exists a partition of unity, subordinate to an atlas, on some neighborhood of a compact. Let Ii' be a compact in M and D be a locally differential operator on M . Then there is a differential operator E which equals D on some neighborhood of Ii', and hence
(Df,v)=(Ef,v)
for
~ E C ~ ( M , )~ E V .K
We can construct an open set N containing K and a measure p with the properties described in Lemma E.10 . Then it follows from this lemma that we have a differential operator Et on N such that ( E f , g . p ) = (f , Etg . p ) . Thus we see that "the dual of D is a differential operator on V K", i.e., there is a linear operator D' on V K such that
(Of,.)= (f,D*v)
for
f E C"(M)
, ~ E V K,
and if we identify v = gp with g then D ' is the restriction of a differential operator defined on a neighborhood of Ii'. This implies that this D' is continuous from V K into itself. Now V ( M ) is the inductive limit of the V K ' S , therefore the formula
(Df,v) = (f, D'v) defines a continuous operator D'
for
f
E C"(M)
, v EV(M)
.
Definition E.ll. If D is a locally differential operator on the manifold A4 then we extend it to D ' ( M ) by setting
D := D"
, i.e., ( D u ,=~(u, ) D*v) .
Manifolds, Distributions, Differen tial Opera tors
253
The complex conjugate of a locally differential operator D is defined by D f _ = (of), and it is clear that this formula remains valid for distributions, too. If D = D then D is called real; this is equivalent to the requirement that the coefficient functions of each + D '' be real.
-
It is evident from this definition that the algebraic structure of the set of locally differential operators is preserved. In particular, if [f] denotes the multiplication by f then V [ f ]= [ f ] V [Vf]for any vector field V and smooth function f. It is also evident that [ f ' v = f . v , and hence if u is a locally integrable function and f is a smooth function then [ f l u = f u , where on the right we mean the product as a locally integrable function.
+
Definition E.12. We say that a distribution u is function-like up to the order k if Du is a locally integrable function for any compactly supported differential operator D of order 5 k . The set of these differential operators is a module over the ring C" ( M ) if f D means [f]D and clearly it is enough to know that Du is a locally integrable function for a subset of D's generating this module. Thus we see that u is function-like up to k if and only if u o cp-' is such for some atlas of cp's, and on cp(V) this can be decided by considering the differential operators with
IcrI 5 k .
'a
We say that u belongs to the 'local Sobolev space' HEi,':(M) if Du E L p ( M ,p ) for the D's as above and for any smooth measure p. Of course, if there exists a positive smooth measure p then it is enough to check this definition with that one. We see that u E H::(M) if and only if u is function-like u p to k and ( P ( uo Po-')('is locally integrable (or P ( u o cp-') is locally essentially bounded if p = $00 for some atlas of cpb and for all la1 5 k . Versions of Sobolev's lemma can evidently be generalized to this setting. We record here the version which is most important for us: if f E H:d,2(M) with k 2 [ t]+ 1 (where n is the dimension of M ) then f is a continuous function; and if Zi' is a compact in M , A l , . . . , A , are vector fields such that A l ( x ) , . . . , A , ( x ) is a basis for M , for every 2 E Zi' and p is a smooth measure which is strictly positive when restricted to a suitable neighborhood of L', then we can find a constant C (independent o f f ) such that
where L is any compact in the interior of Ii' (the constant C depends on L ) . The (global) Sobolev spaces H k J ' ( M , p ) are usually defined when M is a n open subset of some R" by requiring that P u E L P ( M , p ) for la1 5 k , and if p is the Lebesgue measure then it is omitted from the notation. Now global Sobolev spaces can also be defined naturally by replacing Rn above with a Lie group and using invariant vector fields in the place of the aj's of R" .
254
Appendix E
Note that it is possible to consider more general local Sobolev spaces. With the help of the Fourier transforms F and - on R" (cf. Chapter 7 ) , locally we can construct operators of the type f -+ (k Ff)-where k is some nice function. If k is a polynomial then we obtain a differential operator. Thus operators of this fashion can be called pseud-differential operators. Now some nice sets of these operators can define local Sobolev spaces similarly as in Definition E . 1 2 . Another version is obtained when we require k .Ff E Lq(R") instead of ( k . F f ) - E LP(R") where 1 = 1 . This definition yields a smaller, a greater or the same space when ; p > 2 , p < 2 or p = 2 , respectively. The interesting problem of multiplying distributions is also tractable by the local Fourier transform technique, and leads t o delicate questions about the convolution on R" . +
+
I n the remainder of this chapter we discuss linear partial differential equat i o n ~i t~. , equations of the form Du = f where D is a differential operator and u , f are distributions. Proposition E.13. Let M be a manifold endowed with a positive smooth measure p , D be a differential operator on M , and denote by D+ the differential operator which 'equals D' through p ' , i.e., for which
Let 2 = { ( 2 ,y) E M x M ; t # y } and assume that we have a smooth function A' on 2 which is a "fundamental kernel for D+ ", i.e., for any v E V ( M ) the integrals dv h ( t ) = K ( t , .) dv exist for all 2 , h is smooth and D+h = - . dP Then any distribution u satisfying the equation Du = 0 is a smooth function; arid whenever y E M and 11, E C F ( M ) are such t h a t 11, = 1 on a neighborliood of y then
where T can be any p-measurable set with compact closure whose interior contains supp 11, ( 0 : means that the differentiations are performed in the variable 2: ).
PROOF:Let v = p . p E D ( M ) such that I)= 1 on a neighborhood of suppv and 9 E C r ( M ) such that 9 = 1 on a neighborhood of supp11, . Then writing f(tl
Y) = 9(1) D,+[q.l Y ) ( l - I)(.))]. '
P(Y)
we obtain a compactly supported smooth function on M x M and so f . ( p x p ) E D ( M x M ) . Now test the distribution u x 1 with this measure, where u is a
255
Manifolds, Distributions, Differen tial Opera tors
distribution with Du = 0 and 1 stands for the function which identically equals 1. Then Theorem E.9 yields
(E-9)
/
(%
Y) . P ) d A Y ) =
f(.l
(uz
1
[/
Y) dP(Y)] . P
)
.
Denote by U the interior of the set where $ = 1 and for y E U set
!?(?/I
= (u,d . D + [ I % Y ) ( l
-$)I +>
Then clearly g(y)'p(y) = ( u , f(.,y) ' p ) . T h e latter is smooth for any 'p with s u p p ' p c U (see Theorem E.9) and hence g E C"(U) . We can see that the left hand side of (E-9) is ( 9 , v ) . The integral in the right hand side can be written in the form d(x)D,+ K ( x , .)( 1 - y',(x)) dv
J
(we can use here the simplest method of differentiating under the integral sign). Now writing /I(.) = Z<(zl.) dv we see from (E-9) that
s
( g , v )= ( u , 2 9 . ( D + h - D + ( $ h ) ) ./ I > Now Dth = p by the assumption, d p = p and d D + ( $ h ) = D + ( $ h ) because of the relations between the supports, and D t e . p = D * ( e . p ) for any e E C,"O(M), so the right hand side above reduces to ( u ,v ) , for Du = 0 . This is true for any v with s u p p v c U , i.e., uIu = g . Now this is true for any $, hence u E C " ( M ) . Therefore the definition of g can be given as an integral. Thus we have a formula of the form u(y) = d b d p instead of the desired ~ ( y = ) s T b d p , but we know that b is smooth on M . It follows from the assumptions on T that we can find an increasing sequence F, of compacts and a decreasing sequence Gn of open sets such that s u p p$ c int(F1) the closure of G1 is compact, F n C T C Gn and p(Gn \ F,) < 1/n . Then b is bounded on GI and choosing 19, for Fn,Gn by Proposition E.3 we have
,s
so the proof is complete.
Notation E.14. Let D be a differential operator of order k and consider an expression of order k which equals it:
D=
C [fj]F,~ . . . y,d,
.
j
Using such a notation that dl = . . . = d p = k and the other d j ' s are smaller, set
256
Appendix E
This function x , considered on the cotangent bundle T ' ( M ) (i.e., the disjoint union of the M z ' s ) is called the 'characteristic function' of D. We should check that it is independent of the particular expression of D. Let cp be a chart. Denote by @ the corresponding chart of T ' ( M ) , i.e., for w E M,' let @( w ) = (cp(x), u ) with (u,d,cp(v)) = ( w , v ) for u E M z , where on the left (., .) means the ordinary scalar product in R" . Then it is easy to check that x o @-' is the same as
In the special case when the a,'s are constants, the multiplication by this x" corresponds to the 'principal part' of D" through Fourier transform (this is the reason for the traditional factor i k ) . It follows from the equivalence of (E-10) and (E-11) that the characteristic function can be defined by either of them and is independent of the expression of D as well as of the chart.
Definition E.15. A differential operator D is called 'elliptic' if its characteristic function x does not vanish non-trivially, i.e., x(w) # 0 for any non-zero cotangent vector w . We say that D is 'strongly elliptic'if the real part of the characteristic function is negative for non-zero w's; this is equivalent to saying that for any chart p and any compact I< in the range of 'p we can find a positive 6 such that
We remark that if D is real and the dimension n > 1 then the two notions almost coincide; more precisely, if U is a connected component of M (which is more than one-dimensional) and D is elliptic and real on M then DI, or -DI, is strongly elliptic. Of course, a strongly elliptic operator is necessarily of even order. The archetype of strongly elliptic operators is the Laplacean operator A = C.82 . The good properties of elliptic (and special parabolic) differential equations are crucial in the representation theory of Lie groups. Differently from the former parts of this chapter, from now on we shall discuss statements (concerning such equations) which are far from elementary, and therefore we shall give exact references. Unfortunately, in some cases the author was unable to find a really good reference in the literature; there we include more detailed explanations. J .
'
Theorem E.16. (See [Nor], Theorem 7.4.1 .) Let D be an elliptic differentialoperator oforder lc, and assume that D u E H,'d:(M) for some distribution u. Then necessarily u E H,'Lk'*(M) . In particular, if Du is a smooth function then u must also be a smooth function. Theorem E.17. (A consequence of Theorem 7.4.1 in [Hor], cf. Theorems 4.1.3 and 4.1.8 of [Hor] .) Let D be a strongly elliptic operator on M , I be an open interval in R and V be a
Manifolds, Distributions, Differen t ial Opera tors
257
never vanishing vector field on I . We also denote by D the corresponding differential operator on I x M which does n o t depend on the first variable, and similarly denote by V the corresponding differential operator on I x M which does not depend on the second variable. If V u = Du for some distribution u on I x M then u must be a smooth function. Note that a much more general statement, concerning differential operators which are ‘semi-elliptic in the same fashion at different points’, can be proved by this method. However, Theorem E.17 is quite appropriate for our purposes. The proofs of the two theorems above are based on certain ‘fundamental solutions’ which are the best in some sense. We shall need similar statements based on other fundamental solutions which are best in some slightly different sense. First we discuss some consequences of the work of S. D. Eidelman. Consider the situation of Theorem E.17, but assume, in addition, that M is an analytic manifold and D is analytic, i.e., on charts it has analytic coefficient functions. T h e n Theorem E.17 is improved to the following.
Theorem E.17.a. u ( t ,.) is analytic for any 1 and, moreover, this analyticity is locally uniform in the sense that for any point ( t o , t o ) E I x M we can find a chart p around t o and an open complex ball around cp(z.0) such that for 1’s in a suitable neighborhood of t o t h e analytic function u(1,.) extends holornorphically to this complex ball and the extension is still smooth in 1 , I (in the real sense, i.e., with respect to the corresponding 2n + 1 real variables). It seems this statement could be proved by the methods of [Hor] but not stated there. On the other hand, it is almost stated in [Eid] and, anyway, we need another consequence of Eidelman’s approach (see Theorem E.18 below) which only seems accessible this way. Unfortunately, misprints and inessential oversights hinder those who study [Eid] . These are easily corrigible and I do not know a clearer source to refer to. After formulating Theorem E.18 we dwell a bit on the question of how studying [Eid] yields these results. In the theorem below 6 is some “Dirac’s delta” at a certain fixed ( 2 0 , t o ) E I x M , i.e., (E-13)
(6, .) = $(lO,tO)
where ,Y is a Co measure whose elp,pdoes not vanish at p(i0,zo) and the RadonNikodym derivative in (E-13) is meant just locally. It is easy to check this 6 depends only on p.(to, t o ) and 6 o cp-l equals l / ~( ~ ~ ( 1 0,to)) , ~ times the usual Dirac’s delta at c p ( 2 0 , t o ) .
Theorem E.18. The differential equation (V - D ) u = 6 h a s solutions at least locally around ( 2 0 , z o ) . If u is any distribution satisfying the equation locally then we can find a neighborhood J x B of (0,O) in It”+’ and a locally integrable function v on it which differs from u only by a regular solution (i.e., (V-D)’+’(uocp-’ - u ) = 0 with asuitablechart c p ) and h a s t h e followingproperties.
258
Appendix E
We have an open ball N c C" around 0 such that B = N n R" and we have an extension 6 : J x N H C of v such that G ( t , .) is holomorphic for all t , v' is smooth (in the real sense) on { t E J ; 1 > 0 ) x N , satisfies [(V - D)"]" 6 = 0 there, and (E-14) ~ i j ( t , z + i y ) l ~1 s . t - " l k . e x p ((-~zllill+ + ~ 3 1 l y l l + ) t*) for t > o and v ( t , .) = 0
for 1 5 0
where x,y are real vectors, k is the order of D and C1,Cz, C, are positive constants. The suitable (analytic) chart cp above can be prescribed with the constraints that it should be of the form cp(t,z) = ( f ( t - t o ) , $(I)) with $(zo) = 0 and the sign of f ( t - t o ) must equal the "sign" of the vector field V . Note that the facts that ij satisfies the extended equation for t > 0 and i t is analytic in space make i t possible to estimate any derivative of v' by (E-14). In particular, we can see that v' is smooth on the set
Since our statements are of local nature, in order to prove them one may forget about manifolds and assume that M is a bounded domain in R" . Divide by the absolute value of the coefficient function of V , use the substitution t + f ( t - t o ) as indicated above and, if necessary, reduce both I and M a little in order to modify V - D into a differential operator d a,@, z)-8., possessing the following ilol property. A neighborhood N of M in C" can be found such that the coefficient functions a, are restrictions of functions 6, : I x N H C which are smooth in the real sense with bounded derivatives and for any fixed t b ( t , .) is holomorphic and, finally, for the corresponding characteristic function we have the strong elliptic property uniformly, i.e.,
at xlalsk
lal=k
with some positive constant n. Unfortunately, we have not yet arrived at the setting of [Eid] because it is vital for the construction of Eidelman's 'fundamental solution' Z ( t , ~ , i , tthat ) the coefficients also be extendible to I x R" with good enough properties. Note this localization problem is somewhat slighted in the relevant Section I/9 of [Eid] leaving it t o the reader. Now we show how t o do this. Of course, we just need to extend from a smaller M1 (for our goal is local), and so we obtain a smooth extension with bounded derivatives and uniform strong elliptic property if we can find a smooth function F : R" H M with bounded derivatives and satisfying F is identical on M I . For our purposes it is enough t o consider the case when M1 is a ball around 10 and then F ( I ) = f(x). I (1 - f(z)) . i o will do whenever f E CF(R") is such that 0 _< f _< 1 , f = 1 on M I and suppf is contained in a convex subset of M .
+
Manifolds, Distributions, Differential Operators
259
So we have established the conditions necessary to apply the results of [Eid] . It is a nuisance the results we need are scattered in that book and are formulated slightly differently. Some remarks may smoothen the path for those who want to check the validity of our statements. Though not easily, it can be collect4 (mainly from Section I/9, Subsections I/3.2 and I/3.5 of [Eid]) that Z ( t , ~ , t , < ) is smooth (in the real sense) for 2 > T when considered on ( I x ( N U R“))’ with a suitably small complex neighborhood N and satisfies an estimate similar to the one in Theorem E.18. Then take v ( t , t ) = Z(t,O,z,O) for Theorem E.18; while for Theorem E.17.a infer from Lemma E.10 that a substitution 2 + -t makes the adjoint (used in Proposition E.13 ) of a parabolic operator parabolic (here ‘parabolic’ is meant i n the sense of [Eid] and ‘adjoint’ is meant with respect to the Lebesgue measure). In fact, later parts of [Eid] contain arguments of this type with the extra burden of considering coefficients with minimal smoothness conditions (for us this is of no importance). For more detailed remarks on [Eid] (especially how to correct the oversights i n the crucial Section I/9 of it) see Subsection 1.4 in [Ma4]. We shall discuss some good properties of elliptic and strongly elliptic equations (Theorems E.19 and E.22 below) whose relevance to representations was discovered by R. P. Langlands and can be found in [Lan] as “lemmas” (these are in fact powcrful theorems). Theorem E.19 is essentially proved in [Lan] (it is assumed there from the outset that u is continuous). We include a proof here for two reasons: first because the readers may encounter difficulties when looking for [Lan] in libraries (for it is not a regular publication, just a thesis) and second because the author prefers the proof below to Langlands’ one. We mention that this theorem is a special case of tlie results i n [Bro], though it seems that [Bro] concerns with real elliptic operators in more than one dimension, so essentially with strongly elliptic operators. Unfortunately, [Bro] is a proclamation of results rather than a usual article with proofs even though the ideas of proofs are sketched. Now the proof below is just a n “expansion” of F. E. Browder’s ideas. Theorem E.19. Let D be an elliptic differential operator of order m > 0 on the nianifold M . If u is a distribution such that Du is a continuous function then u E Crn-l(M) .
PROOF:Since the statement is local, we can assume that M is an open subset of R” containing 0 and it is enough to show the C”-’ property in a small neighborhood of 0. Then we write D = Ial<m _ u,aa where a , E C”O(M) (we do not use the usual ( - i ) l a l ’ s here because we want to invoke [John] where they are not used either). Let A = CIaI=, u,(O)P considered as an operator on R” . Write Q(<)= U , ( O ) < ~ , and set
c
cIaI=,
Appendix E
260
if the dimension n is odd or even, respectively, where S = {( E R" ; 11(11 = 1 } is the unit sphere, w is the usual surface measure on it and A is the Laplacean operator on R" . These are the 'fundamental solutions' of F. John, and actually copied from the formulas (3.54) and (3.61) of [John] (the formal deviation from [John]-(3.54) is necessary because we allow complex coefficients and then m may be odd). It is supposed in [John] that A is real, and also that n > 1 , but these restrictions can be omitted without any effect of consequence (the case n = 1 is disposed of by then). We are not interested observing that K ( z , y ) = sign(x - ' (. - ')"-' 2(m - l)! am(0) in the precise forms above, just in the following properties. First of all, it is clear that I< depends only on x - y, and we denote the corresponding function by G, i.e., I<(z,y) = G(z - y) . Then we have the usual formula
(cf. our Chapter 2 ) , and therefore 1;'s being a fundamental solution in the sense of [John] is equivalent to saying that AG = 6 , where 6 is the ordinary Dirac's delta, i.e., the distribution (6, 'p. A) = cp(0) where X is the Lebesgue measure. This fundamental solution G is an analytic function on R" \ {0}, and it has the following rather nice structure. There is an analytic function F : R x S I+ C and another analytic function w : R" H C such that
G(rC) = rm-" . F ( r , c ) + log r . w ( r ( ) for any r
>0,
cES
.
This is proved in [John] (the corresponding local statement for differential operators with analytically varying coefficients is comprised in formulas (3.46) and (3.49) of [John], and thereabouts we also find the niceties of the case of constant coefficients). In fact, w = 0 if n is odd (for the general analytically variable case, too) and also if n > in (in the particular case of our operator A ) . T h e statement about n > m follows from formula (3.48) of [John]. The function w can vanish in other cases, too; and even in the most annoying case when it does not it has the additional good property that w ( z ) = Ilzllm-" . w l ( 2 ) with another analytic function wl (see (3.63a) of [John]). We mention that the proofs of the statements about the logarithmic term are based on the evident formulas
2 llzll > 0 , C = , b is the vector field on S which is expressed in 1141 CjjCja, , s is any number and f,g are coordinates of R" by 4 = 8, - Cy='=l
where r = the
26 1
Manifolds, Distributions, Differential Operators
arbitrary differentiable functions on suitable domains. Now these formulas enable us to state that
with some analytic functions F, and wo . If the logarithmic term is not absent then necessarily n is even and 5 m . Then we have the following estimates for small positive r = 1 1 ~ 1 1:
2 C=. llzll d"(G * f ) = ( P G ) * f
with some constant C , where
for any compactly supported Now let B,f = distribution f (the derivatives are understood in the sense of distributions). It follows from (E-15) that if la1 5 m - 1 then 8°C is a locally integrable function and is the same when interpreted in the classical sense; and then B,f can be calculated by the classical convolution (formula (2-3) ) for any compactly supported integrable function f . If la1 = m then d a = d j @ with some j = 1 , 2 , . . . , n and = m - 1 , so doG is locally integrable, and we have with
where pj is the measure on S which equals the n - 1 dimensional volume of the projection in the j - t h direction if c j 2 0 and the opposite of this if c j 5 0 (it is easy to check that pj = . w ). Now write d"G(e) = r-"F,(O,C) H ( e ) in the first term. Then H is locally integrable by (E-15), so the corresponding term has a limit and can be separated from the rest. The remainder of the first integral under the limit equals
c,
+
(the upper limit of the integration with respect to r is actually finite because 'p is compactly supported). Let c = F,(O, .) dw (we shall prove that it must equal 0) and write y ( r ) = F,(O,C)'p(r(')dw(C) - ccp(0) . Then we have
s,
s,
and thus if c did not vanish then we would have a term of magnitude I IogEI under the limit whenever 'p(0) # 0 while the second integral is bounded by (E-15). Thus c = 0 and the first integral under the limit in (E-16) converges, since
Appendix E
262
is bounded. T h u s t h e second integral must converge, too. Now replacing
by 'p(0) in i t , we have the same limit by (E-15). So eventually we see t h a t there is a number c(a) such t h a t 8°C - c ( a ) b equals the 'principal value of the classical derivative', i.e., its value when tested with 'p equals t h e limit of t h e first integral in the right hand side of (E-16). For t h e moment denote by G, the function which equals 8°C when llzll 2 E and 0 otherwise. Then we know t h a t lim,,o G, = 8°C - c ( a ) S weakly, i.e., in D'(R") . T h i s implies t h a t (P(E<)
(E-17)
Baf = c(a)f
+ lim(G, * f ) c-0
for any compactly supported distribution f . NOW if f E Lp(R") with 1 < p < $03 then we are exactly in a position to apply the results of [C-Zy] to the singular integral S~-"F,(0,(')f(y - z ) d x (we have already verified t h a t F,(O,.)dw = 0 , and the other assumption of [C-Zy] is trivially satisfied, for Fa(O,.), being analytic, is Lipschitz). T h e n Theorems 1 and 7 of Chapter 1 of [C-Zy] show t h a t this singular integral converges in Lp(R") and the norm of the limit is bounded by Co(a,p)11 fIIp . Now recall from (E-15) that d a G ( z )- r-"F,(O,C) is a locally integrable function, and infer by (E-17) and by the Hausdorff-Young Inequality that if we fix a compact neighborhood Zi' of 0 then we have
1,
(E-18)
Baf E Lp(R") and llBafllp 5 C ( a , p ) Ilfllp if la1 = m , f E LP(R") , supp f c Zi' and 1 < p < +03
(we have used that the LP topology is stronger than the weak distribution topology, and even the latter is Aausdorff). Now fix a p > R and choose an open neighborhood R of 0 such t h a t R c M n Zi and R is so small that
(note t h a t the existence of such an R is ensured even if the coefficient functions a are just continuous at 0). Denote by Eo the restriction of Bg t o LP(Q), i.e., if f E Lp(R) then consider i t as 0 outside R and let Eof = ( G * f ) l , . T h e n (E-18) o EO is a bounded linear operator in Lp(R) for any and (E-19) imply t h a t la1 5 m and 11 (D- A)ln Eoll < 1 . Denote this last operator by R , and set
aff
j =O
+
Since Al, Eo = Z , we have DI, Eo = Z R and so DI, E = I . O n t h e other hand, the operators E, = da o E = (8. o & ) ( I R)-l are bounded in P ( R ) for la1 5 m . T h u s
(E-20)
+
E E B( LP(R),HmiP(R))
Manifolds, Distributions, Differential Opera t ors
263
and therefore the range of E is contained in Cm-'(52) because p > n (for a proof of the relevant theorem of Sobolev see [D-Sch], pp.1680 et seq. and especially p.1686 for the statement we need). Now consider our original distribution u for which Du is continuous. Since the closure of 52 is compact in M , the restriction f = (Du)ln is bounded. Hence f E LP(f2) for any p and so Ef E C"'-'(52) . But DI, E = I , and thus in 52 we have D ( u - Ef)= 0 . Then an application of Theorem E.16 completes the proof. It is well known that if n = 1 then we have u E Cm(M) (and this can also be proved by the method above and without invoking [John] and [C-Zy], for then G(") = 1 6 and = &I is bounded in any space). In contrast to arn(0) this, 11 is not necessarily a C"' function for any dimension n 2 2 (even if D is the Laplacean operator on R" ). Note that (E-20) is valid for a "non-smooth, just continuous differential operator L)". Now there are theorems in the literature ensuring that any solution of Du = 0 is C" (i.e., a classical solution) provided that certain functions involved in the construction of D are continuously differentiable certain times (then, of course, the original distribution u should be such that Du makes sense). Then the conclusion of Theorem E.19 is also valid (cf. [Bro]). Our next objective is Theorem E.22 below, which is essentially taken from [Lan] and is also related to (Eid] . The proof is naturally divided into an analysis of differcntial operators with constant coefficients and an application of that to the particular problem. The tool necessary for the latter is the following lemma. a, (-i)I"lP be an operator of order k > Leimila E.20. Let D = Clalsk 0 on an open subset U of R" such that a, E Clal(U) ( D will be applied to functions f E C k ( V )). Let A ( w ) = C, a,(w) (-i)lalc?F , considered as a family of differeribial operators with constant coefficients on R" , depending on the parameter w E U . Assume that we have a family of fundamental solutions C(z,w) of A ( w ) (i.e., locally integrable functions u = G(.,w) for which A ( w ) u = 6 ) such that G is Ckon (Rn\ (0)) x U and there are positive constants K , V (independent of w ) with which
(E-21)
laP:G(xc,
W)l
5 K . I 1 O g I I ~ l II 1141k-n-IPI '
if 0
< 11x11 < 7
+
whenever 1/31 IyI 5 k . Let f E C k ( U ) and 11 E C r ( U ) . Then for any x E U we have
where lc(*)I = 1, the sum of the lengths of the orders of differentiations in the terms of is = la(*)l for any * and lp(*)I 5 k - 1 for any * .
Appendix E
264
PROOF:Denote by B, the ball around z with radius E in R" and consider the integrals
G(z - Y, Y) . $(Y) . Df(Y) dY .
I ( & )= L B .
Since the function G(z - y , y ) . $(y) is integrable in y by (E-21), we have limc_.o I ( & ) = I ( 0 ) . Now, of course, we want to integrate by parts. If u, v are C' functions on R" \ {z} such that supp(uv) c U then we have (by Fubini's theorem) for any E > 0
<,
where p, = . w is the corresponding surface measure on the unit sphere S (cf. the explanation after (E-16) ). Therefore if
then the expression in (E-23) tends to 0 with E . Now apply (E-21) t o see that I ( 0 ) = lim,,o J ( E ) where J ( E ) is the result of shifting a t most k- 1 differentiations from f to a,G$ in each term, so we have
with such that the functions h j are of the form f.Cc.$[ $(y). a, (y). G (z-y, y)] 1/31 = k - 1 for each term, I,L?(*)l 5 k - 1 for every * , and the c's and c(*)'s are of absolute value 1 . J 2 vanishes because even its integrand vanishes (since G ( . ,y) is a C ksolution of A(y)u = 0 on R"\{O}). I t follows from (E-21) that lim,+O J ~ ( Eis) an integral of the form C , of (E-22). So i t remains to prove that L = lim,_.o Jo(E) (which necessarily exists) actually equals $ ( z ) f ( ~.) Because of (E-21) the only terms of the integrand of JO which may contribute to the limit a r e o f t h e form c.f(y).$(y)-a,(y).(dfG)(z-y,y). Ifwe replace these by c.f(z).$(z).a,(t).(a~G)(z-y,z)then the integral of the difference tends to 0 (use (E-21) and Lagrange's inequality). Thus we see that L depends only on A ( z ) , $(z) and f(z) (and linearly on the latter two). To calculate L we may then assume z = that D = A(z) , U = R" , f E CF(Rn)and consider a sequence $ ~ ~ ( y) $I(. + $) with $1(z) = 1 . With these data we obtain liw+m J l ( 0 ) = 0 and
+
Manifolds, Distributions, Differential Operators
265
limm-+mI(O) = JRm G(z-~,z).(A(z)f)(y)dy, i.e., L = (G(.,z) (A(t)f)(z- .)) (we have used the substitution y + 3: - y and identified D(R") with C,OO(Rn) as usual). Since a: = -13, in this identification and is interchangeable with translations, we get L = (A(z)G(.,z),f(z - .)) = (6, f ( x - .)) = f(x) and the proof is complete.
a,
Notation E.21. We call an open set R c C a 'U-domain' if it is defined by the angles - x / 2 5 a < 0 < /3 5 x / 2 and the real number B in the following way:
(E-24)
R = { z E C ; 36 E (a,@): Re(zei') > B }
Such sets look like the greater one of the regions into which the plane is divided when we take out a "U-nail)) (a curve consisting of two half-lines and a circular arc). This U-nail becomes an angle if B I 0 and if, in addition, -a = /3 = x / 2 then it is a half-line (for B = 0 ) or the empty set (for B < 0 ). U-domains arise naturally in connection with holomorphic semigroups, cf. Theorem 1.19 . Let D be a strongly elliptic differential operator of Theorem E.22. order k > 0 on the manifold M . Fix a point xo E M . Then we can find a U-domain R in the complex plane with the following property. For any neighborhood W of x0 there is a constant C such that for any function f which is ckon we have
w
for all X E R , where .(A)
is the distance of X from C
\R.
PROOF:Since the statement is local, we can assume that M is an open 1 and let A ( w ) = subset in R" and 20 = 0 . Let D = Caaa-aa 214
C, a,(w) (-i)I"la: . We shall construct a U-domain R and a nice fundamental solution G(.,w,X) for the operator X - A(w) if X E R and if w is small. Then Lemma E.20 will imply the result. First, observe that the usual change of the path of integration implies the following. Let C, e, F, p be positive constants, p > 1 , and denote by E the set of entire functions f : C" I-+ C for which
Consider the inverse Fourier transform
Appendix E
266
for f E 8 . Then these are entire functions and satisfy the estimate
5
(cf. Eidelman's where C1 , e l , F1 are some positive constants and q = Lemma 1.1 in 5 I/1 of [Eid]). Fix a neighborhood U of 0 such that the closure of U is compact in M . Then the operators A(w), w E U , are uniformly strongly elliptic, i.e., denoting the characteristic function of A(w) by x w lwe have a constant po > 0 such that
and with some constant M we also have in U that
Then set (E-28) where
d -(3g) dt
z E C" and t > 0 . This is motivated, of course, by the facts that
= F ( A ( w ) g ( z ) ) ('3'means the Fourier transform in the variable z ) and that limt,03g = 1 in a suitable sense (we shall be more precise soon). Now consider the Laplace transform
1
00
(E-29)
G ( z ,w,A) =
e-A'g(r, w ,t ) dt
If all formal manipulations are justified then G( ., w, A) is a fundamental solution for the differential operator A - A(w). We want t o make this statement more precise and, a t the same time, to obtain certain estimates for the derivatives of this function G. We have the elementary inequality c;' I p j c j where p j , cj > 0 and Cj pj = 1 . From this we infer that for any c > 0 we have a positive number F ( E ) such that
nj
where z = t + iy , z , y E R" and t
> 0 . Write
cj
Manifolds, Distributions, Differential Operators
267
and compare ( E 3 0 ) , (E-26) and (E-27) to obtain the existence of positive constants el F, C such that
+
where, as usual, z = 2: iy with z,yE R" . Then Proposition (2.12 can be used to check that the derivatives 8f8: ( e f ) l (the inverse Fourier transform is in the first variable) can be obtained by differentiating under the integral sign (where /3 2 is arbitrary and IyI I m ). Now observe that for any E > 0 the function eEr is bounded on the positive half-line, use a substit,ution u =
[
(as usual, z = z + i y ). Here the positive constaiit B depends only o n h4, m ,k, and QO , and l i ( d ) is a constant depending, in addition, o n d . Hence if A ( w ) is replaced by e " A ( w ) such that the angle 29 is small enough to have (E-26) for this modified family (with different eo maybe) then (E-31) is still valid. Thus, with other positive constants, (E-31) holds if we allow t to vary in a suitable sector I arg tl < 90 and replace t by It1 on the right hand side. It follows (e.g., from Proposition C.14) that this extended g is holomorphic in the variable t .
Let zo E R" \ (0) and let z vary in a suitably small neighborhood of zo in C" . Then we see from (E-31) that afa,'g(z, w,2 ) tends t o 0 uniformly in z , w when 2 --+ 0 . This fact and the fact that g is holomorphic in t imply the following. The Laplace transform G ( z ,w, A ) can be interpreted as
(E-32) whenever Re(A() > B and Iarg(1 < 9 0 with = 1 ; and the derivatives in z and w can be calculated under the integral sign. Denote by R the U-domain consisting of those A's for which we can find an above ( . We have arrived at the following stage: C(xlw, A) when considered with domain ( R" \ (0) ) x U x R is a smooth function which is analytic in the first and third variables, and its derivatives with respect to the first and second variables can be computed under the integral sign in (E-32) (as well as with respect t o A but that is not important for us).
268
Appendix E This implies first of all that for
+ E R" \ (0)
e-xc'(&g)(x, w,C2) C dt =
A(w),G(+,w,A) =
1"
(E-33)
d
. -[g(z, dt
w , Ct)]dl
=
we have
1
"
d - ~ [ e - .~g (~x , w, ~ ]c t ) dt =
X . G(+,w,A) (the integration by parts should be performed on compact subintervals and then observe that the first factor kills the singularity a t 00 while the second one at 0). Compare (E-32) with (E-31) to obtain the following estimates for G. If V c R" is some region and h : V H U is any measurable function then
+
I(dfd:G)(+, h ( + ) ,X)l dx 5 li' . r ( X ) (E-34)
-1
+
if [PI 5 k - 1 and IPI IyI L k where K is some constant and r(X) is the distance of X from C \ R . These calculations were based on Fubini's theorem which is justified by (E-31). T h e same implies that
so we see that for any A E R
(E-35) We also need pointwise estimates near 0. Split the integral of (E-32) into three parts a t 2 = 11x11' and t = 1 (for small x ) and estimate FlF2F3 with F2F3, Fz and F1 on the thirds, respectively, where F1 = e-'' with s < .(A) ,
F2 = t
-181-" k
and F3 = exp
(-PI
.
")
($)
. Thus we get (for 11x11 5 1 )
+
w,XI1 I C l ( P , Y ) e(lPI, 11. ) CZ(X, P,r) where e ( j , s) = s k - " - j if j # k - n and e ( j , s ) = Ilogsl if j = k - n . We can now infer by (E-33) and applying the usual integration by parts that (A - A(w))G(.,w, A) must be a constant times 6 . We assert that this constant c equals 1 , Let cp E CF(R") such that cp(0) = 1 , and let cpy(x) = c p ( : ) , then for any v we have I(@:G)(x,
'
(E-36)
c = ((A - A(w))G(.,w, A ) , P I / ) = (G(.,w,A), (A - A ( W ) * ) c p o , ) and the sequence ( A - A(w)*)cp, tends t o the constant function X - a e ( w ) locally uniformly and under a bound. Then (E-35) shows that c = 1 . Putting together the information obtained we see that X - D satisfies (on U ) the conditions imposed on D in Lemma E.20. Choose a II, E C r ( V n W ) satisfying $(O) = 1 and apply (E-34) t o complete the proof.
269
F. Locally Compact Groups, Lie Groups General reference: [H-R], [Kap], [Rudl]; [Boul], [H-Sch], [Hell], [Serl,2]. Since this book is concerned with representations of Lie groups, it is natural that the readers must be familiar with the rudiments of the theory of Lie groups themselves. Nonetheless, we collect a lot of important facts below. Proofs can be found in the literature. Definition F . l . A ‘locally compact group’ is a locally compact space G endowed with a group structure such that the division I ,y + xy-’ is continuous from G x G into G. A ‘real Lie group’ is an analytic manifold G endowed with a group structure such that the division is analytic. A ‘complex Lie group’ is the analogous object arising from complex manifolds (then the division is required to be holomorphic). Since any complex Lie group of dimension n is also a real Lie group of dimension 2n, we have the natural hierarchy of these three notions. In this book the expression ‘Lie group’ will be used a s an abbreviation of ‘real Lie group’. We note t h a t for (real or complex) Lie groups it is enough t o require that the multiplication be analytic (or holomorphic) because the stronger assumption of the definition follows from the Inverse Function Theorem then. For locally compact groups the analogous statement does not hold. We shall discuss the relations between these three concepts below, but first recall what is the ‘Lie algebra of a Lie group’. It has several equivalent definitions. The one which is also applicable to locally compact groups is t o describe it as the set go of the so called one-parameter subgroups, i.e., continuous homomorphisms from R into G I where R is considered as a Lie group with addition as its group operation. Unfortunately, it is not so clear from this description that this set has a n y algebraic structure (in fact, if it has then the locally compact group G is more closely related to Lie groups). Now if C is a Lie group then denote by g1 the tangent space of G at the identity 1 (we always denote the identity of a general group by 1, while if the operation of a commutative group is denoted by ‘+’ then its identity will be denoted by 0). It is known that any one-parameter subgroup of a Lie group is actually analytic, and taking its tangent vector at 0, i.e., := do7 (&) , the mapping y + i is a bijection from go onto g1 . Thus we have a linear structure on the Lie algebra. To obtain the so called Lie product we must consider a third realization. Before that we fix some notation.
+
We denote by L, and R, the left and right translation N o t a t i o n F.2. by the element g on any group G, i.e., L,(x) = g x and R,(x) = zg . On a Lie group these are analytic diffeomorphisms, thus their tangent mappings are diffeomorphisms of the tangent bundle T ( G ) ,and we use the “module notation” gv := dL,(v) , v g := dR,(v) , which is justified by the corresponding associative property.
270
Appendix F
Now a vector field V on G is called left invariant if V(g) = gV(1) for any g E G , and it is called right invariant if V(g) = V(1)g for any g E G . Denote by gt and gp the sets of left and right invariant vector fields, respectively. I f f is any function on G then we use the notations [L(g)f](z) = f(g-’x) , [R(g)f](z) = f ( z g ) . Then a vector field is left invariant if and only if it commutes with each L(g) when considered as a differential operator on CDO(G).Any differential operator commuting with each L(g) will be called a left invariant differential operator, and right invariant differential operators are defined similarly. We denote by Ue and 4 the sets of left and right invariant real differential operators, respectively (cf. Definition E . l l ) . It is easy to see that t,he operation [ A , B ] := A E - B A , considered on the set of differential operators, results a vector field if A , B were such. On the other hand, Ue and 4 are evidently (real) subalgebras of the algebra of all differential operators (endowed with operator-multiplication). Thus and gr become Lie algebras with the Lie product [., .] . It is known that they generate the objects Uc and 4 ,respectively, if the latters are considered as real associative algebras with identity. Consider the mappings
(these are, in fact, the tangent mappings of the regular representations L and R on C”(G) , but representations are discussed in the main text). Then we obtain two Lie algebra structures on g1 through these linear bijections, and they actually coincide. It is also known that 4 realizes the universal enveloping algebra of the Lie algebra gp and the case is similar with left invariant operators. Thus, of course, we have a natural isomorphism between 4 and Ue which can explicitly be given by D + J D J with J f (x) = f (z-’) . Evidently, this isomorphism extends to the complex hulls, i.e., to the sets of all right and left invariant differential operators. We shall use the expression ‘the Lie algebra g of the Lie group G’ to mean any of the four realizations go , g1 , ge or gp but, at heart, we prefer to think that “ g = g1 ” because of its natural identification with some Lie algebra of matrices in the special case when G is a ‘matrix Lie group’ (see Notation F.4 below). If L1 is “the” universal enveloping algebra of g (i.e., any realization which is fixed in the context of a particular discussion) then we use the notations d L and d R to denote 4. and U H Ue ; and also t o denote their complex linear the isomorphisms U extensions to C 8~ U .
Locally Compact Groups, Lie Groups
271
The algebraic operations of go (for a Lie group) can equivalently be defined by Xv(t) = v(Xt)
,
(u
+ t ) ) ( t )= n-cc lim
(F-3) [ u , v ] ( t )= n,m-cc lirn ( u
(A) );( );( v
u
1)
(;))^“‘
Notation F.4. Let G L ( n , F ) denote, as usual, the set of invertible n x n matrices over F,considered as a Lie group or complex Lie group if F = R or C , respectively. We also use the notation G L ( V ) for the Lie group of invertible linear operators on the finite dimensional (real or complex) vector space V . Let B ( V ) denote the ring of all linear operators on V as is customary i n functional analysis. We call G a ‘matrix Lie group’ if it is a strong Lie subgroup of some G L ( n ,R) , i.e., it is a subgroup and a strong submanifold a t the same time (strong Lie subgroups can be identified with closed subgroups in any Lie group by a theorem of h. Cartan). ‘Complex matrix Lic groups’ are defined analogously, though in this case we must consider complex strong snbmanifolds and Cartan’s theorem does not apply. Since G L ( n ,F) is an open subset of the set of all matrices, and the latter is, in a sense, Fna, it is customary to identify the Lie algebra of a matrix Lie group with the corresponding subspace of matrices through the “identical” chart of T ( G L ( n , F ) ) . This is further motivated by the fact that the Lie product turns into the cornmutator A B - B A of mat.rices then. We sometimes also use, somewhat loosely, the term ‘mat,rix Lie group’ to denote a Lie group G which is just isomorphic to a matrix Lie group, especially when G is a strong Lie subgroup of some G L ( V ) . In this case we also use the corresponding identification of g wit,h a Lie algebra of operators (the difference from matrices is that we do not distinguish a particular basis in V then). Notation F.5. We use the abbreviation g = L, o R,-I , i.e., g ( t ) = gtg-’ . As usual, Ad(g) := dlg E GL(g) and ad := dl Ad where the Lie algebra of GL(g) is identified with B(g) as in Notation F . 4 . Then Proposition F . 3 implies that ad(u) v = [ u , IJ] . We write exp(v) := v(1) where v E go and use this exponential mapping on the other realizations of the Lie algebra accordingly. This notation comes from, and is justified by, the fact that in the case of matrix Lie groups we have
Since do exp is bijective, and exp is analytic, we have an analytic inverse mapping ( explu)-’ on some open neighborhood U of 0. If U is also connected then we shall call such an inverse mapping ‘a logarithm’ on G . A related concept is that of the ‘product chart’. Let a l , . . . ’ a , be a basis of g , then exp(tla1) .. .exp(t,a,) is an
Appendix F
272
analytic mapping from R” into G whose derivative a t 0 is invertible. Taking the analytic inverse of it on a suitable neighborhood of 1 we obtain a so called product chart with respect to the basis a l , . . . ,Q, . If f : G I-+ H is a morphism of Lie groups (which is equivalent, by Cartan’s theorem, to saying that it is a continuous homomorphism of groups) then we use the abuse of language df := dl f (which is motivated by the fact that df(zv) = f (z)df ( v ) with the notation of Appendix E and similarly with right translations). It is evident from the definitions that
However, this becomes a powerful theorem through the various identifications. A special case of it is the important formula (F-5)
exp( eadu v ) = exp u exp
exp(-u)
.
Now return to the relations between locally compact groups, Lie groups and complex Lie groups. The relation between Lie groups and complex Lie groups is relatively simple. A Lie group G admits a complex structure if and only if g can be endowed with the structure of a complex vector space in such a way that Ad(g) be complex linear for any g E G . If such a complex linear structure is given on g then it, together with the real structure of G, determines the corresponding complex Lie group. The exponential mapping is holomorphic for a complex Lie group. If G is connected then the complex linearity of every Ad(g) is provided by the complex linearity of ad, i.e., by the possibility of turning g into a Lie algebra over C . Cartan’s theorem implies that if a locally compact group G admits a Lie group structure then the latter is unique. So it has sense to ask when a locally compact group is actually a Lie group. An answer was suggested in the famous 5th problem of Hilbert. The serious efforts of the mathematical community in solving this problem ended in complete success (see Theorem F.6 below). First discuss the much easier question of relations between Lie groups and “smooth Lie groups” (the latter ones being defined analogously but requiring the manifold and the division be just smooth). Here the answer is simply that they are the same, i.e., the maximal atlas of a smooth Lie group contains the maximal atlas of a unique analytic Lie group. The proof is based on the facts that the exponential mapping is already defined by the smooth structure and the multiplication in logarithmic coordinates is analytic. T h e last statement is a consequence of the Hausdorff-Campbell-Baker Formula: log( exp u exp v ) = u (F-6)
+
I’
)vdt
e $(eadu
for small u , v
tadu
r logr with $(r) = r-1
where the definition of 1c, means that consider the analytic mapping defined around I in B ( g ) which has analogous Taylor-series as the function written in (F-6) has
273
Locally Compact Groups, Lie Groups
around 1. We mention that the so called ‘version of Dynkin’ expresses the multiplication in logarithmic coordinates by a series of iterated Lie products in a quite explicit manner. We record here the key step of the proof of (F-6) (cf. [H-Sch]) because it is important in its own right. Namely, if y : R Q g is a smooth curie then dt(exP 07)= (exp Y(l)) . a(- ad y ( 4 ) .Y” %J
-, the tangent mapping of a curve (on the left) is ( j l)! identified with the corresponding tangent vector as usual, the first ‘ . ’ on the right where
d(z)
=
+
refers to translation in T ( G ) and the second one to application of the linear operator to the vector in g . We know that exp - E = (exp z)-’ and the tangent of the inverse-taking operation maps g x to -zg-’ for E E g . Thus the identity above with -y yields an analogous formula with right translation. In short, we have
(F-7) (d, exp)(z) = (exp u ) . d ( - ad u ) z = d(ad u)2: . (exp u )
ez - 1 with d ( z ) = -
z
for all u , E ~g . Some not too complicated topological considerations and (F-6) imply that the functor f 4 df establishes an imbedding of categories from the category of simply connected Lie groups into the category of finite dimensional real Lie algebras. The fact that this imbedding is actually an equivalence (i.e., “each algebra has a group”) is usually referred to as ‘Lie’s third theorem’ even though it was first proved by E. Cartan in 1936. A related, and essentially stronger, theorem of Ado asserts that each finite dimensional real Lie algebra can be imbedded into B(R”) with some n . Note that the connected Lie groups are exactly the factors of the simply connected Lie groups by discrete normal subgroups; the latters are necessarily contained in the centrum of the corresponding Lie group. Turning to Hilbert’s 5th problem we note that go and exp are defined for any locally compact group G (though with no good properties perhaps) and one naturally tries to use (F-3) to obtain a Lie algebra at least. T h e main difficulty lies with constructing enough one-parameter subgroups. We refer the reader t o [Kap] for a nice exposition of this topic (as well as of some others concerning Lie algebras).
Theorem F.6 ( Montgomery-Zippin-Gleason-Yamabe Theorem, S o l u t i o n of Hilbert’s 5th P r o b l e m ) . Let G be a locally compact group without small subgroups, i.e., there be a neighborhood U of 1 in G such that U contains no subgroups other than { 1) (this is true for any Lie group). Then G must be a Lie group. Any connected locally compact group is a ‘Lie-limit’, i.e., if U is a neighborhood of 1 in it then there is a closed normal subgroup contained in U such that the factor group is a Lie group. If a locally compact group is locally connected and of finite dimension (with respect to the inductive dimension, see [Kur]) then it h a s no small subgroups and
Appendix F
274
consequently is a Lie group. In particular, this is true for locally compact groups which are ‘topological manifolds’, i.e., for which an ‘atlas’ exists with continuous change of coordinates. Recall that the factor group GIN of the locally compact group G with respect to its closed normal subgroup N is the set of cosets endowed with the usual multiplication and with the factor topology, which in this case is the unique topology with which the factorization is a continuous open mapping. If G is a Lie group then these factor groups are also analytic factor manifolds. A Lie subgroup of a (real or complex) Lie group is an (analytic or complex) submanifold and subgroup such that it is also a Lie group with the inherited multiplication. Strong Lie subgroups are those Lie subgroups which are strong submanifolds; in the real case these coincide with the closed subgroups.
Notation F.7. Let G be a locally compact group or a (real or complex) Lie group and H be a closed subgroup or a strong Lie subgroup of it, respectively. Then we write
G/H = { x H ; x E G} ,
(F-8)
H\G = { H X ; x E G}
where these sets are considered as endowed with the action of G and with the factor topology or the structure of an (analytic or complex) factor manifold. It is not hard to check by Proposition A.3 that these homogeneous spaces are paracompact. For Lie groups the action of G is analytic (holomorphic) and a chart cp is obtained in the following way. Let b be the Lie algebra of H (considered as a subalgebra of g ) and V b e any linear subspace such that g = f~ @ V as linear spaces. Writing v = v1 212 according t o this decomposition, let f ( v ) = expv2 expvl and g(v) = expvl expv2 . Then c p ( f ( v ) H ) = v2 or cp(Hg(v)) = 212 with small v yields a chart for the space of left or right cosets, respectively (more precisely, one should fix a basis in V to obtain charts). We shall call these charts ‘logarithmic with respect t o V’.
+
Maybe the most important tool in studying locally compact groups is the Haar measure. Recall that if G is a locally compact group then there exists a positive Radon measure p on G satisfying
and any Radon measure satisfying the above is a scalar multiple of this p . T h e positive multiples are called ‘left Haar measures’ on G. Of course, the situation is arialogous with right Haar measures. We mention, as a curiosity, that the Haar measures are completion regular, i.e., any negligible set which is contained in a denumerable union of compacts is covered by a negligible Baire set. For Lie groups a Haar measure can simply be constructed by considering p * ( g ) = g( p,(1)) with an arbitrarily chosen positive ~ ~ ( 1(cf. ) Notation E.6).
Locally Compact Groups, Lie Groups
275
This also implies that Haar measures are C” for Lie groups. In fact, Haar measures on Lie groups correspond to (left or right) invariant differential n-forms ( n is the dimension of G), and also coincide with the volumes of (left or right) invariant Riemannian structures. It follows from (F-7)that if the eigen-values XI,. . . , A, uf some ad u satisfy 29(Xj) # 0 (note the set of roots of the entire function 29 equals 2 4Z \ { 0)) ) then exp is regular a t u , and in the corresponding “logarithmic chart” the functions
(F-9)
pl(z) = Idetd(-adz)l
and
e,.(z)
= Idetd(adz)l
are exactly the Radon-Nikodym derivatives of any left and right Haar measure, respectively, with respect to the measure p4(1) on g if p is the Haar measure in question.
Notation F.8. Let G be a locally compact group. If p is a left Haar measure on it then it is clear that
defines another left Haar measure, and hence v = c . p with some positive number c, which depends only on g and independent of p . Following the tradition, we write c = A (g-’) so that
(F-10) and call A the ‘modular function on G’ (we mention that some authors call d(g) = A (g-’) the modular function, which practice is equally justifiable). If the modular function identically equals 1 (i.e., if the left Haar measures are also right Haar measures) then G is called ‘unimodular’. It is easy to see that the modular function is a continuous homomorphism into the Lie group of positive numbers with multiplication as group operation. Therefore any compact group is unimodular. If C is a Lie group then A is analytic and we have the formulas
(F-11)
A(g) = I det ( A d (9-’)
) I , d A ( v ) = - tr(ad v)
Thus it is not hard to decide the unimodularity of a Lie group, especially when it is connected. If p is a left Haar measure on the locally compact group G then clearly (v,f)= ( p , f (z-’) ) defines a right Haar measure. We have
(F-12)
Appendix F
276
Now consider the question of quasi-invariant measures on homogeneous spaces G / H . A theorem of A. Weil states that we have a non-zero Radon measure on G / H which is invariant under the action of G if and only if the restriction of the modular function of G to H equals the modular function of H (and then this invariant measure is unique up to a constant factor). In particular, if H is compact or if H is a normal subgroup then this always holds. If G is a Lie group then the above condition is equivalent to saying that d e t ( P Ad(h)l,) = f l for any h E H , where P is any projection in the Lie algebra g with kernel f~ , the Lie algebra of H , and V is the range of P . If H is connected then this is equivalent to the requirement that tr ( P ad = 0 for any v E b .
.Iv)
N o t a t i o n F.9. Let G be a locally compact group, H be a closed subgroup of it, and X = G / H . We use the term ‘quasi-invariant measure’ to mean a positive Radon measure p on X which “answers with a continuous Radon-Nikodym derivative to the action of G ” , i.e., for which we have a continuous function F : G x A’ H R such that (F-13)
Note that there are slightly different versions of the concept of quasi-invariant measures in the literature. Most notably, G. W. Mackey, the father of this concept, required only that we have measurable F ( g , . ) for any g but he considered just M2 groups. It was discovered a little later (by L. €I. Loomis according to the author’s knowledge) that there are quasi-invariant measures in our stronger sense on any homogeneous space X . For Lie groups there are even smooth quasi-invariant measures. For a short exposition see [ M a l l . N o t a t i o n F.lO. The ‘conditional expectation’ of a (nice enough) function f on G with respect t o the closed subgroup H is the function Ef on G / H defined by the formula (F-14)
E f ( P ( Z ) )=
J f ( Z h ) dh
where dh means a fixed left Haar measure on H and p : G H G / H is the factorization. A ‘pfunction’ with respect to ( G ,H ) is a locally p-integrable function e on G ( p is some Haar measure on G ) such that
(F-15)
If Y
e
is a pfunction and p is a left Haar measure on G then the measure z1, v of compactly supported bounded
= e . p has the property that for any pair
Locally Compact Groups, Lie Groups
277
measurable functions
(F-16) If e is continuous and positive then this v gives rise to a quasi-invariant measure d x through
(F-17)
Proposition F . l l . The left (right) Haar measure of a locally compact group G is finite if and only if G is compact. Proposition F.12. Let G be compact, H be a closed subgroup of it, p and v be Haar measures on G and I!, respectively, and let u be a G-invariant Radon measure on X = G / H . Denote b y p the factorization and let E be the conditional expectation with respect to v. Then we have
and, consequently, if G is a Lie group then
where the tangent space of X at p(1) is identified with some subspace V of g (cf. Notation F.7 ,) and the product measure is interpreted with respect to the decomposition g = fj @I I/ . The following proposition essentially follows from Theorem E.7 and is applied in the proof of Weyl's formula (see Lemma 6.15). Proposition F.13. Let G' be a Liegroup consistingofan at most denumerable number of connected components, H be a closed subgroup of G, X = G / H and let P be a projection in the Lie algebra g of G with kernel 6 , the Lie algebra of H . Suppose that X admits a G-invariant Radon measure and consider such a measure u satisfying u. ( p ( 1)) x v, (1) = p* (1) where p and v are some fixed left Haar measures on G and H , respectively, and the product is interpreted with r e spect to the decomposition by the projection P . Let a C' function f : X x H H G be given, and let m(g) denote the cardinality of the set f - ' ( ( 9 ) ) If any of the two integrands below is integrable then
Appendix F
278
where a ( p ( g ) ,h ) = I det F ( g , h)l and F ( g , h ) E B ( g ) is defined by
F ( g , h ) 21 = [ f ( P ( g ) , h)-'1 d(p(g),h)f( g p v , h(v - p v ) ) (we identified the tangent space of X a t p ( 1) with the range of P in g , and we use the module notation for the action of G on T ( X ) and T ( G ) ,so the expression means the action of this element).
[..I
Let G be a connected Lie group and consider a left invariant metric on G which is constructed from some left invariant Riemannian structure (see Appendix E). Denote by 191 the distance of g from 1 with respect t o this metric. It is clear from the construction that if 191 < ( n + 1)' with some positive r and positive integer n then we can find hl,h2 such that lhll < nr , lhzl < r and g = hlhz (simply choose a suitable hl from a short enough curve connecting 1 with 9 ) . So if we write B, = {g; 191 < n r } then B,+1 c B,B1 . Since 1 has a compact neighborhood, we can find an r > 0 such that the closed ball around 1 with radius 2r is compact (as a matter of fact, every closed ball in a complete Riemannian manifold is compact, so the above is true for any r , but we obtain a much simpler proof for that shortly). Then the closure of the set B1 B1 is compact, hence covered by some finite collection of left translations a,Bl of B1 . Thus we infer by induction on n that B, B1 is covered by the union of ai, . . . ai, B1 . This implies that, denoting a left Haar measure by p , p ( B , + 1 ) 5 p" . p ( B 1 ) if p is the number of the ai's (the compactness of any closed ball also follows). Note that B,+1 c B I B , also holds, thus a similar inequality (with another p ) can b e obtained for the right Haar measure. Thus we can see that the Haar measures are a t most exponentially increasing, i.e., there is a number K: such that
(F-18)
J
exp(-+I)
dx
< +oo
if dx means a left or right Haar measure (possibly with different I E ' S ) . Another easy but important consequence of the relation B,+1 c B,B1 is that i f f is any locally bounded submultiplicative function on G (the latter means that If ( t y ) l 5 If(x)l . If(y)l ) then with some positive constants M , C we have
(F-19)
1j(x)l 5 M
. eclzl
for all x .
Of course, (F-18) and (F-19) are valid (with other constants) for a right invariant metric , too. Notation F.14. When studying distributions on Lie groups, it is customary t o identify D(G) with CT(G) through the bijection cp . p -+ cp where p is some fixed Haar measure. In this book we mostly use a left Haar measure to this purpose. If L) is a locally differential operator on G then D' becomes a locally differential operator D+ through this identification, i.e., D+ is the locally differential operator for which
(F-20)
Locally Compact Groups, Lie Groups (it follows from the suitable density of C,oO(C) that cf. Theorem C.2).
279
Dt is determined by (F-20),
Observe that the operation D D+ is independent of the choice of the left Haar measure p . Clearly, this operation is an involution on the algebra of locally differential operators. Lemma E.10 shows that if V is a vector field then V + = -V [f] with some smooth function f . It is easy to see that -+
+
Vt = -V Vt = -V - ( V ( l ) , A )
(F-21)
for V E gr for V E gt .
and
We shall denote by b b+ the corresponding involution on the abstract enveloping algebra L1 and also on C@R,LIthrough d L . Thus ci . ai,l . . . a;,k,)' = C;(-l)knci.ai,k,. . . ai,l if ai,j E g , so this involution coincides with the one usually considered on the enveloping algebra of an abstract Lie algebra. -+
(xi
The LCvi-Malcev Theorem asserts that any finite dimensional Lie algebra over F is the semi-direct product of its radical by some semi-simple subalgebra (we refer to the literature for the basic definitions). We mention that this theorem forms a key step in a proof of Lie's third theorem as well as in a proof of Ado's theorem. Thanks to the Ldvi-Malcev Theorem, the theory of semi-simple Lie algebras and their representations can be applied to obtain information about general Lie algebras and Lie groups. The theory of semi-simple Lie algebras became a gem of mathematics due t o the work of several great scientists, most notably E. Cartan and H . Weyl. We have enough textbooks on the topic by now to omit here the rudiments. On the other hand, we must fix notation. First recall that a real finite dimensional Lie algebra L is semi-simple if and only if its complexification A = C @ R L= Lei.1, is a complex semi-simple Lie algebra. Semi-simplicity is equivalent to the non-singularity of the Killing-form, which we always denote by (., .) , i.e., ( u , v ) = tr(ad u . ad v ) .
(F-22)
Notation F.15. Let A be a fixed finite dimensional complex semi-simple Lie algebra and M be a fixed Cartan subalgebra of it. Denote by R the set of roots of M in A (we also consider 0 as a root), let Ro = R \ (0) and denote by A , the root space corresponding to the root a E Ro , so we have [u, v] = a(.) * v if u E M and v E A , . For each a E Ro denote by h , the unique element of M satisfying ( h a , u )= a(.) for all u E M and denote by H , the dual system, i.e., n
L
Ha = h, . The real vector space spanned by these elements is denoted by ( h a ,ha) Mr. Let now 17 be a fixed simple set of roots, i.e., h ' is a basis of the space of linear functionals on M , taken from R , such that a or -a has non-negative integer coordinates with respect to this basis for any root a . Then the altitude of a root
Appendix F
280
is defined to be the sum of its coordinates with respect to 17. Roots with positive altitude are called ‘positive’, their set is denoted by R+ . The ‘Dynkin diagram’ is a graph whose set of vertices is 17, two different vertices a , P are connected with a ( H p ) . ,B(Ha) lines, and where we have multiple lines there we give a direction to them from the longer root to the shorter root ( a is called longer than ,L? if (herha) > ( h p , h p ) ). Then the Cartan matrix (which consists of the numbers a ( H p ) ) can be read from the Dynkin diagram. The set { u E M , ; a ( u ) # 0 V a E Ro} is such that its connected components, the so called Weyl sectors are in a bijective relation with the possible 17’s; namely, if 17 is a simple set of roots then its Weyl sector is the set { u E M , ; a(.) > 0 va E n}. A closely related concept is that of the Stiefel diagram: this is the set (F-23) We call the connected components of the complement of the Stiefel diagram in i . M,. ‘cells’; the cells are congruent, and those cells whose closure contains 0 are of the form
where S is some Weyl sector and D is the set of highest roots with respect to the simple set IT corresponding to S (each simple component of A contributes one highest root). In particular, if A is simple then the cells are simplexes. The Stiefel diagram is useful in studying compact groups, cf. [Dy-0] . The Weyl group W is the group of operators in B ( M ) which are restrictions of inner automorphisms of A . This group is generated by the Weyl reflections s, (where s,u = u - a ( u ) H , ). W permutes the Weyl sectors one-transitively. It is known that another Cartan subalgebra can be mapped onto M by some inner automorphism. Thus different Dynkin diagrams of A are obtained from each other by inner automorphisms (use the Weyl group, too). On the other hand, nonisomorphic Lie algebras can not have isomorphic Dynkin diagrams. More precisely, we have the following theorem.
Theorem F.16. Let A’ be another semi-simple Lie algebra with M ’ , 17’ and suppose that f : 17 I+ I l l is an isomorphism between the Dynkin diagrams. Choose x, E A, \ (0) and yo E A$(,) \ (0) for a E 17 . Then there exists exactly one isomorphism F from A onto A’ such that f ( a )= (Y o F-’ and y, = F x , for every a E 17 . Notation F.17. It is known that if we choose non-zero elements I, E A, for each a E Ro then with some non-zero complex numbers c m , d a , p we have [x,, x - ~ ]= c,H,
and
[x,,
t p ] = d,,px,+p
if a
+ P E Ro
.
Locally Compact Groups, Lie Groups
28 1
This collection of 2,’s is called a ‘Chevalley basis’ if we have c, = 1 and d,,p = -d-a,-p everywhere in the formula above. The existence of a Chevalley basis essentially follows from Theorem F.16, and for a Chevalley basis the numbers d,,p are integers whose absolute value depends only on a , P .
- Since the structural coefficients of a Chevalley basis are real, we also have d,,p = -d-,,-p and c, E R . If a collection 2, satisfies at least these weaker properties then consider the unique conjugate linear mapping T which is identical on i . M,. and takes x, to -x-,. This T preserves the Lie product, evidently T’ = I , and ( u ,v) = - ( u , r v ) defines a positive definite scalar product on A . The T constructed above is the most general mapping with these properties which, in addition, leaves M invariant, while any ‘posit,ive conjugation’ (i.e., conjugate linear mapping with these properties) leaves some Cartan subalgebra invariant. The set where a positive conjugation is identical is a so called compact semi-simple real Lie algebra. If we fix a T then we can consider adjoints with respect to the corresponding scalar product, and then F’ = T F - ’ T for any automorphism F , which enables us to take the polar decomposition
F = U . eadx
,
Ur=rU ,
TX=
-x
Now if L is a real semi-simple Lie algebra and A is its complexifcation then we have a conjugate linear automorphisin u on A defined by u(x i y ) = x - i y where z , y E L . We can find (e.g., with the help of the polar decomposition) a positive conjugation T which commutes with u. This T establishes a so called Cartan decomposition on L . Then one naturally looks for a Cartan subalgebra M of A which is in “good relation” with u and r . Of course, we always require that they leave it invariant. One good thing is when UT is identical on a largest possible part of M , while another one when it equals -I on a largest possible part of M . A concept arising from the latter is the Iwasawa decomposition, see an exposition of it in [Hell]. The existence of the former follows from a theorem of F. Gantmacher and is useful, e.g., in classifying the real semi-simple Lie algebras. For the exposition of this particular topic we refer to [Ma2]. The different connected Lie groups having the same semi-simple Lie algebra are also classified, a nice account of the results (without proofs) can be found in [Tits]. We use the customary notations of the classical groups, i.e., S L ( n ,F) is the group of n x n matrices with determinant 1; U ( n ) is the group of unitary matrices, S U ( n ) = U ( n ) n S L ( nC , ) ; O ( n ,F) , Sp(n,F) , O ( p ,q ) are the symmetry groups of the corresponding bilinear forms; etc. The Lie algebras of these groups are denoted by the corresponding lower case German letters, e.g., s ~ ( n )s o, ( p , q ) .
+
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29 1
Index of Notation First we list those notations whose first letter is fixed, then those beginning with a non-letter, and finally those with a variable beginning (e.g., in Dt the exponent is the notation). Bold expressions refer to numbered items of the book, expressions in parentheses to displayed formulas. The pages on which these are found follow after a colon. Single numbers mean pages.
d(A) 1.11: 12 d(s,A) 1.11: 12 Ad F.5: 271 ad F.5: 271 B(p)
(3-4): 41
C 206 C(X) 142, C . l : 219 Cc(X) C . l : 219 Co(X) C . l : 219 Ck(G) E.2: 242 C"(M) E.2: 242 C"(M) E.2: 242 Ck(T) 3.16: 53-54, 3.19: 55-56 Cm(T) 3.16: 53-54, 3.19: 55-56 C"(T) 3.16: 53-54, 3.19: 55-56 Ch 6.19: 130 Cha 6.20: 130-131 Chs 6.20: 130-131
206 192 244 (F-1): 270 (F-1): 270 3.12: 51 245 244 C(A) etA
exP
F
F G(T) G'(T)
1.11: 12 1.2: 4
F.5: 271 206 143
(3-10): 52 3.11: 50 GL(n,F) F.4: 271
Hlkd,"(M) E.12: 253 H k J (M , p ) 253 'If0 (8-1): 155 'HI (&la): 156 3-11, 8.1: 156 1-1, 8.1: 156 'Hh 8.1: 156 'HCP (8-5): 158 inl(A)
J
207 (7-4): 142
L 3.25: 58-59, F.2: 269-270 LP(X,S,p) c . 3 : 220 LP(X,X) c . 3 : 220 LP(X) c . 3 : 220 Lyo,(x,X) c . 5 : 221 Lyoc(x) c . 5 : 221 P(X,S,p) (2.3: 220 P(X,X) c . 3 : 220 P(X) c . 3 : 220 cyoc(X,x) c . 5 : 221 ,q0,(X) c . 5 : 221 N
No Nl
N2
206 (2-8): 28 (2-9): 28 (2-12): 30
O(n,F) 281 O(P1Q) 281 Pk (3-13): 55 p t (used in 5 4 ) p+ (used in $ 4 )
R R
R
76 4.7: 82
206 3.25: 58-59, F.2: 269-270
206
Index of Notation
292
a''
R(A) 18 r (spectral radius)
140
SL(n,F) 281 sl(n,F) 281 SO(n,F) 281 so(n,F) 281 SO(p,q) 281 so(p,q) 281 Sp (spectrum of an element in an algebra) 140 Sp(n,F) 281 sp(n,F) 281 SU(n) 281 su(n) 281 T 206 T(A4) 244 U(n) 281 ~ ( n ) 281 u 51, 270, 279 Uc ( = C @ R U )
Z Z,
51
206 206
p (strong topology of functionals) c.9: 222 A (Laplacean) 256 A (modular function) (F-10): 275 d d,
206 206
dT
206 56
1~
206
[u]
3.21: 57
[fl
251
A (closure)
207
T
(contragredient) 3.10: 49 G (spectrum of a group) 6.7: 117-118
T(G)' (commutant) 6.6: 116 b+ 279 D+ E.13: 254, F.14: 278 A# 223 A* 223 X' c.9: 222 W-defin able
3.2: 40
T(y) 4-5,30 T(cp,W) 39 To(i.,W) 39 Te(y,M.') 51 G ( c P , W ) 51
M,
A"
244
z"
(D-6): 237 206
,u*
E.6: 247, 274
293
Index We use the same conventions as in the Index of Notation. Absolutely continuous (with respect to another measure) 213 absolutely continuous (on a manifold) 246 (B-2): 211 absolute value (= total variation) of a pre-measure abstract discrete cocycle 184, 185 (0-1,2): 1 action adjoint of a n operator 223 of an element in an enveloping algebra GO Ado’s theorem 279 algebra of sets 21 1 almost everywhere 212 almost periodic representation 113 Analytic Inverse Function Theorem 239 analytic function D.2: 236, 244 manifold 241, E . l : 242 vector (for an operator) 1.11: 12 vector (for a representation) 3.16: 53-54 anti-isomorphism (of extensions) 178 anti-unitary operator 173 approximation of Dirac’s delta (2-14): 31 atlas 24 1 Baire measurable set 215 measure 215 topological space 210 Banach-Steinhaus Theorem 223 barycenter 225 base of a topology 207 Boolean algebra 211 Bore1 measurable set 214 measure 215 structure of a topological space bounded (in a locally convex space) Carathbodory extension (B-3): 212 Cartan decomposition 28 1 matrix 280 subalgebra 279, 280 Cauchy-net 208 CCR group 151, 153, 204
167 c.9:222
294
Index
cell 280 central disintegration 149 function 122 E.14: 255-256 characteristic function of a differential operator 6.12: 122 character (of a representation) chart 24 1 Chevalley basis F.17: 280-281 Choquet 's theorem 226 Closed Graph Theorem 223 closed set 207 closure 207 cocycle, commutator 180 cocycle of a Lie algebra 181 of a selection (= discrete cocycle) (9-15): 183 178 combined isomorphism class (of extensions) commutator cocycle 180 compact Lie algebra 124, 281 I15 operator (= completely continuous operator) topological space 209 complete uniform space 208 completion of a measure space 212 complex conjugate of a differential operator E . l l : 253 of a distribution E.8: 249 of an element in an enveloping algebra GO complex Lie group F . l : 269 manifold 241, E.l: 242 measure 21 1 F.lO: 276 conditional expectation (on a homogeneous space) 179 conjugation of a Hilbert space continuous function 208 projective representation 175, 9.3: 177 represen tation 1 contraction type, one-parameter semigroup of 1.G: 7, 17 contragredien t 3.10: 49 convergence 207 convolution 25-30 general 2.6: 28 of a distribution and a compactly supported smootli function
2.5:
27 2.1: 25 of finite complex Radon measures of locally integrable functions 2G countably compact topological space 209 C kfunction 244 C kmeasure E.6: 246 C kvector 3.16: 53-54 C*-algebra 219
Index
295
Daniell-Stone Theorem 215 dense 207 derivative (infinite dimensional, strong) (C-5): 226 derivative, weak (C-6): 226 differential operator 251, E.ll:252 dihedral group 192 Dirac’s delta (E-13): 257 Dirac’s delta, approximation of (2-14): 31 discrete cocycle (= cocycle of a selection) (9-15): 183 series 119-120 distribution E.8: 249 dual Gdrding subspace 3.11: 50 object (= spectrum) of a topological group 6.7: 117-118 Dynkin diagram 280 Eidelman’s theorems 257-259 elliptic differential operator E.15: 256 element in an enveloping algebra 62 entire function D.2:236 vector (for an operator) 1.11: 12 enveloping algebra of a Lie algebra 51, 270, 270 equivalence of projective representations 9.4: 177 of unitary representations 6.5: 116 extension of a Lie group 9.5: 178 Factor group 274 manifold 245 represen tation 118, 148, 150 filterbase 207 210 first category, set of (= meager set) Fourier transform 142-143, 254, 266 Frkchet space 223 Fubini’s theorem 214 for Radon measures 216-217 Galilean groups 189- 190 Gauss-kernel 73, 243 Girding subspace (3-10): 52 GCR (= tame = type I ) group 115 Gelfand map 142 Gelfand-Raikov Theorem 114 generator (of a one-parameter semigroup) Gleason’s theorem 172 GNS construction 141, 226
1.2: 3-4
296
Index
H a a r measure 274-277 Hahn-Banach Theorem 222 Hausdorff-Campbell-Baker formula (F-6): 272 Hausdorff (= TZ ) topological space 208 Hausdorff-Young Inequality 2.12: 32-33 Heisenberg groups 111, 194 highest weight 132, 6.22: 133, 135-136 Hilbert’s 5th problem, solution of F.6: 273-274 Hille’s formula (1-8): 9 Hille-Yosida Theorem 1.8: 8-9 holomorphic semigroup 1.15: 17 homogeneous polynomial (on a Banach space) D . l : 235 space F.7: 274 homotopy 92, 126 Holder’s inequality (C-2): 220 hull-kernel (= Jacobson) topology 118, 141
Ideal of sets 211 imprimitivity, system of 7.10: 152, 159, 165-166 Imprimi t ivi t y Theorem 8.3: 159 induced representations for finite groups 155 for locally compact groups 8.1: 156 induced system 8.2: 158-159, 8.5: 165-166 inductive limit of locally convex spaces c . 7 : 222 infinitesimal weight 6.18: 130 integrable function 213 morphism from a Lie algebra 5.1: 91 integral 213 interior 207 Intertwining Theorem 8.4: 162 invariant differential operators F.2: 269-270 F.2: 269-270 vector fields inverse Fourier transform 146, 254, 265 Inverse Function Theorem (for Banacli spaces) 230 irreducible 114 Iwasawa decomposition 281
Jacobson (= hull-kernel) topology Jordan decomposition 216 Killing form (F-22): 279 Krein-Milman Theorem 223 Krein’s Theorem, generalization of
118, 141
C.10: 224
Index Laplacean operator 256 lattice 172, 211 Lebesgue’s theorem 214 Leibniz Rule C.17: 230 Ldvi-Malcev Theorem 279 Lie algebra of a Lie group 269-270 Lie generating subset 58, 97 Lie group F . l : 269 Lie’s third theorem 273 Lie subgroup 274 1, 1.1: 3 linear representation locally compact group F . l : 269 compact space A . l : 209 convex space 221, C.6: 222 differential operator 251, E.11: 252 finite A.2: 209-210 in t egr a b le fu n ct ion C.5: 221, 246 logarithm on a Lie group 271 Lorentz groups 190 Luzin’s theorem 217
Manifold 241, E . l : 242 mass of an elementary particle 200 matrix elements (of a representation) 6 . 8 : 118-1 19 Lie group F.4: 271 maxi ma1 tor us 125 weight 135 meager set (= set of first category) 210 measurable function 212 set 211 space 21 1 structure of Mackey 118, 166, 8.7: lG9 measure 211 space 211 211-212 of a pre-measure measure unit of time 10.3: 193 Milman’s Converse Theorem 223 modular function (on a locally compact group) F.8, (F-10 ... 12): 275 motion with a constant velocity vector 189 multiplicity theory 144 M I topological space 207 M Z topological space 207
Negligible
212
297
298
Index
negligible (in a manifold) 246 neighborhood 207 base 207 Nelson condition 95, 106 net 207 non-commutative multi-index notation (D-6): 237 nondegenerate representation (of a *-algebra) 139 non-meager 210 normal subgroup analysis 148-153, 165-169 nowhere dense 210 One-parameter semigroup 1.1: 3 subgroup 269 one-point compactification 209 open mapping 208 set 207 orbits of the Lorentz groups 202 Orthogonality Relations [of matrix elements)
G.9: 119
Paracompact topological space A.2: 209-210 partial differential equation 254 partition of unity A.2: 209-210 Peter-Weyl C-Theorem 6.11: 121 6.10: 120 Theorem Plancherel transform 7.6: 147 Poincar6 groups 190 polar decomposition of automorphisms of a semi-simple Lie algebra Polish topological space 151, 169 Pontryagin’s Duality Theorem 142 positive conjugation (of a semi-simple Lie algebra) 281 Radon measure 216 power series (on a Banach space) D.2: 236 pre-measure 211 “probability” 171 product chart of a Lie group 271-272 manifold 24 1 measure space 214 topology 209 projection valued measure (= spectral measure) C.1G: 229 projective representation 171, 9.1: 173 space 171 pure state 141, 226 Quantum physics, central assumption of
171-172
281
Index quasi-complete locally convex space 224 quasi-invariant measure (on a homogeneous space) quasi-spectrum of a locally compact group 148
299
F.9: 276
Radical (of a Lie algebra) 9.7: 181-182 Radon measure B . l : 215 Radon-Nikodym derivative 213 Theorem 213 for Radon measures 216 rank (of a compact Lie group) 125 regular Bore1 measure 215 element (in a compact Lie group or algebra) 125 represen ta t ion 3.25: 58-59, 270 Regularization Theorem 2.11: 31 relativity, special theory of 190 remaining integral term (D-12): 239 represen tat ion 1 Riemannian dist a w e 246 manifold 246 Riemann integral (infinite dimensional vector valued) 224 Riesz' theorems about representing functionals 215, 220 ring 211 root system F.15: 279 rotation (in %space) 189 (in 2-space) 192
Sard ' s theorem B.2: 217 Sard's theorem, generalization of E.7: 247 Schrodinger representation 111, 10.4: 198 111 Schwartz space semi-ring 21 1 separable action 7.8: 150 topological space 207 separated uniform space 209 sequentially compact 209 complete 209 Simon condition 94, 106 simply connected topological space 91 singular element (in a compact Lie group or algebra) 244 smooth (= C" ) function Lie group 272 manifold 241, E.l: 212 vector 3.16: 53-54 Sobolev's lemma (E8): 253
125
300
Index
Sobolev space, global 253 local E.12: 253 special theory of relativity 190 spectral measure (= projection valued measure) C.16: 229 Radon measure C.16: 229 spectrum (= dual object) of a topological group 6.7: 117-118 spin of an elementary particle 200 standard measurable space 151, 166, 168-169 state 14 1 step-funct ion 212 Stiefel diagram (F-23): 280 Stone-Naimark-Ambrose-Godement (SNAG) Theorem 7.4: 143 Stone-von Neumann Theorem 148, 10.4: 198 Stone-Weierstrass Theorem, versions of the C.2: 219 strictly integrable morphism from a Lie algebra 91 positive measure 247 strongly elliptic differential operator E.15: 25G element in a n enveloping algebra 62 strong topology of functionals C.9: 222 submanifold 245 subnet 208 subrepresen tat ion 6.3: 114 sufficient subspace of functionals 3.2: 40 support of a distribution E.8: 249 of a function 209 of a locally differential operator 25 1 system of imprimitivity 7.10: 152, 159, 165-1GG Tame (= type I = G C R ) group 115, 144, 148, 151-153, 166, 8.7: 169 tangent bundle 244 mapping 244 of a representation 3.12: 51 space 244 tensor product of distributions E.9: 250 time, measure unit of 10.3: 193 Tomita-Takesaki theory 117 topological space 207 vector space (= linear topological space) 221 totally bounded 209 simple 6.17: 129-130 u-fi ni t e 213 type I (= tame = GCR) group 115 TZ (= Hausdorff) topological space 208
Index U-domain E.21: 265 uniform continuity 208 space 208 unimodular F.8: 275 unique left division property (of a semigroup) Unitary Trick 6.1: 113 Uryson Lemma 208 Vector field 245 von Neumann algebra
30 1
40
117
Weak derivative (C-6): 226 integral 223 topology (of locally convex spaces) c.9: 222 weak* topology of functionals c.9:222 weight 6.18: 130 Weyl group 280 sector 280 Weyl's dimension formi~la 6.23: 135-136 for mu I a 6.22: 133 integral forinula 6.15: 127 W igner 's theorem 173 wild (= non type I ) group 115, 154 Yegorov 's theorem Yosida's formula p-measurable p-negligible gbounded gfunction a-algebra a -fini t e a-ring
214 (1-7): 9
212 2 12 41 (F-15): 276
21 1 213 211
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