To professor MIRON NICOLESCU, on his 70th anniversary
ION SUCIU
FUNCTION ALGEBRAS translated from the Romanian
by Dr. Mihaela Mihailescu
Editura Academiei N oordhoff Republicii Socialiste Romania 1975 International Publishing Bucure~ti Leyden
The monograph is a revised translation of the original Romanian version
"Algebre de fUDctii" Editura Academiei, 1969 Work awarded the "Simion Stoilov" Prize of the Romanian Academy
ISBN: 90 286 0445 6 (Noordhoff)
EDITURA ACADEMIEI REPUBLICn SOCIALISTE ROMANIA Calea Victoriei 125, Bucure~ti All rights reserved PRINTED IN ROMANIA
Preface
Under the title of Function Algebras we may now include a very large number of H'orks, published mainly in the last decade; which confititute one of the important chapters of functional analysis. This chapter has grown up from various problems, permanently furnished to mathematics, by the theory of functions, using modern methods of algebra, topology and functional analysis and presenting large possibilities of applications in operators theory. Herefrom proceeds its living character, the variety of obtained results, the variety of forms and contexts in which these results can be found. This also explains the difficulty of an exhaustive exposition of these problems. The purpose of the monograph is to present a coherent exposition of the fundamental results of this theory with an orientation to their applicability to the theory of operator representations of function algebras. The idea of such a work appeared during the seminaries on function algebras held at the Mathematical Institute in Bucharest, under the direction of C. Foia~ and at the Faculty of Mathematics and Mechanics under the direction of N. Boboc. It is a pleasure for the author to express his gratitude to C. Foia~ for assistance in his efforts, in general, and for the large contribution the discussions and cooperation with him had brought in the elaboration of this monograph. I also would like to thank N. Boboc for the clear discussions we hal'e had during the seminaries and the elaboration of some chapters. I am grateful to G. Gussi for the useful discussions on various problems and to all my seminary colleagues. My thanks are due to the Publishing House of the Ron1anian Academy for the graphic presentation of the monograph. I. S.
Bucharest, September 1968
Preface to the English edition The English l'ersion of this monograph contains several changes and completions reiatil'e to the Romanian edition. I mention some of them: the proofs to the theorems on spectral and semispectral measures in paragraph 7.1, the Jf70 ld decomposition for subspectral representations (paragraph 8.4), the decompositions with respect to the Gleason parts (paragraph 8.5), the results relating Theorem 9.15 to the known predication theorems (paragraph 9.5). Chapter 10 was entirely rewritten in order to have a frame for the example of a function algebra representation with no spectral dilation. I take this opportunity to thank professor B. Sz.-Nagy for the interest he took in my book as well as for his useful suggestions relative to completions. In writing the completions I had continued to benefit from c. Foia{ advice. I thank him once more. I also like to thank my colleague Mihaela Mihdilescu for translating this book into English. I. S.
Contents
CHAPTER 1 Preliminaries 1.1. 1.2. 1.3. 1.4.
Commutative Banach algebras. Measures . . . . . . . Convexity . . . . . . . Holomorphic functions of several complex variables Notes ....
11 14 15
16 18
CHAPTER 2 Boundaries
2.1. 2.2. 2.3. 2.4.
Function algebras Representing measures The Choquet boundary The Shilov boundary 2.5. Geometric characterization 2.6. Representing theorems Notes . . . . . . .
19 20 22 28 31 34 37
CHAPTER 3 Algebras on the maximal ideal space 3.1. The maximal ideal space . . . . 3.2. Locally analytic functions. . . . 3.3. The local maximum modulus principle . . . . . 3.4. Gleason parts Notes
39 40 43 47 53
CHAPTER 4 Approximation and interpolation 4.1. Restrictions . . . . . . . . 4.2. The case of the intersections of peak sets
55
59
4.3. Antisymmetry . . . . . . . . 4.4. Some characterizations of C(X) Notes
65 67 73
CHAPTER 5 HP.spaces 5.1. Definitions and basic lemmas . . . . 5.2. The theorem of F. and M. Riesz and Szego theorem . . . . . 5.3. The factorization theorem. 5.4. The characterization of the functions in HP . . . . . 5.5. Invariant subspaces . . 5.6. The aigebra H"JJ (dm) Notes
75
82 92 97 101 106 107
CHAPTER 6 Special classes of function algebras 6.1. Dirichlet and 10gmoduJar algebras 6.2. Algebras generated by inner functions ." " . . . . . . 6.3. Maximal algebras. . . 6.4. Functions algebras on compact sets of the complex plane . . . . . . 6.5. The standard algebra and H"JJ algebra Notes ..............
109 120 127 130 143 146
CHAPTER 7 Operator representations of function algebras 7.1. Positive definite maps on C(X). Spectral and semispectral measures 7.2. Representations of function algebras 7.3. Representations of the algebra C( X). Notes ...."........
147 156 158 166
CHAPTER 8 Elements of spectral theory of representations of function algebras 8.1. The canonical decomposition . . . . 8.2. The spectral dilation and attached
spectral measures
167 171
9
8.3. Szego measures and natura] representations . . . . . . . . . . 8.4. The Wold decomposition . . . 8.5. Decompositions with respect to Gleason parts Notes CHAPTER 9 Elements of prediction theory generated algebras
184 193 203 209
on S-
9.1. Semigroups of contractions 9.2. The Wold decomposition . 9.3. The semigroup of unilateral translations . . . . . . . . . . . 9.4. Representations of S-generated algebras 9.5. Prediction theorems Notes ......
211 221 225 230 234 242
CHAPTER 10 Some examples in the spectral theory of non-normal operators 10.1. The case of a single contraction . . 10.2. Operators having spectral sets with connected complement . . . . . . . 10.3. Finite systems of commuting contractions . . Notes Reference list
245
248 251 264 265
CHAPTER 1
Preliminaries 1.1. Commutative Banach algebras Throughout this book we shall denote by C the field of complex numbers and by R the field of real numbers. A Banach algebra over C is a non-empty set A endowed with a vector algebra structure over C and a norm relative to which the linear space A is a Banach space, with the property (1.1.1)
\labll
~
lIallllbll
(a, b E A).
Moreover, it is supposed that multiplication in A is commutative and that there exists a unit element e E A which satisfies I e II = 1. From (1.1.1) there results lIanll ~ lIall n so that we can easily verify that for any a E A with lIall < 1, (e - a)-1 exists. A complex homomorphism of A is a linear and multiplicative map from A into C. Proposition 1.1. The set
nonzero homomorphisms of A is a weak*-compact subset of the unit sphere of A *, A * being the dual of A. Proof. Let cp E 8'lL. cp =F 0 implies cp(e) # 0 and cp(e) = cp(e) cp(e) implies cp(e) = 1. Let a E A, I a \I ~ 1 and I cp(a) I > 1. Setting
h
=
j)lt, of complex
_f!_ we have IIbll < I, so that (e cp(a)
b)-1 exists.
Function algebras
]2
Then I
= epee) = ep[(e - b) (e - b)-I] = ep(e - b) ep[(e - b)-I] =. =
[epee) - ep(b)] [epee - b)-I]
=
0,
which is not possible. It follows that lIepll = 1 for any ep E.9IL Since the unit sphere of A is weakly compact, it remains to show that ffi is weakly closed. Let a* E A* be a point in the weak*-c1osure of ,3Jll, Ila*11 ~ 1. For an 8 > 0 and 0, bE A, let V";Q,b,ab(a*) be the neighbourhood of a* defined by Ve: b. ah • We have 1a*(a) - b*(a) I < 8, 'a*(b) - b*(b) I < 8, I a*(ab) - b*(ab) I < 8 for any b* E Ve;a,b,Qb(a*). But a* is in the weak*-c1osure of gm, so that there exists a ep E .JIll such that ep E V";Q,b,ab(a*). Then Q.
I a*(ab) - a*(a) a*(b) I ~ I a*(ab) - ep(ab) I + I a*(a) i I ep(b) -a*(b) I +
+ I ep(b) i j cp(a) -
a*(a) I <
80 + Iiall + IIbll).
Since 8 is arbitrary, a*(ab) = a*(a) a*(b) for every a, b E A, i.e. a* is multiplicative on A. It remains to prove that a* =I O. Suppose a* = 0 and let ep E c'1ll such that cp E V1i2;eCa*). We have
I = I a*(e) - epee) I < 1/2 which is impossible. It follows that a* =F 0 so that a* E c">R, which completes the proof. The compact Hausdorff space 3''[ constructed this way is called the maximal ideal space of A. The elements of.9IL may be put into a one-to-one correspondence with the maximal ideals of A by the map ep ~ Ker cp, cp E c'1lL Let C(.5l1l) be the Banach space of all continuous complex valued functions on 3)[, with the supnorm. With the usual multiplication for functions, C(cl1l) becomes a commutative Banach algebra with unit element. For a E A we define on ~ the function by
a
(1.1.2)
a" (ep)
=
ep(a)
(cp
E
cJJl).
Ch. I. Preliminaries
a
. It is easy to verify that
Iia II
(1.1.3)
E
C( ~l1t) and
sup I o(qJ) I ~
=
13
Iiall
(a
A).
E
q: E~R
The map a
-+
a is
a continuous algebra homomorphism from A
1'\
into C(.mt). Let A be the range of this homomorphism. Proposition 1.2. (i) For any a E A we have
110 II II
(ii) the homomorphism a
=
II
lim Ilanlll/ n
A.
-+
" I I '
a is injectil'e
if and only if (a
E
A),
A.
(iii) the homomorphism a -+ a is a topological injection there exists a constant K such that (a
(iv) the homomorphism a
-+
E
E
and only
if
if
and only
if
A),
a" is an isometric injection (a
if
A).
a
Proof. As is easily verified, (S11L) is just the spectrum of a so that (i) results from the spectral radius formula. The statement (ii) follows immediately from (i). We now prove (iii). If IIall 2 ~ Klla 2 11, then
so that
Kilo I
=
K lim Ila2"111/2" n-+ 'JC
Conversely, if
Ilall
~
Milo II,
then
~
Iiali.
Function algebras
14
To prove (iv) we follow the above proof for (iii) exactly; we have for (iv) in addition: K = M = 1. The proof thus is complete. In the case when all elements of A satisfy (1.1.4)
the homomorphism a
-+
ais an isometric embedding of A
in C(811L).
1.2. Measures Let X be a compact Hausdorff space and C(X) the Banach space of all continuous complex valued functions on X, with the supnorm. By a measure on X we mean a linear continuous functional on C(X). We call a measure p. on X real if for any real valued f E C(X), p.U) is real. The measure J.l on X is called positive if for any positive function f E C(X), J.l(f) is positive. Let J.l be a positive measure on X. The theory of integration relative to p. together with that of the spaces LP(dp.) is well known. For 1 ~ p < 00, LP(dp.) is a Banach space with the norm
L
(dJ.l) is a Banach algebra, with the ess supnorm. If f E LP(dp.) and g E Lq(dp.) with lip + llq = l, then fg and Holder inequality holds: 0Cl
E
Ll(dp.)
The equality sign holds if and only if f and g are linearly dependent. For p ~ 1, any function in LP(dp.) is also in Ll(dp.). The set of all functions of C(X) forms (modulus dp.) a dense subspace in LP(dp) for] ::::;p < 00, and a weakly*-dense subspace in LOCl(dJ.l). For 1 ::::; p < 00 and I < q ~ 00 with lip + Ilq = I, the dual space of LP(d/l) is isometrically isomorphic to Lq(dp). The characteristic function Xw of a Borel set 0) C X belongs to LP(dJ.l). We put p(0)
= JXwdp..
Ch. 1. Preliminaries
15
We say that the positive measure Jl has a support in the Borel set (J) if p.(X - ro) = o. Any measure Jl can be uniquely written as Jl = III + i1l 2, Ill' 112 being real measures. For any real measure Jl one can uniquely define the positive measures Jl + and P- with supports in two disjoint Borel sets and such that Jl = Jl+ - Jl-. For a measure Jl of the form Jl = JlI + iJl2 and a Borel set ro, we put
Jl(OJ)
=
Jli (w) - Pl(OJ) + i[Jlt(OJ) -
Jli{OJ )].
For any measure j.i there exists a unique positive 111easure 1 III and a uniquely determined function FE L!(dJl), such that I F I = I JJlJ-almost everywhere and dJl = Fdl JlI. We have IIJl" = I JlI(1). If Jl is real, then IJlI = Jl+ + Jl-. A measure Jl is positive if and only if IIJlII = Jl(1). We say that a measure JlI is absolutely continuous with respect to the measure Jl' if for any Borel set ro C X with Jl( OJ) = 0 we have Jll(ro) = O. If JlI is absolutely continuous with respect to the positive measure Jl, then there exists a function f E Ll(dJl) such that dJll = fdJl (RadonNikodym theorem). The measures PI and Jl2 are called mutually-singular if there exist two disjoint Borel sets OJ! and OJ 2 such that for any Borel set OJ one has JlI(W nWI) = Jll(OJ), P2(OJ n ( 2) = Jl2(OJ). If Jl is a positive measure and JlI an arbitrary one, then JlI can be uniquely written in the form JlI = Jla + Jls' where Jla is absolutely continuous with respect to Jl and Jls and Jl are mutually-singular (Lebesgue's decomposition theorem). Let K be a closed subset of X. Any measure Jl on K can be considered as a measure on X by putting Jl(f) = Jl(fJK) for g E C(X). If Jl is a measure on X we put PK. == XKJl· We note that the space Jt(X) of all measures on X is the dual of the Banach space C(X).
1.3. Convexity Let B be a real normed space and B* its dual space. The space B* endowed with the weak*-topology is a locally convex space and B is (isomorphic to) its dual.
Function algebras
16
A subset E of B* is called convex if it is non-empty and for each a, bEE the segment [a, b] = {c E B*: c = ta + {l - t)b, 0 ~ t ~ I} is included in E. It is clear that any non-empty intersection of convex sets is a convex set. Thus for a non-empty set Fe B* there is a smallest closed convex set which contains F. We denote it by (F) and call it the closed conrex hull of F. The set of all convex combinations of elements from F is evidently convex, so its closure contains (F). The converse is clearly also true, thus (F) is the closure of all convex combinations of the elements of F. Proposition 1.3. Let K be a compact subset of B*. The closed convex hull of K is compact and we hare (K) = {b
E
B*: b(u) ~ sup k(u);
U E
B}.
keK
Let E be a convex set in B*. We call e E E an extremal point of E if there is no segment of E containing e in its interior. Proposition 1.4. (Krein-Milman theorem). Let K be a convex compact set in B*. Then K is the closed convex hull of the set of its extremal points.
1.4. Holomorphic functions of several complex variables We shall denote by en the cartesian product e x ex... x e of n copies of the complex plane c. The points in C n will be written as Z = (Zb ••• ' zn). If Z E en then we put
Let WE (11 and r = (r1 , ... , rn) ERn, rj > O. By an open polydisk of polyradius r and center w we mean the set ~(w, r) defined by ~(w; r)
= {ZE cn: IZj - M11 < 'j, 1
~j ~
n}.
Ch. 1. Preliminaries
17
en.
Let D be an open set in A function f defined on D and with values in C will be called holomorphic in D if for any WED there exists a neighbourhood V of w such that W EVe D and for any Z E V 00
f(z)
t
=
ak lt k2' ... , kn(Zl
-
W1)k t
•••
(zn - wn)k n
klok2' ... ,kn=O the series being convergent for every Z E V. Let Ph P'}" ... , P, be a system of polynomials in n variables Zl' Z2"'" Zn The polynomial polyhedron of radius ~ defined by these polynomials is the open set o
Q(P1 ,
•••
{z E cn: Izi <
P,; ~) =
IP/Z) I <
~, 1 ~j ~
~, 1 ~ i ~
n,
r}.
en.
Let D be an open set in A system of Cousin data on D is the system {V;, h ij }, where {V;} forms a finite open covering of D, that is U i are open sets Vi C D with D = U Ui and h jj are functions defined and holomorphic on V j Vj, which satisfy
n
on U·nU. I J h·· Ii
+ h'J'k +
hk I•
= 0
Proposition 1.5. (Cousin's theorem). Let {Vj , hij} be a system of Cousin data on a polynomial polyhedron Q. Then there is a system of junctions h j so that hi is defined and holomorphic on Uj and
For a compact set K of of K as "
K
=
{z E ell: /P(z)
en we define
the polynomial-com'ex hull
~ max IP(w)1 for any polynomial
Pl.
wEK
The compact set K is called polynomially com'ex if K" = K. 2 - c. 437
18
Function algebras
Proposition 1.6. (Oka's theorem). Let K be a compact polynomially convex set of en. Any holomorphic function on a neigbourhood of K is a uniform limit on K of polynomials.
Notes The purpose of this chapter is to define the basic concepts and notations of the elements of commutative Banach algebras, the measure theory, the convexity theory and the theory of holomorphic functions of several complex variables, used throughout this book. The present chapter also contains a number of mathematical results which help readers to quickly gain the required information for the folluwing chapters. Detailed expositions on the mathematical facts presented in this introductory chapter can be found in: I. GELFAND, D. RAIKOV, G. SHILOV, [1] and C. FOIA~ [4] for commutative Banach algebras, N. BOURBAKI [1], N. DINCULEANU [1] and M. NICOLESCU [1] for measure theory, N. BOURBAKI [2], N. DUNFORD J. SCHWARTZ [1] and K. YOSIDA [1] for convexity theory and other elements of functional analysis, and E. GUNNING, H. ROSSI [l] for holomorphic functions of several complex variables.
CHAPTER 2
Boundaries 2.1. Function algebras Let X be a compact Hausdorff space. We shall denote by C(X) the Banach algebra of all complex valued continuous functions on X, provided with the supremum norm
Ilfll = sup If(x)1
(f E C(X)).
XEX
By CR(X) we denote the real Banach algebra of the real valued functions belonging to C(X). A subalgebra A of C(X) will be called a function algebra on X if the following conditions are satisfied: a) A is uniformly closed in C(X), b) A separates the points of X: for any Xl' X 2 E X, Xl ¥= x 2 , there is an I E A such that I(x l ) # f (X 2) , c) A contains the constant functions. Any function algebra is a Banach algebra with unit element e = 1. For a function IE C(X) we denote by Ref its real part and by f its complex conjugate. We also use the following notations
ReA = {u
E
CR(X): u
A = {g E C(X): g
=
= Ref with I
j-
with
f
E
A},
E
A}.
20
Function algebras
The set ReA is a vector subspace of CR(X) and A is a function algebra on X. A very simple example of a function algebra on X is C(X). Another is furnished by the most appropriate model of the entire theory: let X = {z E C: Izi = I} be the unit circle of the complex plane and A the algebra of continuous functions on X which have an analytic extension into the interior of the unit disc. A is a function algebra on X and we have A 'i= C(X). In the following we shall call this algebra the standard algebra.
2.2. Representing measures Let A be a function algebra on X. A positive measure p. on X will be called a representing measure for the point x E X if (2.2.1)
I(x)
=
SIdp.
(j E A).
It follows from (2.2.1) that representing measures are multiplicative on A and that p(X) = 1. A positive measure p. on X is a representing measure for x E X if and only if (2.2.2)
u(x) =
Judp.
E
ReA).
(j E
C(X».
(u
The point mass ex defined by
eil) = I(x)
is clearly a representing measure for x. To simplify the notation we shall write Jl ~ ex for "/lis a representing measure for x." Let us fix an x E X and let S be a closed subset of X such that for any u E ReA, u ~ 0 on S, we have u(x) ~ O. For a function l' E CR( S) we define Q~(l')
= inf {u(x):
UE
ReA, u ~ v on S}.
Ch. 2. Boundaries
21
One easily verifies the following properties of the functional Q~ on CR(S):
a) Q~(l'l
+ l'2)
VI
+ Q~(vJ,
= CtQ~( v)
b) Qi(Ctv) c)
~ Q~(VI)
for Ct ~ 0,
~ V2 implies Q~( VI) ~ Qi( v2),
d) Q~(u)
=
u(x) for any
ReA,
U E
e) - Q~( - v) = sup {u(x),
U E
ReA, u ~
on S}.
V
Theorem 2.1. Let x E X and SeX, closed, such that for any function u in ReA, non-negative on S, we have u(x) ~ O. For any WE CR (S) and any Ct belonging to the closed interval [-Q~(-w), Q~(w)] there exists a measure Jl ~ ex so that p(X - S) = 0 and Jl(w) = Ct. Conrersely, for any Jl ~ ex with Jl(X - S) = 0, we have pew) E[-Qi(- w), Q~(w )]. Proof. As the functional Q~ is subadditive, positively homogeneous and bounded on CR( S), by applying the Hahn-Banach theorem we find a real measure Jl on S such that Jl(~')~Q~(v) for any v E CR(S) and pew) = Ct. ButQ~(l') ~ 0 for v ~ 0, hence p(l') ~ 0 for v ~ 0 and Jl is a positive measure on S. For U E ReA we have p(u) ~ Q~(u) = u (x)
p( -u) ~ Q~ (-u) = - u(x) which means that Jl(u) = u(x) and thus Jl ~ ex. The measure It may -be considered as a measure on X and we have p(X - S) = O. The second part of the theorem follows immediately from the definition of Q~ and property (e). The proof is complete. 0 Corollary 2.2. Let x and S be as in Theorem 2.1. The necessary and sufficient condition for x to admit a unique representing measure with support contained in S is that for any v E CR(S), - Q~( -r) = = Q~(l') i.e. sup {u(x):
U E
ReA, u ~
l'
on S}
=
inf {u(x):
U E
ReA,
U
~
von S}.
22
Function algebras
Corollary 2.3. Let x and S be as in Theorem 2.1. Then for any l' E
CR(S) Q~(l')
= sup {Jl{}'): J1
~
ex, Jl(X - S) = O}.
2.3. The Choquet boundary A non-empty closed set E C X will be called absorbant if for any x E £ and Jl ~ ex we have Il(X - £) = O.
Proposition 2.4. Any closed set which is a union of absorbant sets is an absorbant set. Any non-empty intersection of absorbant sets isabsorbant. Proof. The first statement is obvious. To prove the second one it is sufficient to note that if F is a closed set, such that for any open set V with F C V there is an absorbant set E with the property F C £ C V, then F is absorbant. Hence the family of all absorbant sets is inductively ordered by inclusion and from Zorn's lemma we deduce that any absorbant set contains a minimal absorbant one. Since X is clearly an absorbant set, the existence of minimal absorbant sets results. Proposition 2.5. Let E be an absorbant set and u E ReA, sllch that u ~ 0 on E. Then, the set £1 = {x E E; u(x) = O} is either absorbant or empty. Proof. Let x E El and J1 ~ ex. Let K be an arbitrary compact set in X - E 1 • Since u > 0 on KnE, there exists a positive constant r such that ru ~ 1 on E. We have
Kn
Jl(K)
= Jl(K
n £) + Il{(X ~ Jl{ru)
=
rp(u)
E)
=
n K) = Jl(K n E)
ru{x)
~
= O.
A non-empty closed set K C X is called a peak set (with respect to A) if there is a function f E A with the properties: f(x) = 1 for any x E K and If(x)l < 1 for any x E X - K. The point x is said to be a peak point if {x} is a peak set.
Ch. 2. Boundaries
23
Proposition 2.6. For any peak set K there exists afunction u
ReA, u > 0, such that K = {x E X: u(xj = a}. If u E ReA, u ~ 0 and F = {x E X: u(x) = O}, then for any y E F there is a peak set K such ' that y EKe F. Proof. Let K be a peak set and f E A be such that f(x) = 1 for x E K and If(x) I < 1 for x E X - K. If g = 1 - f and u = Reg then U E ReA, u ~ 0 and K = {x EX: u(x) = O}. Now let U E ReA, U ~ 0 and F = {x EX: u(x) = O}. Take g E A such that u = Reg and put h = e- g • Then we get Ih(x)1 = I for x E F
and Ih(x)1 < 1 for x
E
X-F. For a fixed y
e=
arg hey). \Vc have f only if If(x) I = 1. If we put K set and y EKe F. where
E
F letf
E
+
_I (e- i6 h 1), 2 1, f(x) = 1 if and =
A, Ifi :s; L f(y) = = {x E X:f(x) = I} then K is a peak E
Proposition 2.7. A non-empty intersection of peak sets is absorbant. If a non-empty GlJ-set is an intersection of peak sets, then it is a peak set. Proof. From Proposition 2.5 and 2.6, it follows that any peak
set is absorbant. Hence the first statement follows from Proposition 2.4. Un be a non-empty Go-set which is an intersection of Let F = peak sets. Then it clearly results that for any n there is a peak set Kn such that Fe Kn C Un· We get then F ~ Kn. Let fn E A with fn(x) = 1 for any x E Kn und Ifn(x) I < I for x E X - Kn, and
n
n
We have f E A, f(x) = 1 for x that is F is a peak set. Theorem 2.8. Let x
E
F and If(x) I < 1 for x
E
X - F,
'
X. The following statements are equivalent: a) x belongs to a minimal absorbant set; b) {x} is an absorbant set; c) ex is the only representing measure for x; d) For any neighbourhood U of x and any positive real numbers r1> r2 there is a U E ReA, U ~ 0, such that u(xj < r 1 and u > r2 on X- U; e) For any closed set F of Go-type, containing x, there is a peak set K such that x EKe F; f) For anJ' neighbourhood U of x there exists a peak set K with the property x EKe U; E
24
Function algebras
g) For any y =F x there is a u E ReA so thaI U ~ 0 on X, u(x) = 0 and u(y) > 0; h) {x} is an intersection of peak sets. Proof. (a) --+ (b). Let E be a minimal absorbant set with x E E. We show that for any f E A and y E E we have f (y) = f(x) and, since A separates the points of X it will result that {x} = E = absorbant. It is sufficient to prove that for any u E ReA and Y E E we have u(y) = u(x). Obviously, one can also suppose u > O. Let
Y E} .
= inf { u(x);
r
E
u(y)
E is compact so that there is an x 0 E E with
r= Let
Uo =
u(x) -
rUe
u(x) u(x o)
.
We have Uo E ReA,
Eo = {y
E
E: uo(Y)
=
Uo ~
0 on E. Let us write
O}.
Clearly Xo E Eo. By Proposition 2.5 one gets that Eo is absorbant. But E is a minimal absorbant set, hence Eo = E. Then, for any y E E we have u (x) = = ru(y) and, since in particular the same holds for x, r = 1 and u(y) = u(x) for any y E E. (b) --+ (c) is immediate. (e) --+ (d). Let U be a neighbourhood of x, v E CR(X), v ~ 0, such that vex) = 0 and v ~ r'J. on X - U. By use of Corollary 2.2 one gets from (e)
o=
l'(X)
=
inf {u(x): u E ReA, u ~ v}
so that there exists a u E ReA as required in (d). (d) -+ (e). Let F be a closed Gd·set, F = Vn , with x E F. Using induction we construct a sequence of open sets Un and a sequence of functions gn from A such that .
n
(i)
(ii)
Ilgn+l - gnll Ilgnll
~
~
2 -n+ 1
3(1 - 2- n - J)
Ch. 2. Boundaries
(iii) gn(x)
=
3(1 - 2- n)
(iv) Ign+l - gnl < 2- n (v)
XE
25
Un+ 1
1
on X - Un + 1
C Vn
'1
'2
Taking = - In(l - e), = - Ine, it results from (d) that for any open U with x E U, there is an I E A such that "III ~ I, I/(x) I> > 1 - e and III < e on X - U. Let I E A be such that
!!f!!
~ 1,
3
1
4
4
If(x)! > - and If I < -on
.x -
VI'
We put
We have
hence gl satisfies (ii). At the same time
3 -1 gl(X) = - = 3(1 - 2 ) 2 so gl satisfies (iii). Suppose we have already constructed U1 ,· •• , Uk and gl, ... , gk with the properties (i) - (v). Let
Since gk(X) = 3(1 - 2- k ), We have x E Un Vk' We now write U"+1 = Un Vk' Then x E Uk~1 C Vic' i.e.(v). Let I E A with ·1/1 < 1 ~
3
I/(x) I > -
4
and write h
and
1
III < -
= 3.2- k - 1 [f(X)]-1 I.
4
on X - U"+h
Function algebras
26
Ilhll
We have hex) = 3.2-k- \
~ 2- k + 1 and, for Y EX -
Ih(y) I = 3.2. k-l[f(x)]-llf(y)1 ~3.2-k-l
i . J_ 3
Let gk-"-l 'of 11 we get
= gk
+ h.
=
Uk + 1
2- k- 1 •
4
From (i) - (v) for gk and from the properties
which are just (i), (iii) and (iv). For Y EX - Uk + 1 we have Igk-d(y)1 ~ =
For Y
E
Ilgkll + Ih(Y)1
3(1 - 2.2- k -
2)
~ 3(1 - 2- k -
+ 2.2- k - 2
~
1)
+ 2- k-
1
=
3 (1 - 2 -k-2).
Uk + 1 we have Y E U, hence
= 3(1 - 4.2- k - 2 )
+ 2-
k - 2
+
. 8.2- k - 2 = 3(1 - 2- k - 2 ).
Then, it results that
i.e. property (ii) for gk+l' Thus, the sequence gn with properties (i)-(v) has been constructed. (i) implies that the series oc,
~ (gn-"-l n=l
gJ
27
Ch. 2. Boundaries
is uniformly convergent, hence 00
g
~ (gn + 1
=
-
gn) EA.
=
3 and from (iv) and (v) we get
n=1
From (ii) and (iii) it results that Ig(y) I ~
Ilgnll
+
IIgll
00
~ Igk+l (y) - giy) I < 3 (1-
2- n- 1)
k=n
E
+
~ 2- k -
1
=
k=n
=
for any y
00
}'(2".1.. ... / ... 1 )' < .),
".)
X-F. Writing
f
=
~ (3 + g) 6
and K
=
f- 1
(1), we see
that K is a peak set and x EKe F. (e) -+ (f). Let U be a neighbourhood of x and v E CR(X), l' ~ 0, vex) = 0 and v ~ 1 on X - U. Then F = {x: l'(x) = O} is a closed set of Gb-type and x E FeU. (f) -+ (g) follows immediately. From (g) we have
{x}
=
n {y EX: u(y) = O},
UE6 x
where bx = {u E ReA: u ~ 0, u(x) = O}. Then Proposition 2.6 implies that {x} is an intersection of peak sets. (h) -+ (b) is proved by Proposition 2.7 and (b) -+ (a) is obvious. The proof of Theorem 2.8 is complete. <> A point x E X will be called a Choquet point if it satisfies one of the (equivalent) assertions (a) - (h) of Theorem 2.8. The set of all Choquet points of X is called the Choquet boundary of X (relative to A). We denote it by r = r.(A; X). It will be proved in the following that the Choquet boundary depends only on the Banach algebra structure of A. Point (a) of Theorem 2.8 shows that r is non-empty. Corollary 2.9. Any intersection ofpeak sets contains a Choquet point. Corollary 2.10. For any f
E
A, there is an x
E ~
such that If(x) I= Ilf!i.
28
Function algebras
e it Proof. Suppose il!" = I and let g = -- (1 + f) where t is chosen 2 such that the set K = {x; g(x) = I} is non-empty. Then K is a peak set and hence it contains a Choquet point. But If I = 1 on K, so that the Corollary is proved. Corollary 2.11. Any Choquet point with countable basis is a peak point. Corollary 2.12. If any Choquet point has countable basis then r is the smallest subset of X on which, for any f E A, If I attains its maximum.
2.4. The Shilov boundary U E
A closed subset S of X will be called a determining set if for any ReA, u ~ 0 on S, we have u ~ 0 on X.
Proposition 2.13. A closed set S is a determining set if and only if for any f E A there is an XES such that If(x) I = Ilfll. Proof. Let S be a determining set,f E A and m = sup {If(x)l; XES}. Let y EX, (} = arg f(y) and g = m - e-i) f. The function u = Reg is obviously positive on S, hence u ~ 0 on X. Then, u(y) = m - e- iR fey) ~ ~ 0, so that If(y) I = e- i6 fey) ~ m.
Conversely, assume that the modulus of any function in A attains its maximum (on X) on the closed set S. Let U E ReA, u ~ 0 on S and f = u + iv E A. If we put g = e- i then g E A and Igi = e- U ~ 1 on S. There results e- U = Igi ~ 1 on X, thus U ~ 0 on X. Proposition 2.14. For any absorbant set E and any determining set S we have EnS ¥= 0. Proof. If EnS = 0, then for any x E E and J.l ~ ex one has p( S) = O. On the other hand, the condition of Theorem 2.1 is fulfilled for S and any point x E E, hence there exists a Jl ~ ex with support in S, which contradicts the previous assertion. 0 Corollary 2.1S. Any determining set contains the Choquet boundary.
Ch.
2~
Boundaries
29
From Corollary 2.10 it follows that the closure of the Choquet boundary is determining and from Corollary 2.15 it follows that there exists a smallest determining set. The smallest determining set is called the Shilol' boundary of X relative to A and is denoted by r = r (X, A). Theorem 2.16. The Shilov boundary is equal to the closure of the Choquet boundary. A point x E. X belongs to the Shilov boundary if and only if for any neighbourhood U of x there is a function I E A such that J/I < Ilfll on X - u. . Proof. Since the closure of the Choquet boundary is determining, ii contains the Shilov boundary. Then, fronl Corollary 2.15 there results that the closure of the Choquet boundary is simply the Shilov boundary. The second part of the theorem follows from point (f) of Theorem 2.8. In the case of the algebra C(X) , the Choquet boundary is equal to the Shilov boundary and equal to X; this follows immediately from point (c) of Theorem 2.8. Let A be the standard algebra onX = {z: I z I = I} and Jl ~ Bx , X E X. Then, ·the real measure u = Jl - ex satisfies the relation
Jz"dv =
(n = 0,
0
+ 1, ... )
hence v = O. It follows that Jl = ex and point (c) of Theorem 2.8 yields
~
=
r . x.
The Choquet boundary is generally not closed, and is hence different from the Shilov boundary. To illustrate this point, we shall give an example. Let X = {z E C: I z I = I} and A the algebra of continuous functions I on X, having analytic extensfon into the interior of the unit disc and verifying 1(0) = f(I). .Since X is the Choquet boundary of the standard algebra on X, from Corollary 2.11 it follows that for any x E X there is a continuous function Ix on X, having analytic extension in the interior of the unit disc, such that fx(x) = I and Ilxl < I on X - {x}. Let x E X, x :F 1 and let g E A be defined by
g = z . I~_ - Ix( 1) • ~ - fJI)fx
Function aJgebras
30
Obviously, g E A, I g(x)1 = I, I g I < 1 on X - {x}. Thus x is a peak point for A, hence a Choquet point for A, and this is true for any x E X, x '# I. If J1. is the normalized Lebesgue measure on X, then it is known that f(O)
=
Jfdll
(fE A)
and as f(O) = f(l) for f E A, we have 11 ~ 81 • But clearly 11 ,#81 , hence from point (c) of Theorem 2.8, I does not belong to the Choquet boundary of X relative to A. Then, in this case, we have r = X - {I} and r = X; r is not closed, and is hence different from r. Now let A be a function algebra on a compact Hausdorff space X and Fe X, F closed. We denote by A j F the algebra of all functions in C(F) which are restrictions to F of functions in A. Theorem 2.17. Let seX be a determining set. Then: (i) the algebra A IS is a function algebra on S, isometrically isomorphic to A; (ii) a closed set Kl C S is a peak set for A IS if and only if there exists a peak set K C X for A such that Kl = K S; (iii) The Choquet (Shilov) boundary of S relative to A I S is equal to the Choquet (Shilov) boundary of X relative to A. Proof. (i) Let T be the restriction operator defined on A, with values in A I S
n
Tf=flS
(f E A).
T is linear and maps the algebra A onto the algebra A I S. Since S is a determining set, we have II Tfll = IIfll hence T is isometric. Thus, T is an isometric isomorphism between A and A IS, which gives that A I S is a function algebra on S. (ii) Let K C S be a peak set for A I Sand u E ReA be such that u ~ 0 on Sand Kl = {x E S: u(x) = O}. Take K = {x E X; u(x) = O}. As S is a determining set, we have u ~ 0 on X and K results a peak set for A. Clearly Kl = K S. The converse statement is obvious. (iii) Since S is determining, we get r(A; X) C S. Kjo Suppose XES is an intersection of peak sets for A, {x} = Then one has {x} = {x}nS = n(KinS) and from (ii) there results that x is an intersection of peak sets for AIS. Thus, we get rCA; X) C
n
n
C
rCA
IS;
S).
Conversely, let x E rCA I S, S) and let K be the intersection of all peak sets for A, containing x. (ii) gives SnK = {x}. Since K is an
3r
Ch. 2. Boundaries
intersection of peak sets for A, it results from Proposition 2.7 that K is absorbant for A. Let E be a minimal absorbant set for A, E C K. From Proposition 2.14 we have EnS '# 0. Then
o #= En SCKn S= {x}. Hence x
E
E and point (a) of Theorem 2.8 gives
XE
rCA,
X).
2.5. Geometric characterization Let A be a function algebra on X and A* the dual of the Banach space A. When endowed with the weak*-topology, A* becomes a locally convex space. For an x E X, let ex be the functional defined on A by
exCf) = f(x). It is clear that ex E A* and that the map x --+ ex is a homeomorphic embedding of X into A *. In this sense, we shall consider X as a compact subset of A *. In the following, we shall consider A * as a locally convex space over the field of real numbers. Then its dual is ReA. We recall that for F C A* we denote by (F) its closed convex hull. Proposition 2.18. The closed convex hull (X) of X is giren by
= {a* E A*: a*(1) = Ila*1I = I}.
(X)
Proof. From Proposition 1.3 we get (X) = {a*
E
A*: a*(u) ~ sup u(X) ,
UE
ReA}.
XEX
Let a*
E
(X). We have a*(u) ~ sup u(x) ~ XEX
-a*(u)
= a*( -u)
~
sup (-u(x)) XEX
~
Ilull
(u
E
ReA),
Ilull
(u
E
ReA).
Function algebras
32
Therefore
- lIull l.e.
I a* I
~ a*(u)
< Ilull
(u E ReA),
~ 1.
On the other hand, -a*(1) = a*( -1) Q*(1) ~ I, that is
~
sup (-I) = - I. Hence
a*{l) = 1 = Ila* II· Conversely, let a*
E
A * be such that a*(1) = 1 =
Ila*ll.
Suppose a* ¢ (X), which means that there is a u E ReA such that a*(u) > sup u(x). Let n be large enough to ensure that n + u ~ 0 xeX
on X. Then
lin + ull =
a*(n
n
+ sup u(x). We have xex
+ u) = n + a*(u) > n + xeX sup u(x)
which contradicts the fact that II a* I
=
~
lin + ull
I.
Proposition 2.19. Let a* E A*. Then a* E (X) if and only exists a'positil'e measure Jl on X, Jl(X) = I, such that a*(f) = J/dJl
if there
(I E A).
Proof. If there is a positive measure Jl on X, for which a*(f) = J/dp,
(f E A),
then it is clear that a*{l) = 1 = lIa* I and from Proposition 2.18 it follows that a* E (X). Conversely, if a*(I) = 1 = lIa*/I, then there is an extension 11 of Rea* to CR(X) with the same properties: Jl(1) = 1 = 111111. Let
Ch. 2. Boundaries
33
JI. = JI. + - JI. - be the Jordan decomposition of a real measure p..
We have
1 = p.(I) = p.+(1) - JI.-(l) 1 = 11,u11
/,uI(1) =
=
p+(1)
+ ,u-(1).
Therefore p-(l) = 0, which implies /I,u-II = JI.-(1) = 0 and hence p. - = O. Then J.I. is a positive measure with I1(X) = 1 and a*(f) =
Jfdll.
(I E A).
Theorem 2.20. The Choquet boundary of X relative to A is equal to the set of all extremal points of the convex compact set (X) = = {a* E A*: a*( 1) = 1 = Ila*ll}. Proof. Let x be an extremal point of (X). Since X is a compact subset of A*, it results that x E X. Let 11 ~ Bx and suppose there is a compact K eX - {x} with I1(K) > O. The compactness of K ensures the existence of a point y E K such that for any neighbourhood U of y, I1(U) > O. If for any neighbourhood U of y we have I1(U) = 1, then for any compact F which does not contain y, J1.(F) = 0, hence, p(X) - {y}) = 0, that is, It = BySince JI. ~ ex, f(x) = fey) for any f E A, which is a contradiction because A separates X. Thus there is a neighbourhood U of y with 0 < It(U) < 1. We write r = I1(U) and III = r-1xu Il, 112 = = {l - r)-l (l - Xu)l1. 111 and 112 are positive measures on X and I'I(X) = P2(X) = 1.
If we put a~(f)
= Jfd,u1
(f E A).
JId,u2
(I E A)
a:(/) =
then, according to Proposition 2.19 we get two functionals at and ai from (X). If x = ai = a;, then x E (K) which is not possible since x is extremal and K ex - {x}. On the other side we have
3 - c. 437
Function algebras
34
and so x = rar + (1 - r)a: which contradicts the extremality of x. Thus p,(X - {xD = 0, hence {x} is absorbant and point (b) of Theorem 2.8 implies that x belongs to the Choquet boundary of X relative to A. Conversely, let x be a Choquet point and suppose x = raT + + (I - r)a: with 0 < r < I and a~, a: E (X), af i= a:. By Proposition 2. 19 there exist positive measures JIh Jl2 on X, Jl1(X) = P,2(X) = I, such that
a!(j) = Jfdp,l a:(j)
=
If we write p, = rJII on X, JI(X) = I and f(x)
=
Jf dll
+ (I
2
(f E A), (fEA)
- r )P,2' then 11 is a positive measure
Jfdp,
(fE A)
then p, ~ ex. Since aT ¥: a:, we have JII ¥: P,2' Hence III i= ex say, which means JII( {x}) < 1. It follows that
i.e. p, ¥: ex. Thus we have constructed a measure 11 ~ ex, different from ex, which contradicts the fact that x is a Choquet point. The contradiction follows from the assumption that x is not extremal. Therefore x is extremal and the theorem is completely proved. <)
2.6. Representing theorems Let A be a function algebra on X, :r its Choq uet boundary and r its Shilov boundary. Since for any determining set S and any point x E X the conditions of Theorem 2.1 are fulfilled, we have the following. Theorem 2.21. For any x E X there exists a positil-'e measure Jl on ..¥ such that Jl ~ ex and p,(X - r) = o.
Ch. 2. Boundaries
35
One can investigate the existence of representing measures for points in X which are supported by the Choquet boundary. The problem is dificult in general as the Choquet boundary is an arbitrary set (sometimes even not a Borel set). In this case there is a result establishing the existence of representing measures for the points of X, which vanish on any Baire set contained in X - r (Bishop - de Leeuw). Here we treat only the case when X is metrizable. Theorem 2.22. Let A be a function algebra on a compact metric space X. The Choquet boundary of A is a Go-set. Proof. Let d be the metric on X and Vn the set of all points x EX for which there is a function f E A such that (i)
II!II
~
1
(ii) If(x) I > 3/4 (iii) If(y) I < 1/4 for any y
E
X with d(x, y) > lin.
n
Vn are obviously open sets and L = Vn • Indeed, since any Choquet point is, in the metrizable case, a peak point, we have Len Vn • The converse inclusion results from point (d) of Theorem 2.8 0 Theorem 2.23 (Choquet). Let A be a function algebra on a compact . metric space X and L its Choquet boundary. For any x E X there is a measure J.l ~ ex such that Jl(X - L) = O. Proof. Suppose X is embedded in A* as in §2.5. Clearly we may assume X = <X). For a function WE CR(X) we define w(x) = Qf(w) = inf {u(x): u E ReA,
U
~
w}.
wis upper semicontinuous and w ~ W, W = w if W E ReA. Since X is metrizable, ReA is separable. Let Un E ReA be such that lIunll = 1 and {un} be a dense set in the unit sphere of ReA. Let W = L 2-nu~, WE CR(X). Let B denote ReA Rw the subspace generated by ReA and W in CR(X). Let us fix an Xo E X. Since the functional Q~o is a seminorm, by the Hahn-Banach theorem there is a functional Jl on CR(X) such that J1(w) ~ Q~O<w) and J1(u rw) = u(xo) rQ~(w). If g E CR(X) and g ~ 0, then 0 ~ Qfo(g) ~ J1(g), and thus J1 is a positive measure. Obviously J1 ~ exo '
+
+
+
Function algebras
36
Since w ~ W, we have pew) ~p(w). On the other hand, for U ~
wand so
U ~
U E
ReA,
w. Therefore, U(Xo) = p(u) ~ pew).
Then
pew) ~ w(xo) = Q~(w) = I1(W) and pew) = pew). But w ~ w, so that Jl vanishes outside the set {x E X: w(x) = w{x)}. On the other hand, this set is contained in r. Indeed, let x be such that w(x) = w{x) and x
~z+~y
with z ¥: y in X. ReA 2 2 separates X and since {un} is dense in the unit sphere of ReA, there is an n such that lIn(y) # un{z). Then =
2[ 1 + -1y] = -1 Un')() + -1 Un')( y) + -1 unCz)uly) < 22442
Un -
Z
Z
<
., 21 Un9()z + 21 Un{Y)
and, therefore, w( -1 z
+ -1y) <
-1 w(z)
222
+ -1
2
w(y).
Hence w{x) <
~ w{z) + ~ w{y) ~ ~ w(z) + ~ w(y) ~ 2
2
2
SInce
w( ~ z + ~ Y) = inf {U ~ ~ inf {u{z), U E ReA, U ~ 2
=
2
C + ~ Y): z
w}
»{x)
UE ReA, U ~
w} ~
+ ~ inf {lI(Y): U E ReA, U ~ w} = 2
~ w(z) + ~ w(y). 2
2
We have arrived at a contradiction, which shows that x is extremal and therefore x E r. The theorem is completely proved. 0
Ch. 2. Boundaries
37
Notes This chapter presents, in the frame of function algebras, different results of integral representations. on minimal boundaries; the origin of these results can be found in contexts of: commutative Banach algebras, convexity theory in locally convex vector spaces, potential theory, convex cones, etc. E. BISHOP [1] was the first to remark the connection between the theory of minimal boundaries for function algebras and the Choquet theory. Theorem 2.1 and its corollaries have been exposed by many authors in different contexts (cf. R. ARENS, I. M. SINGER [1], H. BAUER [1], G. E. SHILOV [l], etc.) The form presented here was given in a more general frame in N. BoBoc, A. COR~EA [1]. Theorem 2.8 contains several characterizations of the points of the Choquet boundary, which appear in different mathematical fields in works as: H. BAUER [1], E. BISHOP [1], E. BISHOP, K. DE LEEUW [1], N. BoBoc, A. CORNEA [1], D. A. EDWARDS [l], R. PHELPS [1]. The characterization of the Shilov boundary points presented in Theorem 2.16 appeared for the first time in G. E. SHILOV [1]. Theorem 2.21 belongs to G. E. SHILOV [1] and Theorem 2.23 derives from the well-known classical Choquet theorem (cf. G. CHOQUET [1]). The present chapter has been written following mainly N. BoBoc, A. CORNEA [1], K. HOFFMAN [3] and R. PHELPS [1].
CHAPTER 3
Algebras on the maximal ideal space 3.1. The maximal ideal space Let A be a function algebra on X. A is a commutative Banach algebra with unit element e = 1. Moreover
(f E A).
(3.1.1)
Let Slll be the maximal ideal space of A. We have already seen in paragraph 2.5 that the map x -.. ex, ex E A*, defined by
eif) =f(x)
(fE A)
is a homeomorphic embedding of X into A*, A* being endowed with the weak*-topology. Since ex are obviously multiplicative, by this embedding X is a compact subset of ~lR. 1\
From (3.1.1). the homomorphism f -+ f of A into C(cllt.) is an isometric isomorphism and because of this A will be considered as a subalgebra of C( ~JrL). Of course, A is a function algebra on S'rL. (3.1.1) also shows that X is a determining set of ..!llL. Proposition 3.1. The following assertions are equh'alent: (a) X = SllL, (b) for any finite system fl,"" fn of functions of A, which have no common zeros on X, there exists a s}'stem gv ... , gn' of functions of A, such that rfigi = 1.
Function aJgebras
40
Prool. (a) -+ (b). Let I be the ideal generated by iI'···' fn. If I or/:. A then there is a maximal ideal M = Ker qJ, qJ E jJ[ such that Ie M. Therefore, cp(/j) = 0, i = 1, 2, ... , n, which is not possible. Thus I = A and there exist gI,' .. ' gil in A such that
(b) -+ (a). Let qJ E &nt and suppose qJ ; X. Since X is a compact subset of mR, there is a neighbourhood Ue;f1 •. .!n(
n
3.2. Locally analytic functions From now until the end of the chapter we shall suppose that A is function algebra on the maximal ideal space ~Jll. We call the com1non spectrum of the elements iI"'" fn E A the set in C n defined by
By U(fh' .. , ill) we denote the convex polynomial hull of U(ih···' In)· Both (J(fI"'" In) and o.(fl"." III) are compact subsets of C". Lemma 3.2. Let A be a function algebra on j)1[. Let fh ... ,fll E A and U be a neighbourhood of U(fh ... ,III) in C". Then there exist In+l'··· ,1m E A such that n[o. (iI'" 1m)] C ;, (11,. , fn) u, where n m 1r: cm -+ C is the natural projection of c onto C".
n
Ch. 3. Algebras on maximal ideal space
41
Prool· Let a = a(/l , . , In). If U C V nothing remains to be proved. Thus let ZO E V, ZO = (zf, ... , z~). The functions II - z~, 12 - zg, .. , In - z~ then have no common zeros on sm.. Proposition 3.1 asserts the existence of gh . . , gn E A with the property rg;(li - z?) = 1. Let a' = a(/h . . , In' gl'.' gn) and n: C2n -+ C n be the natural projection. Clearly n(a') C a and hence n(a') Ca. Since rZn+ i(Zi - z?) = = 1 on a', the same is true on We have
a-
u'.
{z E c n : ~ Zn +l(Z i
-
z?) = 1, zi
=
z?, 1 ~ i ~ n} = 0, thus
But fr' is compact so that there is a neighbourhood Uzo of ZO such A, that n(a ) Vzo = 0. Hence, V, as a compact set can be covered with a finite number of neighbourhoods VI' .. ' V k constructed in the same way as above. Letfn+l,"" 1m be the finite system of functions which appears when constructing VI,·., Uk' Then
n u-
Theorem 3.3. (Shilov-Arens-Calderon).' Let A be a function algebra on ~t-the maximal ideal space, and fl,"" In E A. If F is a holomorphic function on a neighbourhood U of the common spectrUln a(fl,'" ,fn} of (fh' '" fn), then F(II' f2,.", fn} EA. Proof. Lemma 3.2 asserts the existence of fn+1"'" fm E A for which n[u(Il"'" fm)] C u. Then F 0 n is holomorphic on a neighbourhood of the compact polynomially convex set a(fh""/"')' From Oka's theorem (Proposition 1.6) we find that there is a sequence of polynomials P"(Z1,.,,, zm) uniformly convergent to Fo 1t on u(fl"",/m). Hence P,,(/l"'" 1m) converges uniformly to F(fb"" fn) on fr and therefore F(f1"'" In) EA. <>
un
Theorem 3.4. (Shilov). Let A be a function algebra on 8fTt-the maximal ideal space. Let K be a closed and open set in 5)[. Then X" EA. Proof. Let x E K and y EE K. Since A separates the points of 5)[ there is an I E A such that I(x) = 1, I(y) = o.
Function algebras
42
Let
el [
K is compact and, therefore, it can be covered with a finite number of neighbourhoods ViI"'" Vin constructed as above. Let -
n
ViI"'" Vin be their corresponding neighbourhoods and Vx
=
n VIi'
;=1
Hence we have constructed for any x E K a neighbourhood Vx, the functions II" .. , fn and the open s~ts ViI" .. , V In , such that
I/il <
1
4
on VIi'
Ifi - 11 <
1
4
on Vx'
N ow we cover K with a finite number of open sets U1,· .• , Un constructed as above, and let fn,.·., Ilnt,f2l,"" 1211~"'" Imt, ... ,fmntn be the finite system of functions which results from the construction. Let us number these 11"'" Ip. Let
Uk
{z E CP: z = (fl(X), ... ,I/x», x
Uk
=
=
{z E CP:
E
K}
z = (fl(x), ... ,fp(x», x E.9lL -
K}.
n
The properties of 11"'" Ip give (Jk Uk = 0 and, therefore, obviously, (J(fl,· .. ,lp ) C Uk U Uk. Let U and V be two open sets in CP such that V V = 0 and Uk C U, U~ C V. The function F defined by
n
F(z)
=
{I
o
n
for for
Z E
U
Z E
V
is clearly holomorphic on V V, i.e. on a neighbourhood of U(fl"'" Ip). Then from Theorem 3.3 it results that Xk = F(fl,oo.,lp ) belongs to A. The theorem is completely proved. 0
Ch. 3. Algebras on maximal ideal space
43
Corollary 3.5. If &JR is totally nonconnected then A = C(JXl,). Corollary 3.6. If x is al1 isolated point of Jl1l then it is a peak point for A.
3.3. The local maximum modulus principle First we prove the following lemma on functions of several complex variables. Lemma 3.7. Let K be a compact in cn, P C K a closed set and U a neighbourhood of P. Let h be holomorphic in U, h = 0 on P and Reh < 0 on U - P. Let f be holomorphic on a neighbourhood V of K(U C V), invertible on K - U and f = hegh in U, with g holomorphic on U. Then there exists a holomorphic function F in a neighbourhood of K such that F= 1 on P and IFI < Ion K- P. Proof. Since f = he gh on U, we have h
fe - gh
=
=
f
+ g'f2
with g' a holomorphic function on U. It follows that, on
o>
Re h
=
where M = sup u
Ref + Ref2g'
~
Ref -
Ifl21g'I
~
Un (K -
P)
Ref - Mlfl2
Ig'l. Then, on Un (K - P)
and, therefore, , - -1 if i 2M =
i(Ref)2 + (Imf)2
\j
i
=
~( Ref -
- 1 )"'-+ (lmf)2 2M
+ (_1 )2 _ Ref> 2M
M,
/lfl 2
=
+ (_1 )2 -lfl 2 = 2M
_1
2M
Function algebras
44
But f does not vanish on K - U, and hence there exists an e > 0 such that
If - el > e. This implies the existence of an e > 0 such that
If - el > e on K - P. Since f = 0 on P, (f - e)-l exists and is holomorphic on a neighbourhood of K. Let
F is analytic on a neighbourhood of K, F = 1 on P and IFI < 1 on K - P. The proof is complete. 0 Let A be a function algebra on 8'R-the maximal ideal space. A closed subset F of J11[ is called a locally peak set if there exist a neighbourhood U of F and a functionf E A such thatf = 1 on F and III < 1 on U - F.
Theorem 3.8. (H. Rossi). Any locally peak set is a peak set. Proof. Let F be a locally peak set, i.e. there exist a neighbourhood U of F and an I E A, with f = 1 on F and III < 1 on U - F. Let go = f - 1. Then go = 0 on F and Re go < 0 on U - F. Let x E F and V be one of its neighbourhoods of the form
such that x EVe U.
We have x E Vx C V and since F is compact, it can be covered with a finite number VI'"'' Vp of such neighbourhoods:
Let t = tp ' M = max IIgill, a = a(go, gJ,'." g,). l~i~t
Ch. 3. Algebras on maximal ideal space
45
Denote by z = (zo, Zl,' .. , z,) the points of CHI; we define the continuous function G:.9l'L ~ C t + 1 by
Let
{z E C1+l: there exists a j such that IZil < 1 for and IZil < M for other i}.
A=
fj
_1
~
i
~ Ii'
One can easily verify that:
2) A nun {z E C1 -+- 1: Re Z 0 = O} 3) G(F) CAn
=
A nun {z
E
C' -+- 1: Z 0
= O}
u n {z E ct -+- 1 : Zo = O}
n A) C U The compact set (0' n A) -
4) G-l (u
A is included in {z E Ct+l: Re Zo < OJ. Indeed, if Z E unA, then from 4) there exists an x E U such that Z = = G(x), hence Re Zo = Re go(x) ~ O. If Re go(x) = 0, then x E F and from 3) we get x E A. Let Dl be a neighbourhood of u - A and D2 a neighbourhood of (u A) - A, such that D2 C {z E Ct + 1 : Re Zo < O} and Do = - = (Dl - A) U D 2• If Z E u - A and Z E A, then Z E (u n L\) - A C D 2 , and therefore u - A C Do. Moreover Do n A = D2 n A C {z E C t +l: Rez o < OJ. Let D = A U Do. Since D is a neighbourhood of u, from Lemma 3.2 we can suppose C D and follow the same construction in a cn with n ~ I + 1. There exists .a polynomial polyhedron Q such that C QeD. A • Indeed, let r > 0 be such that u C A(O, r). Let V be a neIghbourhood of with V C D and V compact. For any pEa V (the topological boundary) there is a polynomial P such that
n
u
a
u
IP(p) I > max IP(z) I 1\
zEa
Function algebras
46
u
since is polynomially convex. By multiplication with a constant we can suppose IP(p)1 > r > max IP(z)l. A
z Ea
Let Vp
=
{z
E
en: IP(z)1 > r}.
a
We now cover U with a finite number of such open sets and let PI, P2 , ••• , Pm be the corresponding polynomials. The polynomial polyhedron Q = J-V(PI , ... , Pm, r) satisfi.~s q C QeD. Let Qo = Do Q, Ql = A Q. Qo, Ql are open sets and Q = Qo U Ql' Qo Q1 CDo A C {ZE E en: Re Zo < OJ. Then, log Zo is a holomorphic function on Qo Ql and by putting h12 = - h21 = Zo 1 log Z 0 we obtain on Q the system of Cousin data {Qi' hij}' Cousin's theorem (Prop. 15) asserts the existence of holomorphic functions go, gl, on Qo, Ql, respectively, for which go - gl = ZOI log Zo on Qo Q1' Let
n
n
n
n
n
n
Clearly fl is holomorphic on Q. If in the preceding lemma we put
since all conditions are easily verified, there exists a holomorphic function F in the neighbourhood of such that
a
F(z) = 1 for
Z E
a-
nAn {z
E
en: Zo
= O} and IF(z) I < 1
for the other points of U. From Shilov-Calderon's theorem (Theorem 3.3) there follows the existence of a function f E A with f(q» = = F(G(q»). Then f(q» = 1 for q> E F and If(q» I < 1 for q> e F. Therefore F is a peak set and the theorem is completely proved. 0
Ch. 3. Algebras on maximal ideal space
47
Corollary 3.9. For any f E A and t ~ 0, the intersection of each connected component of the set {cp E ~llI: If(cp) 1 ~ t} with the Shilov boundary is Ilonvoid. Corollary 3.10. If V is an open subset of ~lll with V C ~l1l - r, then the maximum of the modulus of any function from A on V is attained on the topological boundary of V. Corollary 3.11. The functions of A can not attain strictly local maxima in SllI - r.
3.4. Gleason parts Let A be a function algebra on X, and r its Shilov boundary.
8)1I
the maximal ideal space
Theorem 3.12. Let CPl, 2 E 8)1I. The following assertions are equivalent: (i) IIcpl - CP211 < 2. (ii) There exists a C < 1 such that for any f E A with IIfli < 1 and f(CPl) = 0 we have If(CP2) 1 ~ C. (iii) For any sequence hn in A, with IIh n ll ~ 1 and lim Ih n(CP2)1 = 1 we have lim Ihn(CPl)I = 1. (iv) There exists an a> 0 such that for any strictly positive function uERe A
(v) There exists a c < 1 and the representing measures Ill' 112 for CPh CP2 respectively, with their support in r such that Cll l ~ 1l2' Cll2 ~ Ill' Proof. Suppose (ii) is not true. For e > 0 we define the function F on the unit disc, by F(z)
=
z - (1 - e) 1 - (1 - e)z
(lzi
~
1).
For sufficiently small e, F is analytic on the closed unit disc, IF(z)1 ~ 1 for any z with Izl ~ 1, F(l) = 1 and F(O) = - 1 + e. Hence, there
Function algebras
48
is an 11 > 0 such that for Izl ~ 1, 11 - zl ~ 11, 11 - F(z) I < 1. Since (ii) is not true, there is an f E A with IIfll ~ 1 for which f( CPt) = 0 and !(CP2) = 1 - 1]. Let g E A be defined by g(x) = F(f(x» for x E X. Then IIgl1 ~ 1 and
Ig(CP1) - g(CP2) I = 1-1 -1)1
~
+8 -
F(l - 17)1 = 1-2
2 - 8 - 11 - F(l - 17)1
~
+8 -
(F(1 - 17) -
2 - 28.
Therefore
i.e. (i) is false. Suppose now (iii) is not true. Then there is a C < 1 such that for any 8 > 0 we can find an
For a fixed 81 > 0 there exists an 111 > 0 and an m > 0 such that 1-
Iz - zol
81 ~ - - - - - - - ~
1
m holds for any IZol ~ C and any Izl ~ 1, 11 - zl < '11. F or any 8 2 > 0 we can find a function F, continuous on the closed unit disc and analytic on its interior (i.e. in the standard algebra), such that IIFII ~ 1, F(l) = 1 and IF(z) I < 8 2 for any Izi ~ 1,11 - zl ~ ~ 111. Such a function indeed exists since z = 1 is a Choquet point for the standard algebra. Then, there is an 112' 0 < 112 < 111 with 11- F(z) I < 8 for Izi ~ 1, J 1 - zl ~ '72. Since (iii) is not true, there is a function f E A for which 11/11 ~ 1, If(CP1) I ~ C and f(CP2) = 1 - 112. Let Zo = j(CP1) and G the function in the standard algebra defined by G = z - Zo F. m
Ch. 3. Algebras on maximal ideal space
If we choose a sufficiently small G(Zo)
Let g
=
=
0 and IG(z) I > (l -
£2'
£1)2
IIGII
~
for
Izl
49
1; we have also ~
I,
11 - zl
~ '12.
G(f). Then one easily verifies that g E A,
which contradicts (ii). Therefore (ii) -+ (iii) is proved. The assertion (i) is symmetric in (Pv ~"2' hence the same is true for (ii) and (iii). Suppose (iii) true and let 0 < c < I be such that there exists a strictly positive function U in ReA with U«({J2) = I, and U«({J1) < c. Now let v E ReA with u + iv E A and! = e-(u+iV ).
Since we can take c as small as we sequence (fn) of functions in A such Ifll(lfJ2) I = e- 1 which contradicts (iii). Therefore, there exists a 0 < c < any strictly positive function U in ReA
like, then, obviously, there is a that 1I!1I11 ~ 1, !II(({Jl) -+ 1 and 1 with the property that for we have
From symmetry there follows that there exists a c with
hence there exists an a < 1 such that
i.e. (iv) is true, and thus (iii) 4 - c.437
-+
(iv) is proved.
so
Function algebras
Suppose now that (iv) is true. Then there is a c such that
=
a- \ 0 < c < 1, .
for any positive u in ReA. We define on A the linear functionals
which are obviously positive, L I (1) = L 2(l) = 1 and therefore IILIII = = IIL211 = 1. Let mb respectively m2 be extensions of L I , respectively L2 to C(r), with the same norm. Since mi(1) = Ilmill, i = 1, 2, the measures m i are positive and
f«({)l) - Cf«({)2) = (1 - c) S fdm 1 , f«({)2) - Cf«({)l)
=
(1 - c) S fdm 2 •
According to these relations we get
f«({)l)
=
Cf«({)2}
+ (1 -
c) Sfdm l = c2 f«({)l)
- c) Sfdm 2
f «(()2) that
=
c2f«({)2)
+ c(l
+ (1
+ c(l
- c) S fdm 1
- c) Jfdm 1
+ (1
- c) Jfdm 2
IS
f(CPl)
=
(1
f(CP2) = (1
-
+ C)-l [c Jfdm + Jfdm 2
+ C)-l [cJfdml + Jfdm
1]
2 ].
Ch. 3. Algebras on maximal ideal space
51
If we write
then the measures Ph P2 are representing measures for tively, with the support in r and we have
qJl
and
qJ2'
respec-
This proves (iv) -+ (v). The implication (v) -+ (iv) is obviously true. It remains to prove that (iv) -+ (i). Let (i) be not true, that is, IIqJl - qJ211 = 2. Then it is clear that for any e > 0 there is anf E A, II/II ~ 1 such thatf = U + iv implies U(qJl) < - 1 + e, U(qJ2) > 1 - e. If we write Uo = U + I, Uo is a strictly positive function in ReA and
2-e which contradicts (iv). The proof is thus complete. 0 The points qJl' ({J2 E 5JR. are said to be Gleason equivalent if they satisfy one of the (equivalent) assertions of Theorem 3.12. As follows from 3.12, this relation is an equivalence relation. An equivalence class relative to it is called a Gleason part of j)R. The principal part of Theorem 3.12 is the following theorem of E. Bishop. Theorem 3.13. Let qJJ, qJ2 E 8m, and III be a representing measure for qJl with its support in r. If ({Jl and qJ2 belong to the same Gleason part, then there is a representing measure 112 for l{J2 with support in r and a constant c, 0 < c < 1 such that Clll ~ 112' Proof. Since qJl and ({J2 belong to the same Gleason part, there exist two representing measures m l , m 2 for qJl' qJ2 respectively, with supports in r, and a constant 0 < c < 1 such that cm l ~ m 2 • If we put
Function algebras
52
then, of course, /12 is a representing measure for qJ2' with support in and CPl ~ /12'
r
Corollary 3.14. The points of the Choquet boundary of A form point Gleason parts. To close this chapter we prove the following theorem on representing measures.
Theorem 3.1S. Let A be a function algebra on X and 3JIL the space of its maximal ideals. Let ({J E 3JIL and v be a complex measure on X such that qJ(f) = Jfdv .
UEA).
Then there exists a representing measure /1 for ({J with support in X which is absolutely continuous with respect to v. Proof. Let H be the Hilbert space obtained by closing A in L2(dl vI) and Ho be its subspace generated by all the elements of A which vanish at qJ. Since for any f E A with qJU) = 0 we have
it follows that Ho '# H. Let hE H orthogonal on Ho such that
and dJ1.
=
Ihl 2dl vi. For any I E A, ((J(J) = 0 we have Ih E Ho and so
JIdJ1. = J Ilzl?fdlvl = Jfhhdl vi
=
o.
Since J1. is a positive measure and J1.(X) = 1, J1. is a representing measure for qJ. II. is obviously absolutely continuous with respect to v and therefore the proof is complete. 0
Ch. 3. Algebras on maximal ideal space
53
Notes The maximal ideal space is one of the basic elements of Gelfand theory of commutative Banach algebras. The study of the structure of this space imposed itself also in the theory of function algebras through the deep results due to mathematicians such as R. ARENS and A. CALDERON, H. S. BEAR, E. BISHOP, E. A. GLEASON, H. HOFFMAN. H. ROSSI, G. E. SHILOV, J. WERMER, etc. Theorem 3.3 has been enounced by G. E. SHILOV [3] and proved by R. ARENS and A. CALDERON [1]. Proof of Theorem 3.4 is given in G. E. SHILOV [2] and Theorem 3.8 is proved by H. ROSSI [1]. The present version follows mainly E. GUNNING, H. ROSSI [1]. The equivalence relation which defines the Gleason parts has been introduced by A. GLEASON [1]. Other characterizations of this equivalence relation, which can be found in Theorem 3.12, are due to H. S. BEAR [2] and E. BISHOP [4]. Theorem 3.15 belongs to K. HOFFMAN [3].
CHAPTER 4
Approximation and Interpolation
4.1. Restrictions Let A be a function algebra on X and F a closed subset of X. Let AIF be the algebra of continuous functions on F which are restrictions to F of functions in A. Denote by kF the closed ideal of A defined by kF= {/EA:!= 0 on F}.
Let AlkF be the factor algebra of A relative to the closed ideal kF. A/kF is known to be complete in the norm
III + kFIl For an x
E
=
inf {IIgll, g = 1+ h, hE kF}.
F, we define on A/kF, the functional Lx by
LiJ + kF) =f(x). Lx is well defined, linear and multiplicative on A/kF. From Proposition 1.1 we get IILxl1 ~ 1, hence If(x) ! ~ II! + kF11 for any x E F and so (4.1.1.)
IIf1K"
~
IIf + kFII.
Function algebras
56
Let T be the restriction operator with values in A IF defined on A by
T/=/I
E.
Clearly T is a linear, multiplicative map from A onto AIF with Ker T = kE. Thus Tinduces an isomorphism T between AjkF and AIF. From (4.1.1) we get
111'(1 + kF)1I = II!I KII ~ 111 + kFII
(4.1.2)
--
and the closed graph theorem asserts that T is a topological isomorphism if and only if A IF is closed. Concerning the restriction algebra AIF there arise the following questions: when is AIF closed, or equivalently, under which conditions is the isomorphism between AjkF and AIF topological? When is this isomorphism an isometry? When does A IF = C( F) hold? The conditions we give below will be expressed in duality terms. Even if some of the theorems are simple reformulations, they help to establish quite concrete results. We recall now some notations . .,II(X) is the dual space of C(X) and is considered as a space of measures . .,II(F) is naturally embedded in Jt(X) and for any /l E .A(X) we can define the restriction measure /IF
=
XFJl·
We denote by Ai = {Jl E Jt(X): /l(f) = 0, / E A} and similarly, (AIF)i. The dual of A (respectively AIF) can be identified with .A(X)jAi (respectively Jt(F)j(AIF)i) and the norms on these last spaces are those of a quotient space. Let T* be the adjoint of T. We have
T* (v
+ (AIF)i) =
Since for any v E .,II(F) tion and
(v
+ Ai)
(v E .A(F».
n Ai we have v
E
(v
(AIF)l, T* is an injec-
E
.,II(F».
From the closed graph theorem we obtain T* has a closed graph if and only if T*-l is continuous. Since T has a closed graph if and only if T* has a closed graph, we have the following
Ch. 4. Approximation and interpolation
57
Theorem 4.1. The algebra AIF is closed in C(F) if and only if there exists a c ~ 1 such that (4.1.3)
II v
+ A IF)1
II ~
c II v + AlII
(v E J/(F».
-
c = 1 if and only if T is an isometrical isomorphism between AIF and AlkF. The proof follows from the above considerations. 0 Theorem 4.2. The algebra AIF is closed if and only if there is a k < 1 such that (4.1.4)
IIJlF
+ (AIF)111
~
kllJll1
AIF is isometrically isomorphic to A/kF if and only Proof. If AIF is closed, then from (4.1.3) we obtain
for any Jl Hence (l
Writing k
c ~
E
if
k = 1/2.
A1.
+ c) IIJlF + (AIF)111 =
c/(l
~
II.up + (AIF}ill
+ cllJlFl1
~
clIJlI\.
+ c) we have k < 1 and IIIIF + (AIF)111 ~ kllJlII
Obviously k = 1/2 if and only if c = 1. We assume now (4.1.4) to be true. We prove first that there exists a 1 for which
(4.1.5)
IIIIF + (AIF)111
~
cllJlx-FIl
(p. E Al.).
Let Co be the smallest real number (co = 00 if there is no such number) which satisfies (4.1.5) and r < co. There exists.u E A1 such that
Function algebras
58
For any
> 0 we choose v E (AIF)l with
t
IIJIF
Let
IIlili
Ji =
= IlJIF
JI
+ v.
+ vII
Ii E A.1
Then
+ vii + IlJIx-FII ~ k I Jill
+ -kr
+ (AIF).LII + 1.
< IlJIF
and
< IIJIF + (AIF).LII
I Jill
+ t + _1r IlJIF+(AIF)1-11 ~
r +1= k 1+ I Jill + 1. r
There follows k >
(\1 Ii II -
t)
r
r
IlJIII (1
+ r)
r
t 1 - --
~
IlJIII 1 + r
r
~
r
+1
If Co = 00 then k ~ 1, which is impossible. Hence there is a c < for which (4.1.5) holds. Let r > c, v E A(F) and JI E Ai. From (4.1.5) we get
and therefore there is
II E (AIF)J.
such that
We have r
IIv + IlI1 = rllv +
JlF
+ JIx-FII
=
rllv + JIFII +
rllllx-FII ~
Since this inequality holds for any r > c, we have (v
i.e. (4.1.3). The theorem is proved.
<>
E
.R(F»
00
Ch. 4. Approximation and interpolation
Corollary 4.3. AIF = C(F) if and only
if there
S9
exists a k < 1 such
that
AIF = C(F) and both isometrically isomorphic to A/kF
if
and only
if
Corollary 4.4. If Jl E AJ. implies JlF E Al., then AIF and A/kF are isometrically isomorphic. If for any Jl E Al. we have JlF = 0, then AIF = C(F) and is isometrically isomorphic to A/kF.
4.2. The case of the intersections of peak sets Proposition 4.5. Let F be an intersection of peak sets for A. Then AIF is isometrically isomorphic to A/kF.
Proof. From (4.1.1) we know that
II/IFII
~
II! + kFIl
(fE A).
Let U be the open set on which If I < II/IFII +6 for a fixed f EA. Obviously, FeU and therefore there is a peak set K such that F eKe U. Let g E A with g = 1 on K and Igi < 1 on X - K. Since for any n, gnf Ef + kF, there results
IIf + kFIl
~ lim sup
IIgjll ~ IlflFIl
+ 6.
(>
Proposition 4.6. Let F be an intersection of peak sets. For any function g E AIF there is an f E A such that f = g on F and IIfll = IIgll. Proof. Let g E AIF with IIgl1 = 1. Since AIF and A/kF are isometrically isomorphic, there is an h E A such that h = g on F and I h II ~ ~ 3/2. Let K = {x: Ih(x) I ~ I}. It is obvious that K is of G«5-type, say K = Vn • As Fe K C V,. then, for any n there exists a peak set K,. such that F C K,. C V,..
n
Function algebras
60
Write Ko
n Kn; Ko is a peak set. Let pEA with
=
K0
IIpll = 1 and
= {x EX: p(x) = I}.
Let
n
Ko C K and hence Fn Ko = 0 and therefore for any n we find an rn such that Iprnl < 2- 2n on Fn. We write
Obviously f
EA.
For x
E
F we have
f(x) = 1: 2- nh(x) = h(x) = g(x). If Ih(x) I ~ 1 then If(x) I ~ 1. Let x there is an In such that x E F m and we get
If(x) I ~ Ih(x)l( ~ 2- n
+ 2-
m
l p rm(x)1) ~ (1
with Ih(x)1 > 1. Then,
EX
+ 2-
m)
(1 - 2- m
+ 2-
3m )
=
n=/=m
Hence IIfll ~ 1 on X and so IIfll = IIgli = 1. The proposition is proved. 0
Corollary 4.7. Let F be an intersection of peak sets and K a compact set such that K F = 0. For any 6 > 0 and g E AIF there exists an f E A with f = g on F and III < 6 on K.
n
gE
Proposition 4.8. Let F be an intersection of peak sets. For any A and 6 > 0 there is an I E A with Ref = Reg on F and sup IRef I ~ sup IRe gl x
F
+ 26.
Proof. Let T be the conformal mapping of the domain {z E C: : Izl < I} onto the domain {ZEC: Izi < 1, -6 < 1m Z < 6} for
Ch. 4. Approximation and interpolation
61
which t(O) = 0, 't(1) = 1. We choose b > 0 such that 't maps the circle {z E C: Izl < b} into the circle {z E C: Izl < 8}. Let U be a neighbourhood of F on which IRegl < sup IReg I + e. F
According to Corollary 4.7 there is an h' E A with h' = 1 on F and Ih'l < b on X- U. Let h = 't(h'). Then hEA, IRehl < e on X- U and IRehl ~ 1 on X. Now, for f = gh we have Ref = Reg Reh - Img Imh and therefore Ref = Reg on F. Moreover IRefl ~ sup IRegl F
+ 2e
on Uand IRef I < 2e on X- U. 0 Corollary 4.9. If B is the closure of ReA in CR(X) then BIF is a closed subspace of CR ( F). If BF = {u E B: u = 0 on F} then BIF and Bj B F are isometrically isomorphic. Proof. Let u E ReA and v E ReA with u = l' on F and II vII < < sup lui + e. F
Since v lIu
U E
BF we have
+ BFII ~ I u + v -
ull
Ilvll
=
~ sup F
lui
+e
hence
Ilu + BFI!
~
sup F
lui·
On the other hand, there is a v E B F such that
lIu + BFlI
~
lIu
+ vII -
8 =
sup lu X
+ vi -
8 ~
sup
lui -
8.
0
F
Theorem 4.10. Let F be an intersection of peak sets and p E eR ( X), P > 0 on X. If g E AlE is such that Igi ~ p on F, then there exists an f E A with f = g on F and Igi ~ p on X.
Function algebras
62
Proof. It is no loss of generality to assume that Ilpll ~ 1. Let ~, {3 > 0 be such that (X < inf p(x), (3 ~ p(x) - (X for any x E X. As x follows from Proposition 4.6, there is an fl E A with fl = g on F and IIflll = Ilgli. Let KI = {x E X: Ifl(x) I ~ p(x) + {3/23 }. We then find an f2 E A with f2 = g on F, IIf211 = IIgll and If21 < (X on KI (Corollary 4.7). Thus we can construct by induction the sequence (fn) of functions in A and the sequence (Kn) of compact sets in X such that fn = g on F, IIfnll = IlglI, Ifni < (X on K n _ 1 and n-l
U
Kn _ 1 =
{x
E
.X: i!j(x)!
~
p(x)
+ f3/2iiTl}.
j=1 00
We write f = (X/(2 - Ct) ~ (rx.i/2 i )fj. Since Ifni ~ 1 we get f EA. j=1
nK r1J
Let x
E
A. If x
E
n,
then If,.(x)1 <
(X
for any n and therefore
n=1
If(x) I < (X < p(x). If x ¢ Kn+l - Kn for any
+
(3
2n + 1
for any n, hence
Ifix) I ~ p(x)
17,
then Ifix) I < p(x)
+
for any j, and therefore
If(x)/ ~ p(x). Suppose now that there exists an n such that x E Kn + I - Kn. In this case Ifix) I < p(x) + P/2 n +1 for j = 1, 2, ... , nand Ifix) < (X for j ~ n + 1. Then we have
~
p(x)
+ (3(I/2nTI -
1/22n+l) - (3/2 n
=
=
p(x)
+ P(I/2
1/2 2n
~
n+1 -
-1-
1 -
1/2")
p(x),
Ch. 4. Approximation and interpolation
63
which completes the proof. 0 Note that the converse assertion is also true.
+
Proposition 4.11. If AIF is closed in C( F) then (kF)l. = Al + Jt(F). Proof. One has of course Al. .-H(F)C(kF)l.. If fEC(X) is orthogonal on Al. .II(F) then it belongs to A since it is orthogonal on Ai and vanishes on F since it is orthogonal on JI(F), i.e. f E kF. Therefore Al. + .A(F) is weakly dense in (kF)l.. It is now sufficient to prove that Al. + ..,H(F) is weakly closed in ..,H(X). Hence we have to show that the unit sphere of Al. .A(E) is weakly closed in .A(X). Let J1 E .,,{I(X) be in the weak closure of the unit sphere of Al. + .A(F) in .H(X). Obviously 1IJ111 ~ I. Now let (J1i + Vi) with J1i E A!, Vi E .A(F) be a generalized sequence in the unit sphere of Al. + .A(F) , weakly convergent to J1. Since AIF is closed in C(F), then according to Theorem 4.1 there exists c ~ 1 such that
+
+
+
(V E
J/(F».
Hence, for any i there exists a measure Ai E (AIF)l. with
Therefore the generalized sequence {Vi - Ai} in ..II(F) has a limit point v (a weak limit in <-,(I(X»), which belongs to .A(F). Then clearly the generalized sequence
has J1 - V as limit point. Since J1i and Ai belong to Al. which is weakly closed in J1(X), J1 - V E Ai, hence J1 E Al + .A(F). The proposition is proved. 0 Theorem 4.12. The following assertions are equivalent: (i) F is an intersection of peak sets. (ii) For any J1 E A! we have J1F E Al.. Proof. (i) -+ (ii). If there exists a Jl E A! with J1F ¢ Ai then there is a J1 E Al. with lIJ1x-FIl = 1 and IIJlF + (AIF)l." ~ 8 0 > o. Let f E A be such that IIfll ~ 1 and
JFfdJl =
r > 28 0 •
Function algebras
64
Lef K be a compact set, disjoint of F, and such that 11l1(X - (F UK» < 80. According to Corollary 4.7 there is g E A with IIgli ~ 1, g =1 on F and Igi < 8 0 on K. We have
-;:: r -
J
IIgli
80 -
dllli -;:: r -
280
>0
X-(FUK)
which is impossible. 1\
(ii) -+ (i). Let B = {I E A : I = const. on F}. X be the factor space obtained from X by reducing F to a point, and B = {f E C(X): there is an I E A withJtx) = I(x)}. Obviously B is a uniformly closed subal" which contains the constants. We prove now that B" gebra of C(X), separates the points of X. "A Let p E X be the point obtained from F and x E X, x # p. Suppose f(p) = I(x) for any I E B. This means that/(x) = 0 for any I E kF, and hence 8x E (kF)l. From (ii) and Theorem 4.2 we get that A IF is closed and from Proposition 4.11 we have 1\
1\
1\
1\
1\
1\
..
'"
A
where (A.L)X_F = {IlX-F with Jl E Aol}. Hence ex E (Al )X-F C Aol. Then I(x) = 0 for any I E A which is impossible since A contains the constants. A" A Let now Xl # P tF x 2 • If I(x l ) = l(x 2 ) for any I E B, then the measure 8 x } - 8 X2 E (kF)ol and, since it is supported in X - F we get ex! 8 X2 E (Al )X-F CAl. Therefore !(x!) = !(x2 ) for any I E A which is impossible since A separates the points of X. "
,..
1\1\
1\
1\
1\
Then B is a function algebra on X. " We define Jl Let 'I E Bol.
E
Jt(X) by
Jl(q»
=
v(~)
Ch. 4. Approximation and interpolation
for cP
E- C(X)
6S
which are constant on F. J.l(f)
=
0
(fEA)
and then, extending to C(X) by Hahn-Banach theorem. Clearly It E AJ. and (ii) gives J.lE E AJ.. Then we have v({p}) = .v(X(P) = P.(XE) = J.lE(l) =
o.
"
.
Hence for any v E BI we have v( {p}) = O. " Let v ~ Gpo Then G - v E BJ. therefore p
IS v =
(Gp -
v) ({p})
=
0, that
Gn •
"
" From Theorem 2.8 p is an intersection of peak sets for Band then, obviously, F is an intersection of peak sets for A. 0 Corollary 4.13. (Rudin-Carleson Theorem). Let Jl be a measure on X such that any v E AJ. is absolutely continuous with respect to J.l, and F a closed set of X with /1(F) = O. Let p E C(X) with p > 0 and f E C(F) with If I ~ p on F. There exists g E A 'such that g = f on F and Ig(x)1 ~ p(x) on ·X. Proof. For any') E Ai one obviously has VE = 0 and, according to Corollary 4.4 AIF = C(F). From Theorem 4.12 F is an intersection of peak sets and therefore Theorem 4.10 completes the proof. 0
Corollary 4.14. .( Bishop's Lemma). If F is an intersection of peak sets for A and KeF is an intersection of peak sets for AIF, then K is an intersection of peak sets for A. 0 Proof. From Theorem 4.12 there results (AI)E C (AIF).L and [(AIF)i]K C [(A IF) IK].l
= [AIK].l. Hence (Al)K = [(A.l)]K C
C [(A IF).l] X C [A/K]l. and Theorem 4.12 completes the proof. (;
4.3. Antisymmetry A subset E C X will be called a set of antisymmetry of A if any function in A, which is real on E is constant on E. The algebra A is said to be antisymmetric on X if X is a set of symmetry of A. 5 - c. 437
Function algebras
66
The closure of a set of antisymmetry symmetry.
obviously a set of anti-
IS
Proposition 4.15. Any maximal set of antisymmetry is closed. Erery x E X belongs to only one maximal set of antisymmetry. Proof. It is sufficient to notice that the union of two sets of antisymmetry with nonvoid intersection is a set of antisymmetry, and the singletons are sets of anti symmetry . Therefore the family {EJiEJ' of maximal sets of anti symmetry is a family of disjoint closed sets which covers X:
X=
U Ei• iEJ'
Proposition 4.16. Any maximal set of antisymmetry is an intersection of peak sets. Proof. Let E be a set of anti symmetry and K the intersection of all peak sets containing E. We show first that K is a set of antisymmetry. Let f E A have real values on K. Then f is constant on E and therefore we may assume IIfli = 1 and f = 0 on E; indeed, if any f E A with real values on K were constant on X, then X would be a set of antisymmetry and we would have nothing to prove. Let N = {x
E
K:f(x) = O} and g
=
1 - fiK.
We have g E A IK, g = I on N and Ig I < 1 on K - N, N is then a peak set for A I K. According to Corollary 4.14, N is an intersection of peak sets for A. Since E C N, from the definition of K we get KeN, hence f = 0 on K. Thus K is a set of anti symmetry and since E C K is a maximal set of anti symmetry, E = K, i.e. E is an intersection of peak sets. <) Proposition 4.17. (de Branges' Lemma). Let /1 be a measure on X which is an extremal point of the unit ball of AJ.. Then its support is a set of antisymmetry. Proof. Let K be the support of /1 and let f E A, real on K. We may assume 0 < f < 1 on K. We write
67
Ch. 4. Approximation and interpolation
dJl1 = N1IfdJl, dJl2 = Nil(1 - f)dJl. We IIJl211 = I, Jl1' Jl2 E A.L and Jl = N IJl1 + N 2Jl2· Since Jl is extremal we get Jl = Jl1 = /12, hence f
Let
=
have =
IIJl111 =
N1 on K. 0
Theorem 4.18. (Shilov-Bishop); Let A be a function algebra on X. The space X admits a decomposition of the form X = U E i , where i E.1
(E j ) i E.I is the family of I1zaximal sets of anfi~'Ymmetry of A. fVe have EinEj = 0 for i =I j and I) A IEi is closed and antisymmetric, 2) f E C(X) belongs to A if and only if for any i E 5, flEi E A lEi. Proof. We need only prove (2). Let f E C(X) with flEi E AIEi for any i EJ and /1 EAl.. It is sufficient to show that Jl(f) = O. We may, of course, assume IIJlII = 1 and also, using the Krein-Milman Theorem (Proposition 1.4), we may assume that /1 is an extremal point of the unit ball of A.L. According to de Branges' Lemma (Proposition 4.17), the support of Jl is a set of antisymmetry, i.e. it is included in one of the Ei sets. Hence Jl(f) = O. Therefore the theorem is completely proved. 0 If A is a function algebra on X, the decomposition X = U Ei iE.1
given by Theorem 4.18 will be called the Shilov-Bishop decomposition of X (with respect to A).
4.4 Some characterizations of C(X) Let A be a function algebra on X. From the Shilov-Bishop decomposition theorem (Theorem 4.18) one can get some characterizations of C(X) among the function algebras on X, according to the following principle: Theorem 4.19. Let (P) be a property which is defined on any function algebra on X, for any compact space X. It is assumed that: if a function algebra on X has the property (P) and A is antisymmetric,
Function algebras
68
then X reduces to one point. Let A be a function algebra on X and X = U E; be the Shilol'-Bishop decomposition of X with respect to A. , E,?
If A IE; has property (P) for any i E J, then A = C(X). Proof. According to the hypotheses, E; reduces to a single point, for any i. Then it is clear that for any f E C(X) we have fiE; E A lEi and therefore Theorem 4.18 givesfE A, that is A = C(X). 0
Theorem 4.20. (Stone- Weierstrass). Let A be a function algebra on X. If A is symmetric, i.e. for any f E A we have f E A, then A = C(X). Proof. Let A be a function algebra on X which is antisymmetric and symmetric on X. If X contained more than one point, then, since ReA separates X, there would exist a non-constant function U
E ReA, u=Ref withf EA. But A is symmetric, hence u =
~(f+l)EA
2 and, on the other hand, since A is antisymmetric u would be constant on X. We have reached a contradiction and therefore X reduces to a single point. Now, since for any closed set Fe X, A IF is symmetric if A is symmetric, we have the conditions of Theorem 4.19, hence A = C(X). 0
Theorem 4.21. LetA be a function algebra on X. The space ReA is uniformly closed in C R (X) if and only if A = C(X). Proof. If A = C(X), then obviously ReA is closed. The converse assertion will -be proved by using Theorem 4.19. Let A be an antisymmetric algebra on X such that ReA is 'closed.' We fix an Xo E X and put (ReA)xo = {u E ReA: u(xo) = o}. Since ReA is closed, (ReA)xo is closed. For U E (ReA)xo there exists a unique 'IE (ReA)xo' such that u + iv E A. Indeed if u = Ref with f E A, then v = Imf - Imf(x o) E (ReA)xo and u + iv E A. If u + iv E A, u + iv' E A with v, v' E (ReA), then v - v' = = -i[u + iv) - (u + iv')] E A and since A is antisymmetric, we get v - v' = v(xo} - v'(xo) = 0, Le. v = v'. We now define an operator T on (ReA}xo with values in (ReA)xo by Tu =.'1, with v constructed as above. ;T is a bounded operator. Indeed, using the closed graph theorem, it is sufficient to prove that if Un E (ReA)xo' UII --+ u and VII = TUn converges to v, then Tu = v; this follows easily from the definition of T. Assume now that X.' contains more than one point. Then there exists g E A with g(xo).= 0 and g :j:: o. ;Let, g =U + iv.: From the
Ch. 4. Approximation and interpolation
69
antisymmetry we get u ¥= 0, v ~ 0 and therefore there is an Xl eX, Xl ¥= Xo such that v(xI ) = Ilvll. We put Zl = g(x 1). Let N be an arbitrary integer and q, the conform mapping of the interior of the rectangle Ro' = {-Ilull ~ x ~ lIull, -llvll ~ y~llvll} on the interior of the rectangle RN = {Ilull ~ x ~ Ilull, - N ~ Y ~ N} such that fl> can be extended to a homeomorphism between R ..and R Nand q,(0) = 0,
(Zl) = iN. It is known that X may be obtaIned as a limit of polynomials on R. Therefore h = fl>(g) E A and if h = s + it then clearly s, t E (ReA)xo"lIsll ~ Ilull, Iltll = Nand Ts = t. Since N is arbitrary, this contradicts the fact that T is bounded, hence X reduces to a single point. Now let A be a function algebra on X such that ReA is closed and X = U Ei is the Shilov-Bishop decomposition of X with respect i
Eo?
to A. Since for any i E J, Ei is an intersection of peak sets, according to Corollary 4.9, Re(AIE) ~s closed. We thus have the conditions of Theorem 4.19 from which we conclude that A = C(X). 0 Theorem 4.22. Let A be a function algebra on X. The space ReA is an algebra if and only if A = C (X). Proof. If A = C(X) , then obviously ReA = CR(X) is an algebra. Now let A be an antisymmetric algebra on X and assume ReA is an algebra. Let Xo E X; from antisymmetry we get that for any u E ReA there exists a unique f E A such that Ref = u and Imf(xo) = O. We define on ReA the norm N(u) = Ilfll, where f E A, Ref = u and Imf(xo) = O. ReA becomes a real Banach space with respect to this norm. According to the closed graph theorem and the uniform boundedness principle we get N(uu')
~
KN(u)N(u')
(u,u'
E
Re A).
We now show that for any pEReA withp > 0 we have logp E ReA. Let B = {q> E C(X): q> = u + iv, v E ReA}. Since ReA is an algebra, B is also an algebra. For fEB we put N(f) = N(u) + N(v) and IIfll' = = sup N(OiJ f). O~8~2."t
One easily verifies that B is a Banach algebra with respect to a norm equivalent to 11.11' and A, A C B, ReB = ReA. Let 5)[ A' .ffi. B be the maximal ideal spaces of A and B respectively. If q> E j)[ B
Function algebras
70
then eplA E .!11L A- Since pEReA and p > 0 on X, we get p > 0 on ~l1l A.
1
Now let p = -(I 2
peep)
=
+ I)- with lEA.
1 -(f(lp)
2
+ I(lp» =
For any lp
E ~lB
1
-(/«({JIA)+/«({JIA» 2
we have
= p«({JIA) > o.
Hence p > 0 on ~R B. Then log p E B and, since p is real, log p E E ReB = ReA. We now assume, that X does not reduce to a single point. Then there is agE A with g(x o} = 0 and !!g!! = 1. Let x E X be such that IIglI = g(x) = 1. We choose an analytic in Izi < 1, continuous on Iz/ ~ 1 such that 0 < Re ~ 1 in /zl ~ 1, Im(O) = 0 and 1m (g(x» ~ ~ n, where n is a fixed but arbitrary natural number. Let I = (g). Then I E A and 0 < ReI ~ I on X, Im/(xo} = 0 and IIfll ~ n. From the above considerations we deduce that there exists FE A such that ReF = log (ReI). Let V = exp (1/2 F); then V E A and I VI 2 = ReI. Therefore I VII ~ 1. By using the identity (Rez)2
=
1
-
(Rez 2
2
+ /ZI2)
we get
(ReV)' = ReC (V'
+f»).
Now, since for any h E A we have N(Reh) ~
Ilhll -
Ilmh(xo)1
there results
N[(ReV)2] ~
as
II/II
~ n and
I V 2 11
1
-(II V2 + III 2
~ 1.
IImV2(xo)1) ~
1 -en 2) 2
Ch. 4. Approximation and interpolation
~
On the other hand, N«Re V)2) 211 Vll 2 ~ 2. Therefore
1
- (n - 2)
~
~
K(N(Re V»2 and N(Re V)
N[(Re V)2]
~
71 ~
4K.
2
Since K is a fixed constant and n is arbitrary, we have a contradiction. Hence X reduces to a single point. If A is a function algebra on X with ReA an algebra and F a closed subset of X, then it is clear that Re(A/ F) is an algebra. Therefore the conditions of Theorem 4.19 are satisfied and we have A = C(X). 0 Theorem 4.23 Let A be a function algebra on X. If for any closed subset F of X. AlE is closed. then A = C(X). Proof. We show first that if E and K are two disjoint compact subsets of X then there exists f E A with fiE = I and flK = O. Let x E E and y E K; since A separates X, there is a function g E A such that g(x) = I and g(y) = O. Let Vx and Wx be two neighbourhoods of x and y respectively, such that Ig(Vx)1 < 1/4, 11 - g(Wx)1 < 1/4. Consider now a sequence Pn of complex variable polynomials which converges uniformly on {z: Izi ~ 1/4}U{z: II - zl ~ 1/4} to the characteristic function of the subset {z: Izl ~ 1/4}. } Then Pn 0 gl VxU Wx is a sequence of elements in AI VxU ~, whidh converges uniformly on VxU Wx to I on Vx and 0 on Wx. Since A IVxU Wx is closed, there exists e E A with e (Vx ) = I, e (Wx ) = o. We fix y E K and cover F with a finite number of neighbourhoods VI, ... , Vn constructed as above and let WI, ... , Wn and e 1 , .. ·, en be the corresponding neighbourhoods of y and functions in A. Then e~ = e1 + e2 - eIe 2 has the value I on V 1 UV 2 and 0 on WIn W 2 , e~ = e~ + ea - e~e3 has the value I on VI U V 2 U V3 and 0 on WI W2 W3 , etc. Thus we obtain a function fy in A which takes the value I on F and 0 on a neighbourhood Uy of y. Let us cover K with a finite number of such neighbourhoods U h "., Un and let fl' ... ,fn be the corresponding functions in A. Then f = fl, ... ,fn belongs to A andflF = I and fIK=O. Now, let x E X with the following property (P): there is a neighbourhood U of x and a constant M such that for any closed subset FC U
n n
n
F unction algebras
72
and any function f E A which takes only the values 1 and 0 on F, we have IIf + kF11 < M. We prove that for any closed K C U we have A IK = C(K). Let p. E [A I K]l. be a measure on. K, Kl a closed subset of K, G > 0 and Ko a closed subset of K - Kl such that I/l1[(K - KI ) - Kol < e/ M. Let f E A with /1 KI = 1, /1 Ko = 0 and IIfll < M. From the first part of the proof and from property (P) of x it follows that there exists such an / in A. Then
Hence
But G > 0 is arbitrary and therefore p.(K1) = 0 for any Kl C K, hence J1 = O. There results (AIK)l. = 0, that is AIK = C(K). Assume now that A is antisymmetric on X and that there is an x E X with property (P). We choose VICVICV2CV2CV3C·V3C C U, neighbourhoods of x. From the above considerations we can find a function f E A such that f{X - V3) = 0 and f( V2) = I, and a function g E A with gl Va real, g(V3 - v2) = 0 and g( VI) = I. Then it is clear thatfg is a real non-constant function in A, which contradicts the antisymmetry of A. Therefore, no point of X has the property (P). The next step is to show that X has a finite number of points. Assume that X contains an infinite sequence {xn} of distinct points. Since none of them has property (P), there exists a sequence Un of disjoint neighbourhoods of XII' a sequence of closed sets Fn C Vo: and a sequence fll of functions which take only the values 1 and 0 on En and 1I/II+kFIIII ~ n. Let F U F,.. Since for n =F m, V"nFm = 0 there results F = = FnUrUFml and FnnrUFml = 0. Hence there exists a function m=:;:'n
f
E
A with f(F,,}
=
n ~
m=/=n
I and f(F - F,.)
=
O. We then have
IIf,. + kFnl1 = II/fll + kF,,1I
~
IIff,. + kFIl.
Ch. 4. Approximation and interpolation
73
On the other hand, since A I F is closed, the isomorphism between A IF and A IkF is a topological one, hence there is an M such that
IIf + kF11
~ M
IlflFl1
(f E A).
But Ilffnl FII = 1 and we reach a contradiction; therefore X is finite. A separates the points of X and is antisymmetric, hence X reduces to only one point. Now if the function algebra A on X has the property enounced by the theorem, it is clear that for any closed set Fe X the algebra A IF has the same property. We have then under the conditions of Theorem 4.19 and A = C(X). 0 Notes Theorems 4.1 and 4.2 together with their corollaries are to be found, under different forms, in E. BISHOP [3], T. GAMELIN [1] and I. GLICKSBERG [1]. The results of paragraph 4.2 leading to the abstract form (Theorem 4.10) of RudinCarleson interpolation theorem (Corollary 4.13) are given in the same works. Different forms of the Shilov-Bishop decomposition theorem (Theorem 4.18) have been-stated by G. E. SHILOV [1] and E. BISHOP [2]. In the present book~ we follow the proof given by I. GUCKSBERG [1] using de Branges' Lemma (Proposition 4.17). Theorems 4.19 and 4.21 belong to J. WERMER [3] and Theorem 4.22 to K. HOFFMAN and J. WERMER [1]. Theorem 4.23 is due to I. GUCKSBERG [2].
CHAPTER 5
HP·Spaces
5.1. Definitions and basic lemmas -
Let A be a function algebra on X and p, a positive measure on X. For I ~ p < 00 we denote by HP(dp,) the LP(dp)-closure of A, which is a subspace of LP(dp). Hoo(dp,) will denote the weak closure of A in Loo(dp). In this chapter we present a theory of HP(dp,)-spaces in analogy with the classical theory of Hardy's HP-classes. We denote by R(X) the set of Borel functions on X and by M(X) the set of bounded Borel functions on X. For a function FE Ll(dp,), F ~ 0, its geometrical mean will be defined as
I(F; dp) Obviously 0 ~ I(F; dp) < log FE Ll(dp).
00
= exp Ulog Fdp]. and J(F; dp) > 0 if and only if
Lemma 5.1. Let p be a positive measure on X for which p(X) and G E Ll(dp) is non-negative. Then the function
OCt)
=
UF'dp]I/t
(0 < t
~
=
1
I)
is monotonously increasing. If F is not constant, () is a strictly increasing function. Moreover
I(F; dp)
=
lim OCt). 1-+0
Function algebras
76
Proof. Let 0 < 1 ~ s ~ I. By applying Holder inequality for F' and 1 and for p = sit, we find 0(/) ~ O(s). If 0(/) = O(s) for t =1= s, then Holder inequality becomes an equality; therefore, F and 1 are linear dependent, i.e. F is constant. The well-known inequality eX ~ 1 x for any real x, gives
+
F' = exp [t log F] (J(/)
=
+ t log F for any I, 0 < t ~ I, hence ~ [I + t Jlog Fdp]l/I, (0 < 1 ~ 1).
~
I
[J F'dp]l/1
Therefore (J(t) ~ exp
[J log F dp] =
J(F, dp),
(0 < 1
~
I),
that is
J(F, dJl)
~
lim 0(1). 1-+0
Let us now prove the converse inequality. First, we may suppose that F ~ Ct, ( 1 ) 0, since the general case can be obtained from this one by Beppo-Levi Theorem. We then take F ~ I, hence log F ~ o. Let b > 0 and 0 < t ~ b ~ 1. We have
P
= 1 + t log F +
00
tk
l: -
(log F)k ~ 1 + t log F + tb
k=2k! =
1 + I(log F
X)
k=ok!
+ bF).
Then and, therefore, (J(t) ~ [1
+ t( Jlog F dp + b JF dp)]l/1
and we have lim O(t) ~ J(F, dp) exp [b JFdp] 1-+0
for any b, 0 < b
~
I, that is lim O(t)
which completes the proof. 0
~
J(F, dp)
1
l: -
(log Ft =
Ch. S. HP-spaces
77
Lemma 5.2. Let T be a linear subspace of M(X) such that 1 E T, L be a linear functional on T, with L( 1) = 1, and p, be a positive measure on X such thaI p,(X) = 1. lffor p, 1 ~ P < 00 and any FVF2E T
we have then
JFIF dp,.
L(Fl)L(F2) = Proof. Let FE Twith L(F) 1=
IL(l - IF) IP
1=
I L(I - IF)L(I
2
=
O. For any complex number). we have
~
J11 -
lFIPdp,
+ IF)!P ~ J11 -
).2F2 1Pdp,.
Therefore, for any t > 0 and any complex number ). we have
~+ [11 + f).FIP -
I]dp :;;, 0
L~ [t 1 + tlF2 jP -
l]dtL ~ O.
)t
-
On the other hand, for any t, 0 < one can easily verify that
I ~
I, and any complex number z
. 1 [_ hm --[ 1 + IzlP - I] = pRete 1-+0 t
There results
JRe(lF)dp, ~ 0
J Re(lF2)dp, ~ O. for any complex number l; this is possible only if
J F dp. = J F dJl = 2
0"
Function algebras
78
Consider now an arbitrary FE T. Then
+ L(F)]djl = J[F - L(F)]dJl + L(F) = L(F) JF djl = J[F - L(F) + L(F)]2djl = J[F - L(F)J2djl + + 2L(F) J[F - L(F)]dJl + [L(F))2 = L(F)2.
JFdp = J[F -
L(F)
2
2 J Fl F2dp =
= [L(FI
J(Fl + F2)2djl - JFidjl - JF:dp =
+ F2)]2 -
[L(Fl)]2 - [L(Fz)]2
=
2L(F1)L(F2).
The lemma is thus proved. 0Let ~ be the maximal ideal space of A. Until the end of this chapter we shall fix an element ep E~, that is a non-zero multiplicative linear functional on A. Then it is known that there is a representing measure jl for ep, with support in X, i.e. a positive measure p on X such that f(ep)
= Jfdjl
(f E A).
Let S be a linear subspace of R(X) and u a linear functional on S. The couple [S, u] will be called admissible (relative to A and ep) if for any fE A and hE S we havefh E Sand u(fh) = ep( f) u(h)
p
E
(f E A, hE S).
For an admissible couple [S, u], a positive measure jl on X and [1, (0), we denote by DP(S, u, djl)
=
inf {J Ih IP djl, h E S, u(h)
=
I}.
We have DP(S, u, djl) ~ 0, DP(S, U, dJl)< 00 for S C LP(djl) and the function p -+ DP(S, u, dJl) is right continuous on [I, (0). The couple [A, ep] is obviously admissible. We write DP(dp)
=
DP(A, ep, djl).
Proposition 5.3. Let jl be a positive measure on X with jl( X) = 1. Then jl is a representing measurefor cp if and only if DP(dp) = 1.
79
Ch. 5. HP-spaces
Proof. Let fl be a representing measure for lfJ. As lfJ{l) results
I, there
=
DP(dfl) ~ Jdfl = I. On the other hand, iff E A and lfJ(f) 1
= lfJ(j) = Jfdfl
~
1, then
=
JIf I dfl ~ [J If IP]I/P,
hence DP(dfl) ~ I. Assume now DP(dfl) = I. For any f
E
A, with lfJ(f)
, f
1 = DP(dJl) ~ (:
J
~
0, we have
Ip
: dfl, lfJ(f)
i
hence
IlfJ(f) I ~ J If IPdp
(f E A).
From Lemma 5.2 there results lfJ(f) = Jfdfl,
(fEA)
i.e. p is a representing measure for lfJ. 0 Lemma 5.4. Let [S,O'] be an admissible couple with S C M(X), let J..L be a positive measure on X and let FE Ll(dfl) be non-negative. Let 1 < p, q < 00 with lip + l/q = I, and assume DP(S, (1, d,u) > O. Then 1) There is only one function Q E Lq(d,u) such that
JIQ IqdJ..L =
I,
JhQI/Pdp =
O'(h) [DP(S,
0',
Fdfl)]IIP
(h
E
S).
2) dA = I Q Iqdfl is a representing measure for lfJ. 3) The function P E LP( dfl) defined by PQ ~ 0 and
IPIP
=
DP(S,
0',
FdJl) I Q I q
belongs to the LP(dfl)-closure of the set {hFI/p: h E S, O'(h)
=
I}.
Proof. We have SCLOO(dp). We define, on the subspace {hFl/P: hES} of LP( dfl), the functional L by L(hF1/P)
=
[DP(S,O', Fdfl)]I/PO'(h).
Function algebras
80
Then
L(hF1IP) = [DP(S,
U,
Fdp)]l/Pu(h) ~
=
IlhF1/PI/Lp(dl')'
[J ih IP FdJl]1iP =
Therefore, L can be extended to a linear functional on LP(dp), of norm less than 1. There exists then a function Q E L q(dp) such that (5.1.1)
(h
E
S)
and (5.1.2) From (5.1.]), for any IE A with cp(f) = 1 and hE S with u(h) = 1, we obtain
[DP(S, ~
U,
Fdp)]IiP
=
JfhQFl/PdJl ~
[J lfl q I Q IqdJl]I/q[J Ih i FdJL]l/P P
and therefore
[S [fl q: Q [qdp]I/P ~ [ DP(S, u, Fdp) ,
JI IP
SlhlPFdp
for any I E A wi"th cp(f) = 1 and h E S with u(l1) = 1. There results (5.1.3) In particular we have
S IQ IqdJl ~ 1 which, combined with (5.1.3), yields
S IQ Iqdp = 1.
(5.1.4) From (5.1.3) we get
Dq(1 Q lqdp) = inf {Slflql Q !qdp; and, since
I
E
A, ({J(f) = I} ~ 1
Ch. S. HP.spaces
81
there follows Dq(1 Q Iq dp) = 1. Then, according to Proposition 5.3, dA. = I Q Iq dJl is a representing measure for cpo Let be the set of all functions of Lq(dp) with the following property: considered as functionals on LP(dp) they extend Land their norm is at most equal to I. From (5.1.4) n results a closed set in {Q E Lq: liQl1 = I}. As n is a convex set and the points of {Q E Lq: IIQII = I} are extremal points for {Q E L q: I Q II ~ I}, n reduces to only one point. This proves the uniqueness of the function Q. 1) and 2) are completely proved. We now prove 3). Let T be the subspace of functions gEL q(u,u) which satisfy
n
JghFI/P dp =
(h
0
E
S, u(h) = 0).
One can then easily verify that
J
J
u(h 1 ) gh 2F 1/P dJl = u(h2) ghlFI/P d,u
(g
E
T, hb h2 E S)
and if we put L(g)
= JghF1/P dJl
(h
E
S, u(h) = 1)
we obtain a linear functional L on T satisfying L(g) u(h) =
JghF1/Pdp.
(h
E
S, gET)
We have
I L(g) I ~ Jig Ilhl for any h E S with u(h) I
FI/Pdp ~ IIgIlLq(dll) =
[J I h IP FdJ.l]1IP
1, and, therefore,
L(g) 1 ~ [DP(S, u, Fdp)j1/P IIgIlLq(dll)'
From (5.1.1) we get Q E T and L(Q) = [DP(S, u, Fdp)]1IP,
hence
IILII
=
[DP(S, u, Fdp)j1IP.
We now extend L to a functional on Lq(dp), of the same norm, and let G be the function of LP(dp) representing it. Then
JIG IPdp = 6 - c. 437
DP(S, u, Fdp)
Function a]gebras
82
and L(g) =
JgGdp.
(gE T)
Since Q E T we have
JQGdp =
L (Q) = [DP(S, u, FdJl)]l!P =
II QIILq(dJl)11 GIILq(dP)'
Hence, Holder inequality for Q and G is an equality, thereforeI G IP = cl Q Iq where c is a constant and QG ~ O. By integration we obtain c = DP(S, u, Fdp}; then G = P where P is the function defined in 3). Let now g be a function in Lq (dJ.l), orthogonal to {liFlll' : 11 E S} C C LP(dp). There results gET and, therefore,
JPgdp =
L(g) =
JghF1/P dJ.l = O.
(h
E
S, u(h)
=
1).
Thus, P belongs to the closure of the set {hF1/P: hE S} in LP(dp}. Let h,. E S be such that the sequence h,.Fl/P tends to Pin LP(dp}. From (5.1.1) we get
Jh,.QF1IP dJ.l =
u(hn) [DP(S, u, Fdp)]1IP,
hence u(h n ) is convergent and lim u(h n} n-+CYJ
The sequence
h~ =
=
_-=--JP_Q_d_Jl_ =
I
DP(S, u, Fdp)
h
n FliP then converges to P in LP(dp) and, since u(h,.) 1, P belongs to the closure of {hF1/P: h E S, u(h) = I}
u(hn/u(h,.)) = in LP(dp). The proof is thus complete. 0
5.2. The theorem of F. and M. Riesz and Szego theorem Let X be the unit circle {z: I z I = I} of the complex plane and A the algebra of continuous functions on X which may be analytically extended in the interior (the standard algebra). Let m be the normalized Lebesgue measure on X, p an arbitrary measure on X and dp =
83
Ch. 5. HP.spaces
+ dJls
the Lebesgue decomposition of Jl with respect to m. The principal part of the well-known theorem of F. and M. Riesz is the following: If Jl is analytic, that is = dPa
(n
=
1,2, ... )
then Jl s E Al.. Indeed, the measure d)' = zdJls results analytic and, since d)' obviously singular with respect to m, we get
IS
(n = 0, 1,2, ... ).
Repeating this argument we obtain by induction
Jz"dJls =
0,
(n=0,+I,+2, ... )
and, therefore, Jls = 0. This principal part of the F. and M. Riesz theorem can also be proved in the case of an arbitrary function algebra A on an arbitrary compact space X, using some special representing measures instead of Lebesgue measure. Let us consider a function algebra A on X and fix an element cP E IDl.
Proposition 5.5. Let m be a representing measure for cpo The following assertions are equivalent: (1) Any representing measure for cp is absolutely continuous with respect to m. (2) Let Jl be a positive measure on X and dJl = Fdm dJls, FE Ll(dm), the Lebesgue decomposition of Jl with respect to m. Then DP(dJl) = DP(Fdm). (3) Let Jl be a positire measure on X and dJl = Fdm + dJls' FE Ll(dm) the Lebesgue decomposition of Jl with respect to m. Let [S, 0"] be an admissible couple, with S C LP (dp). Then DP (S, 0", dJl) = = DP (S, 0", Fdm). Proof. (I) ~ (3). We obviously have
+
JIh IP Fdm ~ JIh IPdp
(h
E
S),
and, therefore, DP(S,
0",
Fdnl) ~ DP(S,
0",
dJL}.
Function algebras
84
We observe that if DP(S, u, dJl) = 0, then the converse inequality is obvious; hence, assume DP(S, u, dJl) > O. Since the function p -. DP(S, u, dJl) is right continuous, we may suppose p > 1. We first assume S C M(X). Let p > I and q with lip + I/q = 1. According to Lemma 5.4 there exists Q E Lq(dJl) such that
JIQlqdJl =
I
and (h
E
S).
Moreover, dJ.. = IQlq dJl is a representing measure for qJ. From (1) we get J.. absolutely continuous with respect to m, hence Q = 0 almost everywhere with respect to Jls' For any h E S with u(h) = I we have [DP(S, u, dJl) p'P
= JhQdJl = JhQFdm
~
~ [J Ihl PFdm] IIp [J IQlq Fdm]l/q = [J Ihl PFdm] lip.
Therefore DP(S, u, dJl) ~ DP(S, u, Fdm). We now assume that S C LP(dJ..) is arbitrary and let V E S be such that u(V) = I. We define
S*
= {h E M(X): h V E S}
u*(h) = u(h V)
(h
E
S).
We have S* C M(X) and we can easily verify that [S*, u*] is an admissible couple. Applying the first part ofthisprooftomeasurelVIPd/l and couple [S*, u*] we obtain
We have DP(S*, u*, IV/PdJl)
= inf {J IhV/P; h E M(X), hVE S, u(hV) = I}
~ inf{JlhIPdJl;hE S,u(h) =
I}
= DP(S,u,dJl).
Hence DP(S, u, dJl)
~
DP(S*, u*, IVIPdJl) ~
~
JIVIPFdm,
DP(S*, u*, IVIP FdJl)
~
~
Ch. S. HP-spaces
8S
since 1 E S* and O'*(l) = I. Then
DP(S,
0',
dJl)
~
DP(S,
0',
Fdm),
which completes the proof of implication (1) --+ (3). (3) -+ (2) is obvious (2) -+ (1). Let Jl be a representing measure for
= DP(dJl)
=
DP(Fdm) ~
JFdm ~ JFdm + Jls(X) = ,u(X) =
I.
There results JliX) = 0 and, since ,us is positive we get Jls = O. The proposition is proved. 0 Theorem 5.6. (F. and M. Riesz). Let m be a representing measure for <po The following assertions are equivalent: (1) any representing measure for
Jfdll = 0
(fE A, Jfdm
=
0)
and dJl = dJIa + dJls is the Lebesgue decomposition of J1 with respect to m. Then JIs E Ai.. Proof. (1) --+ (2). Let JI E Al. and dJI = Fdm + dll s be the Lebesgue decomposition of JI with respect to m. For any natural number n we I ] put dJI,. = [ 1 + -;;
.. IFI dm + dlJIsl. From Proposition
5.5 we get
D'(dp.) = D'([I + ~ IFI]dm). According to the definition of D2 there exist f,.
E
A such that
Function algebras
86
We have
J lin
(5.2.1)
12
dm
+ _1 J1/12/FI dm + J Iln/ 2 dl Jls ~ I
n
~
,,; D2CdJl.)+ ~ 1+
=
D2 ([ 1 +
~
I F I] dm ) +
~ ,,;
I I I - I FI dm + - = 1 + - K, n n n
J
where K
=
J.-n (l + J F 1
1
dm).
On the other hand
Jlin :2dm + ~n Jlin 121 FI dm ~ D2([1 + I F:l dm) and from (5.2.1) we obtain (5.2.2)
J Iif,n
12 i
dJls < -1. n
According to (5.21) we also have
J lin :2 dm + _1n JI/~ I I F I dm < 1 + ~n K and, since
J:In: 2dm
~ D2(dm) = 1
there results (5.2.3) Once more from (5.2.1), we get
J lin :2 dm < 1 + K and, using also (5.2.3) we deduce (5.2.4)
Ji/n 12 (l + /F)dm
< 1 + 2K.
87
Ch. 5. HP-spaces
Therefore, the sequence (In) is bounded in Lp(dm + I FI dm). We may assume (In) weakly convergent (we eventually consider a subsequence). Moreover, we have
We used the fact that m is a representing measure for cp, cp(ln) = I, and the inequality (5.2.1). Hence, (fn) is weakly convergent to 1 in L2(dm I FI dm) and, therefore, the same is true in L2(l F) dm}. Since J1. E A.1, we have
+
for any lEA. Since JllnFdm converges to JIFJm and, according to (5.2.2), JIlndJ1.s converges to 0, we find
JIFdm
=
that is Fdm E A.1. Therefore, J1.s E A.1 and (I) (2) --+ (3). Consider a measure on X such that
J/dJl = 0
(f E A),
0 --+
(2) is proved.
(I E A, J/dm =
0).
The measure J1. - It{l)m then belongs to A.1 and from (2) there results that its singular part relative to m, which is obviously J1.s' belongs to A.1. (3) --+ (1) is immediate. Indeed, any representing measure J1. for cp satisfies the condition required in (3) and therefore Jls E A.1. Since J1. and m are positive, J1.s is also positive, hence Jls = O. The theorem is proved. 0 We now establish Szego theorem in the context of function algebras. In the classical case, Szego theorem established the distance in L2(F dm), where m is the Lebesgue measure, FE Ll(dm), F ~ 0, from 1 to the subspace generated by A 0 - the set of all functions in the standard algebra A which vanish in the origin - by the formula inf IEAo
JII - 1: 2 Fdm =
exp Ulog Fdm].
Function algebras
88
If cp is the functional on A given by the value of functions at 0, then it is clear that inf J,I
-
fl2 Fdm = D2(Fdm)
/EAo
and, therefore, Szego's formula may be written as D2(Fdm) = J(F, dm).
In this form it will also be established in the general case. Let A be a function algebra on X and cp a linear multiplicative functional on A. Proposition 5.7. Let m be a representing measure for cp such that any representing measure for cp, absolutely continuous with respect to m, coincides with m. Let G E Ll(dm), G ~ 0 and dll = Gdm. Then for any FE Ll(dp) , F ~ 0, FG E Ll(dm), any p, 1 ~ P < 00 and any admissible couple [S,O"] with S C LP(dp), S C LP( Fdll) such that at least one of the numbers DP (S, 0", dp), DP (S, 0", Fdp) is different from zero, we have J(F, dm)
=
DP(S, 0", Fdll) . DP(S, 0", dll)
Proof. Since the function p -+ DP is right continuous we may suppose p > 1. Let q be such that lip Ilq = I. We firstly assume F ~ B > O. We now prove the inequality
+
J(F, dm) ~ DP(S, 0", Fdp) . DP(S, 0", dll)
If DP(S,O",dp) = 0 then, according to the hypothesis, DP(S,O",Fdll) > > 0, and therefore, in this case, the inequality is obvious. We may thus assume that DP(S, 0", dll) > o. We also assume, for the beginning, that S C M(X). Since DP(S, 0", dp) ~ D(S, 0", Fdp), there follows that DP(S, 0", F'dll) > 0 for any t, 0 < t ~ I. According to Lemma 5.4 there exist Qt E Lq(dp) and P, E LV (dp) such that P,Q, ~ 0, ! P, IP = = DP(S, 0", Ftdll) : Qt :q. P, belongs to the closure of {hF'IP, hE S,O"(h) = I} in LP (dp) and
Ch. S. HP -spaces
89
Moreover,
JhQ,PIP dp =
[DP(S, (/, Ftdp)],IP(/(h),
(h
E
S)
and dl = I Q, Iqdp = : Qt Iq F dm is a representing measure for cp with dl = dm as dl is absolutely continuous with respect to m. For any 0 < s ~ t ~ 1 we have [DP(S,
(5.2.5)
(1,
Pdp)]lIPq(h) =
~ hPfP (~ )"P Q.dp
(h
E
S).
We choose a sequence (h,,), h" E S, with (/(h,,) = 1 and h"PIP converging to P, in LP(dp). If we write (5.2.5) for every h", we obtain at limit (5.2.6)
[DP( S,
(1,
Pdp )]liP
= ~ P, ( ~
fP
Q.dp.
Applying Holder's inequality we get t-s
~P, ( ~ ) " Q.dp ~ = [J)l'(S,
nl ~ r' ]"P I P, jP (
u, P dp)]lIP
r'l I mr r
[~( ~
= [DP(S, u, Pdp)]l/P
J Q.:qdp =
dp
Q, qdp )"P =
~
s
dm
where we used the relation
JIQs lq dp =
1,
and the fact that dl = : Qt Iqdp is equal to m, and DP(S, (/, PdJi) I Qt Iq· Then, from (5.2.6) we obtain (5.2.7)
C( )I-S dm.
DP(S, (/, FS dp) ~ ~ DP(S, (/, FI dp) ) F
Hence, for 0 < s < t
~
I we have
I P, lP =
Function algebras
90
According to Lemma 5.1.2 there results 1
F.5dp) Jt-s lim sup [ DP(S, (1 '---t-+s DP(S, (1, Ftdp) for any 0
~
~
s
t
~
~
J ( -I , dm )
F
1. In this case it is known that
for any 0 ~ s < t < 1. If we write the last inequality for t - 1 and s = 0 there results J (F, dm) ~ DP(S,
Fdp) . DP(S, (1, dp) (1,
We now give up the restriction S C M(X). For this, let V E S with a(V) = 1. Since S C LP(dp) and S C LP(Fdp), we have
II V:PdJl
<
00,
Ii V;PFdp
<
00.
We write
S* = {h
E
M(X): hVE S}
(1*: a*(h) = aChY),
hE
S*.
One easily verifies that [S*, a*] is an admissible couple, I (1*(1) = 1; we also have S* C M(X). Then
DP(S*, a*, 1 ViPdJl)
= inf {I I Vh :Pdm: ~ inf
{I :h ;PdJl,
hE
E
S*
=
M(X), hV E S, a(hV) = I} ~
h E S, a(h) = I} = DP(S, (1, d/l) >
o.
Since FE P(: V'PdJl), we get
DP(S*, (1*, Fi V;PdJl) ~
=
inf {; h;P i F; I V ,PdJl, II
I i VIP Fdp
E
S*, (1*(h)
since 1 E S* and (1*(1) = 1.
= l}
~
Ch. 5. HP-spaces
91
By applying the first part of the proof to the couple [S*, 0'*], the measure I V IPdJl = j V :PG dm and the function F, we obtain
J (F, dm) ~ DP(S*, 0'*, Fi V !PdJl) ~ DP(S*, 0'*, : V!PdJl)
Ji V:P Fdll DP(S,
0',
.
dll)
Since this inequality holds for any V E S with O'(V) = 1, there results (5.2.8)
J (F, dm)
We can easily see that
~
~
DP(S, 0', Fdll) . DP(S, 0', dJl)
and FG have the same propertIes as
F F and G, enounced in the theorem. Therefore, if we apply (5.2.8) to them, we obtain
1 J(E,
d~)
=
(1 d) J
F
t
m
~
DP(S, 0', dll) DP(S, 0', Edll}
that is (5.2.9)
DP(S, 0', FdJl) J(F,dm) ~ - - - - DP(S, 0', dll)
which, together with (5.2.8), gives (5.2.10)
J(F, dm) =
DP(S, 0', Fdll) • DP(S, (/, dll)
By applying (5.2.10) to the function F + e, and then tending to limit according to the theorem of monotone convergence, we get rid of the restriction F ~ e > O. The proposition is proved. (> Theorem 5.8 (Szego). Let m be a representing measure for cp. The following assertions are equivalent. (1) Any representing measure for cp, absolutely continuous with respect to m, is equal to m. (2) For any FE Ll(dm), F ~ 0, we have DP(E dm) = J(E, dm).
Function algebras
92
Proof. (l) -+ (2) obviously results from the preceding proposition. Let us prove the converse implication. Let dJl = Fdm with FE Ll(dm), F ~ 0, be a representing measure for cp, absolutely continuous with respect to m. We then have I
= DP(Fdm) = J(F, dm)
~
I Fdm = Jl(X) =
I
where we used Proposition 5.3. and Lemma 5. I. There results
J( F, dm)
=
I F dm =
I
and, according to Lemma 5. I, we get F = 1, that is Jl
=
m.
Theorem 5.9 (Szego-Kolmogorov-Krein). Let m be a representing measure for cp. The following assertions are equivalent. (1) m is the only representing measure for cp. (2) Let Jl be a positive measure on X and dJl = F dm + dJ1.s its Lebesgue decomposition with respect to m. Then
DP(dp.) = J(F, dm). Proof. The proof follows as an immediate consequence of Theorem 5.8 and Proposition 5.5. 0
5.3. The factorization theorem Let A be a function algebra on X and cp a non-zero linear and multiplicative functional on A. Throughout this chapter we shall denote by m a representing measure for cp such that any representing measure for cp, absolutely continuous with respect to m, is identical with m. As we have seen, this is equivalent to the fact that Szego theorem holds for m. We denote by K(dm) the set of functions hE Ll(dm) with the property
Ifh dm =
cp(f) I h dm
Proposition 5.10. Jensen's inequality
:J hdm I ~ J(i h:, dm) holds for any hE K (dm).
(fEA).
Ch. S. HP.spaces
93
Proof. We have
J(I hI, dm)
= inf {J If II h Idm,
f
=
Dl(1 h : dm)
=
A,
E
= inf {I
E
A,
i,
f
E
A,
= I} =
= I} = !Jhdm :.
A function E E K( dm) will be called an outer function if
JEdm =
J(I E !; dm) > O.
Proposition 5.11. Let FELl (dm) with F ~ 0 and J ( F, dm) > O. Then there exists an outer function E E K( dm) such that (1)/EI=F (2) Any function G I
E
K(dm) which satisfies
G 1 ~ F,
JGdm =
J(F, dm)
is equal to E. (3) Let hE Ll(dm) with hE E LI( dm). Then h only if hE E K(dm). In this case we have
E
K( dm)
if
and
JhEdm = Jhdm JEdm. Proof. Let S be the subspace of Ll(dm) defined by
S = {h
E
LI(dm): h = fF, f
E
A}.
We define on S a functional L by L(fF) = DI( F dm)
Of course L is linear on S. We have
I L(fF) 1= DI(Fdm)!
=
"fFllv(dm).
Therefore L may be extended to a linear functional of norm at most 1, on LI(dm). Thus, there exists Q E L ~(dm), with i Q ~ ~ 1, and DI( Fdm) cp(f) =
Let E = QF. Then E
E
J1FQdm
(I E A).
Ll(dm) and
JEfdm = JQFfdm = Dl(Fdm)
Function algebras
94
that is E
E
K(dm). Moreover,
JEdm = JQFdm = ~
and, since obviously ; E: J(E, dm)
DI(Fdm) = J(F, dm) > 0
F, we obtain
= JEdm
~ J(I E
I, dm)
~ J(F, dm).
Hence
JEdm =
J(l E !, dm) > 0
i.e. E is an outer function. We now write
S ={ hE B(X): hFE K(dm)} u: u(h)
=
JhFdm
(h
E
S).
S is a linear subspace of B(X) and u a linear functional on S. If G E K( dm) with : G! ~ F and
JGdm =
J(F, dm) > 0
then G = hF, h E S, Ih I ~ I, since J(F dm) > 0 yields F everywhere with respect to m. We then have
~
0 almost
= JGdm = J( F, dm) > 0
u (h)
hence u ¢ O. We easily verify that [S, u] is an admissible couple. We have Dl(S, u, Fdm) = inf {J! h I Fdm, hFe K(dm), =
inf
JhFdm =
{J :g I dm, g E K (dm), Jgdm =
I}
I}
=
= 1.
From Proposition 5.7 we get
= Dl(S, u, Fdm)
1 D I( S, (J, dm)
D1( S, u, dm)
Hence we have 0 < Dl(S, u, dm) <
1
~
= J(F dm).
00
,
and
J (F, dm) D1(S, u, dm),
which yields : u(h); ~ J(F, dm)
Ji h I dm
(h
E
S)
Ch. S. Hg·spaces
9S
and therefore
JhFdm ~ J(F, dm) J:h i dm
(h
E
Ll(dm), hF E K(dm)).
Thus there exists Q E L ~ (dm) with I Q!
Jh Fdm =
J
J(F, dm) h Qdm, (h
Let now G E K(dm) with; G I
JGdm =
~
~
Ll (dm), hf E K(dm)).
E
F and
J(F, dm)
> O.
Consider hE B (X) with G = hF. Then
Jh Qdm =
1 and
Ih I ~
1 and
J
[J(F, dm)] h Fdm = 1.
There results hQ = 1 and I h : = I Q I = 1. Therefore, there is only one function G E K(dm) such that I G I ~ F.
JGdm =
J(F, dm),
and which, moreover, verifies 1 G I = F. This function is obviously E. (1) and (2) are proved. We now prove point (3). We first observe that
JhFdm = JEdm Jh -EF dm (h E L(dm); Let h
E
hF E K(dm».
Ll(dm) such that hE E K(dm). Then
~hEdm = ~ ~ Fdm = )Edm~hdm. If
f
E
A with cp(f)
= 0, then fhE E K(dm) and JfhEdm =
o.
Hence 0= JfhEdm
= Jfhdm JEdm,
that is Jfhdm
and therefore h
E
K(dm).
= 0,
-
Function algebras
96
Let now h
E
K (dm) such that hE E LI(dm). We show at first that
Jh Edm = Jh dm JEdm. In order to do this we consider
S = {h
E
B( X): h E K (dm)}
(1: (1(h)
=
Jhdm
(h
E
S).
One easily verifies that [S, (1] is an admissible couple, S C Ll(dm) and DI(S, (1, dm) = 1. Then, from Proposition 5.5 there results J(F, dm)
i Jhdm j ~ J i h i Fdm
(h
E
K (dm)).
Hence, there exists Q E L ~ (dm), I Q I ~ 1, such that
J
J(F, dm) hdm
= JEhQ dm
(h
By writing the last relation for h = we obtain QFE K( dm) and
JQFdm =
E
f
E
K (dm), hF E LI(dm». A, cp(f)
= 0 and for h = 1,
J(F, dm).
Since I QF i ~ F, according to point (2) there results QF = E and, therefore,
JhEdm = Jhdm JEdm
(h
E
K dm, hE E Ll(dm).
In particular, hE E K (dm). This completes the proof of the proposition. 0 . A function FE K(dm) is said to be an inner function if m-almost everywhere.
IF; = L
Theorem 5.12. Let hE K(dm) with Jhdm i= O. H may then be uniquely written under the form h = EF, where E is tin outer function and F an inner function. Moreover
Jhdm = JEdm JFdm. ,
Proof. Jensen inequality (Proposition 5.10) yields
J
J(J h I, dm) ~ I hdm I > O.
Ch. 5. HP-spaces
97
According to Proposition 5.11 there exists an outer function E E K(dm) such that I h I = I E I. The function F defined by II = EF satisfies I F I = 1, FE Ll(dm) and EF E K(dm). Therefore, from the preceding proposition there results that FE K(dm) and fhdm
=
fEdm
f Fdm. <>
5.4. The characterization of the functions in H
P
We recall that we denoted by HP(dm) the LP(dm)-closure of A for 1 ~ p < 00. By HX(dm) we denote the weak LOC(dm)-closure of A. Also K(dm)
=
{h
E
f
Ll(dm): fhdm =
f fdm f hdm, f
E
A}.
Proposition 5.13. Let 1 ~ P, q ~ oc> be such that lip + llq = 1. We have (1) HP(dm) C K (dm) LP(dm), (2) HP(dm) = {h E LP(dm): f hgdm = 0 for any g E K(dm) Lq(dm) with gdm = O}, (3) HP(dm) = {h E LP(dm) K(dm) : f hgdm = f hdm gdm for any g E Lq(dm) K(dm)}. . Proof. Since
n
n
f
n
n
n
J
Jfgdm
= ffdm f gdm
(f, g E A)
ffhdm
= ffdm Jgdm
(fE A, hE HP(dm»,
there results
hence point (1) is proved. Relation (2) immediately follows from the duality LP, Lq and from the K(dm) definition. Let hE HP(dm). Then hE LP(dm) K(dm) according to (I), and from (2) there results
n
f h(g - f gdm) dm = for any g
E
Lq(dm)
0
n K(dm), that is f ghdm = Jhdm f gdm.
Conversely, if h E LP(dm)
n K(dm) and
f hgdm = Jhdm f gdm,
Function algebras
98
for any g E Lq(dm)
n K(dm), we then have S hgdm = 0,
for any g
E
K(dm)
n Lq(dm) with I gdm = O.
From (2) we then get h E HP ( dm).
Theorem 5.14. Let h, g hg E K(dm) and we hare
E
K(dm) be such that hg E Ll(dm). Then
S hgdm =
I hdm S gdm.
Proof. Let T= {h
E
L: L(h)
=
M(X): h E K(dm)
n LCXl(dm)}, (h
S hdm
E
T).
T is clearly a linear subspace of M(X), with 1 E T, and L a linear functional on T, with L(l) = l. Moreover, using Jensen inequality, we have
I L(h) L(g) I = II hdm II S gdm I ~ J(I hi, dm) J(lg I, dm). for h, g
E
K (dm)
nL
X
(dm). But Lemma 5.1 yields
J(lhl,dm)J(lg!,dm) = J(lhgl,dm) ~
S Ihgl dm.
Therefore,
I L(h) L(g) I ~ II hg I dm and, according to Lemma 5.2, there results
L(h) L(g) = S hgdm, that is (5.4.l )
S hg dm
Let now h, g
Sfhg dm = for any f
E
E
K (dm)
=
Shdm Sgdm.
n L CXl(dm). From (5.4.l) there results
Sfhdm S gdm =
A, hence hg E K(dm)
Sfdm I hdm S gdm = Sfdm S hgdm
nL
X
(dm).
Ch. 5. HP-spaces
99
Assume now h, g E K (dm). Without any loss of generality we may suppose I hdm # 0, I gdm # O. Let then h = E}F}, g = E2F2 be the factorizations of h, respectively g, given by Theorem 5.12. Since F}, F2 E K (dm) L (dm), from the first part of the proof we get F} F2 E K(dm) and
n
00
I F}F2dm = I F dm I F dm. 1
2
But E} is an outer function in K(dm) and, therefore, Proposition 5.11 yields F}F2F} E K(dm) and
I F}F2E}dm = I F}F2dm I E dm. 1
At the same time, E2 is also an outer function in K(dm) and since F}F2E} E K(dm), the same Proposition 5.11 yields FIF2EIE2 E K(dm) and
I F}F2E E dm = I F}F2E}dm JE2dm. 1
2
Hence hg = E}E2F}F2 E K(dm) and
I hgdm = J hdm Jgdm. 0 Corollary 5.15. We have
HP(dm) = K(dm)
n LP(dm).
Proof. The proof follows from Proposition 5.13 point (3) and Proposition 5.14. 0 Corollary 5.16. Any inner function belongs to HP(dm), for any
1 ~ p ~ 00. If an outer function belongs to LP(dm) if then belongs also to HP(dm}. Corollary 5.17. Let h E H"(dm) with
I hdm
# O.
Then h = EF where E is an outer function in HP(dm) and F an inner function in HP( dm). Corollary 5.1S. Let
Am
=
{f E A: If dm
=
O}.
Function algebras
100
We then have HP(dm) = {h
E
LP(dm): Jlhdm =
0,1 E Am}.
Corollary 5.19. We hal'e L2(dm) = H2(dm) EB H;" where H;' is the L2(dm)-closure of Am and H;' = {Ii: hE H;,}. Corollary 5.20. Any real function in H2 is a constant function. To end this paragraph we give the following characterization of the outer functions in HP( dm). Theorem 5.21. Let FE HP(dm), 1 ~ P < 00. Then F = cE, where c is a nonzero constant and E an outer function in HP(dm), if and only if the subspace FHP(dm) is dense in HP(dm). If E is an outer function in Boo ( dm), the subspace EBer. (dm) is then weakly dense in HOC (dm). Proof. Let E be an outer function in HP( dm), 1 ~ p ~ 00, and let hE Lq(dm), lip + l/q = 1, such that JfEhdm
0
=
(f E A).
Then Eh E K(dm) and
JEhdm =
O.
From Proposition 5.11 there results that h E K(dm) and
Jhdm =
O.
Therefore, Jlhdm = 0
(IE A),
i.e. h is orthogonal to HP(dm). The subspace EHP(dm) is hence weakly dense in HP(dm) for 1 ~ p ~ 00 and, therefore, norm dense for I ~ p < 00. We obviously reach the same conclusion if we take cE instead of E, where c is a non-zero constant.
Ch. 5. HP-spaces
101
Let now F be a function in HP(dm), I ~ p ~ 00, such that the subspace FHP(dm) is dense in HP(dm). Since 1 belongs to the LP(dm)closure of the subspace FHP(dm), there results c=
JFdm
#- O.
1 Let E = - F. We have c JEdm=l
and, according to Jensen inequality, we get 1
= JEdtn
~
JOEl, dm).
On the other hand, from Szego theorem there results J(lEI, dm) = Dl(lEI dm) =
inf { JIfliEI dm,f E A, q>(f)
= =
1} ~ 1
since 1 belongs to the LP -closure of the subspace {Ef:f E A, q>(f) = I}, hence also to its Ll-closure. Therefore,
JEdm =
I(IE/, dn1)
=
I > 0,
i.e. E is an outer function. 0
5.5. Invariant subspaces Let A be a function algebra on X, s)ll its maximal ideal space, and lP E sm.. m will denote a representing measure (on X)' for q> with the property that any representing measure for lP, absolutely continuous with respect to m, is identical with m. A linear closed subspace 5 of LP(dm) (norm-closed in LP(dm) for 1 ~ p < 00, and weakly closed in L ~(dm) if p = 00) will be called imJariant if for any f E A and 1z E 5 we havefh E S. From the duality LP(dm) - Lq(dm), we obtain that a subspace S C LP (dm) is invariant if and only if it is weakly closed and A 5 C 5. If S is an invariant subspace of LP(dm), we denote by 55 the weak closure, in L(dmt, of the set {hf: h E S, f
E
A,
Jfdm = OJ.
Function algebras
102
Obviously So C S. The invariant subspace S is said to be simply invariant if So :1= S. For a subset S C LP(dm), we write Sl = {g
E
LP(dm): Shg dm = 0,
hE
S}.
If S is a simply invariant subspace of LP(dm), then it is easy to see 1) that S& and Sl are invariant subspace in Lq(dm) and Sl C S6, 2) SJ. :1= SJ·. 3) A function g E Lq(dm) belongs to Sd- if and only if
-rgfh . dm =
°
for any hE Sand f E A with ((J(f) = 0, i.e. if and only if gh E K(dm) for any h E S. There follows at once that [S~]oC SJ., hence, if S is a simply invariant subspace of LP(dm) then S~ is a simply invariant subspace of Lq(dm). Theorem S.22. Let FE LfX(dm) with
IFI
1. The subspace S
=
=
= FHq(dm) of Lf(dm) is simply invariant. Then So
= {Fh, h E HP(dm), Shdm = o}
and
Proof. S is obviously an invariant subspace of LP(dm). The set {Fh:
hE
HP(dm), Shdm = O}
°
is weakly closed and, according to the definition of So, it contains So. Since FE S, there results that for any f E A with ((J(f) = we have Ef E So and then, as F is bounded and A is dense in HP(dm), we obtain Fh E So for any h E HP(dm) with
Shdm =
0.
Hence So
=
{Fh:
hE
HP(dm), S hdm
= O}
and, therefore, So :1= S, i.e. S is simply invariant.
Ch. 5. HP-spaces
103
If g = Fh belongs to So, then
Sg F dm
~
S Fh F dm = Shdm =
°
and FE S&. Since Sci- is invariant, there results that FHq(dm) C S&. Let now g E So. We then see that for any hE S we have gh E K (dm). Let hI E K(dm) with Fg = hI. Since FE L oc (dm) and g E Lq(dm), there results that hI E Lq(dm) and since hI E K(dm), we have hI E Hq(dm), hence g = FhI with hI E Hq(dm), that is S~ = FHq(dm}. This completes the proof. 0 Theorem 5.23. Let S be a simply inrariant subspace of L2(dm). Then there exists a function FE S such that IFI = 1 and S = FH2 ( dm) . The function F is uniquely determined up to a cOllstant factor of modulus 1. Proof. Let FE S be orthogonal on So and
Since f FE So for any f
E
A with qJU')
=
0, there results
(f E A, qJ(j)
=
0).
From Corollary 5.18 there results IFI2 E H2(dm) and, since it is real, according to Corollary 5.20, we have IFI = 1. Therefore, the multiplication by F is an isometry in L2(dm) and, since FE Sand S is invariant, we obtain FH2(dm)C S. Let now hE S orthogonal on FH2(dm). Then, for any f E A we have
(f E A), and, in particular, Fh gonal on So, we get
E
H2(dm). On the other hand, since F is ortho-
Sfh Fdm = 0
(f E A, qJ(f)
=
0)
that is hF E H2(dm). Then, according to Corollary 5.20, Fh is constant and, since
Jh Fdm =
0,
we have Fh = 0. But IFI = 1, hence h =
°
and, therefore, S = FH2(dm).
Function algebras
104
If S = FIH2(dm) = F2H2(dm), then FI = F'j,h2' F2 = Flhl with hi' h2 E H2(dm). There results that FIF2 E H2(dm), FIF2 E H2(dm) and from Corollary 5.20, FIF2 is constant. Therefore FI differs from F2 by a constant factor, whose modulus is obviously I. Theorem 5.24. Let
F be a function in L (dm) with IFI = 1. The <1)
subspace FHP(dm) is simply invariant in LP(dm) for any p, J ~ P < 00. Conversely, if S is a simply invariant subspace of LP ( dm), then there exists a function FE L 00 (dm) , IFI = J, uniquely determined up to a constant factor of modulus 1, such that S = FHP(dm}. Proof. The first part of the theorem fol1ow!\ from Theorem 5.22. Let now S be a simply invariant subspace of LP(dm). We first assume 1 ~ p < 2. Let T = S L2(dm). T is obviously an invariant subspace of L2(dm). Considered as a subspace of S, T is dense in S. Indeed, let h E Sand Fn be the function of L (dm) defined by
n
<1)
F = . n
{I1/1-1
if if
III ~ n IfI > n.
We clearly have J(Fn' dm) > 0 and, according to Proposition S.II, there exists an outer function En in H<1)(dm), with IEnl = Fn. Since the sequence En is bounded in norm, in LI(dm), it contains a subsequence, also denoted by En' convergent almost everywhere in Ll(dm). let E be its limit in Ll(dm); we have
JE dm =
J
lim En dm = lim J(IEnl, dm) = J(lEI, dm).
On the other hand, since clearly IEnl = I, there results lEI = 1. Therefore, E is an outer function, and E = I follows from lEI = I and the uniqueness part of Proposition 5.11. Since S is invariant and En E H<1) (dm), we get Err! E S and since IE,JI ~ n, we have Errl E T. On the other hand, it is clear that the sequence Enf boundedly converges to f and, therefore, it converges to f in LP(dm). We thus proved that T is LP(dm)-dense in S. From this fact and the simple invariance of S, T results simply invariant. According to Theorem 5.22, there exists a function FE T with IFI = 1 and T = FH2(dm). Since T is dense in S, there results S = =
FHP(dm).
Ch. S. HP-spaces
lOS
Let now S be a simply invariant subspace in LP(dm), p > 2, and let q < 2 with 1/p 1/q = l. Consider T = So< We then know that T is a simply invariant subspace in Lq(dm) and To C Sl. C T, Sl ¥= T. According to the first part of the proof and to Theorem 5.22, there is an FE T such that IFI = 1 and T = FHq(dm), and
+
To
=
J
{Fh: hE Hq(dm), hdm = o}.
Let g E Sl, g = FhI with hI E Hq(dm). There follows (h
E
S).
Since F E T = SJ- and h E S, we know that Fh E K(dm). By applying the multiplicativity theorem (Theorem 5.14) we obtain
JFh dm Jh dm = 1
0,
(h
E
S).
If JFhdm=O
(h
E
S)
there results that FE Sl, hence T = FHP(dm) C S1. which is impossible. Therefore,
JhI dm = and we get g = FhI rem 5.22 we obtain
E
0
To. There follows that S1. = To and by TheoS=
TJ- =
FHP(dm).
Suppose now that for S C LP(dm), 1 ~ P ~ 00, there exist FH F2 E S such that IFll = IF21 = 1 and S = F1 HP(dm) = F2H (dm). Then FI = F2h2' F2 =. FIhl with hI, h2 E HP(dm). There results that FI£2 = h2 E HP(dm), F2Fl = hI E HP(dm), hence FIF2, F2£1 E H2(dm). Then, according to Corollary 5.20, Fl and F2 differ by a constant factor of modulus 1. The theorem is proved. 0
Function algebras
106
Theorem 5.25. Let g E LP(dm), 1 ~ p ~ 00 and S be the invariant subspace generated by g in LP( dm). S is a simply invariant subspace if and anly if J(lfl, dm) > o. Proof. If J(lgl, dm) > 0 then, by Theorem 5.11, there exists an outer function E E HP(dm) such that lEI = g. Then clearly g = FE with FE L oc (dm), IFI = 1. Since the space S is equal to the weak closure in LP(dm) of the set {gf,f E A}, from Theorem 5.21 we get S = FHP(dm) and from Theorem 5.24, S results simply invariant. Assume now that S is simply invariant. According to Theorem 5.24 there exists IFI = 1 such that SCFHP(dm). Let g = Fh with It E HP(dm). Since the set {gf,j E A} is weakly dense in Sand FE S, there exists f E A such that
"'
I J(fg - F)F-dml < I. We then have
I Jfdm Jhdm - 11 = I J(fh - I)dml = I J(fg - F) Fdln I < l. There results
Jhdm
i= 0
and by Jensen inequality we get J(lgl, dm) = J(lhl, dm) ~ IJhdml >
o.
0
5.6. The algebra HOC (dm) Let A be a function algebra on X and lfJ a nonzero multiplicative linear functional on A. Let m be a representing measure for lfJ with the property that any representing measure for lfJ, absolutely continuous with respect to m, is identical with m. The algebra LX (dm) is a commutative Banach algebra, with unit element, whose norm satisfies
IIf211
=
IIfll2
If we denote by Y the maximal ideal space of LX (dm), then it is known that L X (dm) is isometrically embedded into C(Y). Since L OC(dm) is obviously symmetric, from the Stone-Weierstrass Theorem we get LX(dm) = C(Y).
Ch. 5. HP-spaces
107
The algebra H-:L (dm) is a subalgebra of L -:L (dm), hence a subalgebra of C( Y), which contains the constant functions and is uniformly closed. According to Theorem 5.11, H-:L(dm) separates the points of Y. X H (dm) is therefore, a function algebra on Y.
Theorem 5.26. For any real function u E L -:L (dm) there exists E E H-:L(dm) such that E-1 E HX(dm) and eU = lEI. Proof. The functions F = eU and F1 = e- u belong to LX (dm) and we obviously have J(F, dm) > 0, J(Fl' dm) > O. According to Theorem 5.11 there exist the outer functions E and E1 in HX(dm) such that lEI = F, IEII = Fl' Since EEl is an outer function and I EEl 1 =- 1, there results EEl = 1 and, therefore, E-1 E HX(dm). 0
Notes The results contained in this chapter have their origin in classical problems of the theory of analytic functions of a complex variable and especially in the theory of Hardy HP-classes. A modern treatment of all these classical results can be found in K. HOFFMAN'S work [2]. H. HELSON and D. LoWDENSLAGER [1], [2] were the first to present a theory of HP-spaces under more general conditions than Hardy's theory. There followed a s~ries of works which proved in more and more general conditions, results similar to classical ones. We mention here some of them: P. R. AHERN, D. SARASON [1], I. GUCKSBERG [3], K. HOFFMAN [1], K. HOFFMAN, H. ROSSI [1], H. KONIG [1], [2], [3], [4], G. LUMER [1], [2], etc. The present chapter follows mainly the synthesis work of H. KONIG [4].
CHAPTER 6
Special classes of fu nction algebras 6.1. Dirichlet and logmodular algebras Let A be a function algebra on X. A is called a Dirichlet algebra on X if the space ReA is (uniformly) dense in CR(X). A is called a logmodular algebra on Xifthe set log lA-II = {u E CR(X): u = log If/,/EA, 1-1 E A} is dense in CR(X). Since for any I E A we have e l E A -1 and Rei = log lell, there results ReA Clog IA -11 and therefore any Dirichlet algebra on X is a logmodular algebra on X. Theorem 6.1. Let A be a logmodular algebra on X, 3'1t its maximal ideal space and qJ E 5JJl. There exists only one representing measure lor qJ with support in X. Proof. Let Ill, 112 be two representing measures for qJ, with support in X. If f E A-I, we have f(qJ) f-l(qJ) =
Since f( qJ )f- I ( qJ)
=
=
Jfdlll'
Jf-
I
d1l 2 ,
I f(qJ) I ~ JIf I dpl If-l(qJ) I ~
JIfl-
I
dp2·
1, there results 1~
JIII dill J1/1- 1 dill
(f E A -I).
But the set log lA-II is dense in CR(X), therefore for any function e U with u E CR(X) we have (6.1.1)
1 ~ JeUdll 1 Je- udJl2.
Function algebras
110
We now fix
UE
CR(X) and, for any real t, consider
pet) is a differentiable function and from (6.11) we get pet) any t. Since p(O) = I, there results p'(O) = O. Then
o=
p'(O) =
Judll
l -
J udll
~
1 for
2
hence
for any UECR(X) and, therefore, III = 11£. 0
Corollary 6.2. For any x E X, Gx is the only representing measure for x. Then, the Choquet boundary of A is equal to its Shi/ov boundary and both are identical with X. Due to the uniqueness of the representing measure for the elements q> of sm., the results obtained in the previous Chapter on HP-spaces hold for any q> E ~)ll and its representing measure m on X. As an application of these results, we shall determine the structure of the Gleason parts of the maximal ideal space of a logmodular algebra A on x. It is worth noting that the proofs hold also for a more general case, when any element q> E ~l admits a unique representing measure with support in X. An element q>1 E ~1l is called bounded in H2(dm) if there exists a constant c such that
(f E A). A functional in clll which is bounded in H2(dm) can be continuously extended to a linear multiplicative functional on H2(dm), Let us denote H~
The subspace
H~
= {h E H2(dm): Jhdm =
OJ.
of H2(dm) is obviously invariant.
III
Ch. 6. Special classes of function algebras
Proposition 6.3. Assume the subspace H;lI simply inl'ariant and let H;l = ZH2(dm) be its writing gil'en by Theorem 5.23. For hE H2(dm)
we write (n=0,1,2, ... ).
For any ({J1 E c11I, bounded in H2(dm), and hE H2(dm) we hare I({J1(Z) I < 1 and 00
({J1(h)
=
~
an[(,Ol(Z)r·
n=O
The measure
is the representing measure of ({J1' Proof. We first show that I({J1(Z) I < 1. Indeed, since in H2(dm), there results
({J1
is bounded
for any natural n and therefore 1({J1(Z) 1 ~ 1. We suppose I({J1(Z) I = 1. Since Z is determined, but a constant factor of modulus 1, we can take (,01(Z) = 1. For a natural n let fn = 00
=
~ bkz k be a function of the standard algebra such that fn{l) = n k=l 00
and ~
00
Ib
k l2
k=l
~ 1 *). Then the function h"
=
~ bkZk belongs to H2(dm) k=l
and we have
which contradicts the fact that
Function algebras
112
0()
The sequence an is bounded, hence
~
an[CPl(Z)jn is a convergent
1/=0
serIes. Let now hE H2(dm). Since h - cp(h) E H! we get h - cp(h) = Zh 1 with hI E H2(dm). Hence Z[h - cp(h)] belongs to H2(dm) and we have
H~,
as Zh1 is the projection of h on on ZH2(dm). Thus, the linear operator T Th
= t(h -
the constants being orthogonal
Ih dm)
(h
E
H2(dm»
defined on H2( dm) with values in H2( dm), is bounded and II Til We have
I Thdm = JZhdm - I hdm I ,idm = I Zhdm =
0 1.
Furthermore T 2h = T (Til)
=
I T hdm) =
Z(Th -
Z (Th - a1),
hence 01
+ Z[Ph].
I hdm + ZTh =
a o + al Z
Th = Therefore h=
+ Z2[T2h].
Let us assume
h=
00
+ al Z + ... + an _ 1Z n - 1 + zn[lnh].
We then clearly have
I Tnh dm = I Znhdm =
an
which implies T n +1h = T (Tnh) = Z(rnh -
I rnhdm) =
Z(rnh -
an),
~
1.
Ch. 6. Special classes of function algebras
113
that is
Then
and, for any natural number n, we have
Therefore we obtain n
~
ak [((Jl(Z)]k = ((Jl(h) - [((Jl(z)]n+l[((Jl(Tn+lh)].
k=l ...
But I((Jl(Z) I < 1 and the sequence ((Jl(Tn+lh) is bounded, since ((Jl IS bounded in H2( dm) and I Til ~ 1; hence 00
lfJl(h)
=
~ an [lfJl(z)]n
(h
E
H2(dm».
n=O
Let 1 - I ((Jl(Z)1 2 _ dm. IZ - lfJl(Z)1 2 Since
+ lfJl(Z)Z
00
~ n=O
8-c.437
_. _____ _
zn[lfJl(z)]n
Function algebras
114
there results ~
Sfdml
=
~ [CPI(z)]n S i'1dm
+
,,=0
~
~ [cPI(z)]n+l
S Z"+lfdm =
nU
x'
L
a,,[CPI(Z)]"
=
f(lPI)
11=0
for any f E A, i.e. m 1 is a representing measure for The proposition is completely proved. 0
CPl'
Theorem 6.4. Let L\ be the Gleason part of J11L which contains lP. lVe have L\ i= {lP} if and only if the subspace H;;, is simp~v inl'ariant. If L\ =1= {lP}, there exists an injective map r of the unit disk D = = {z: Izl < I} of the complex plane, into 011l, such that r is continuous and: (a) r(D) = L\ (b) For any f E A, F(z) = f[r(z)] is an analytic function in D. Proof. Assume L\ =1= {lP} and let lPl E L\, lPl ~I= lP and m 1 be its representing measure. According to Theorem 3.13, there exists c, o < c < 1 such that cm 1 ~ m, em ~ mI' Hence, the identity map of A into A generates a topological isomorphism between H2(dm) and H2(dm 1 ) and lP and lPl are bounded in H2(dml) respectively H':(dm). Let S be the H2(dm1)-closure of the set
Am = {f E A : S fdm = O}. Since the H2(dm)-closure of Am is identic~l with H,~" the identity map of Am onto Am generates a topological isomorphism between and S. S is obviously an invariant subspace in H2(dl1l l ). We prove it is simply invariant. Indeed, if So = S, then SInce So is the closure of {fll: hE S,f E A, lP(f) = O} there results
H:n
Shdm1 = In particular
0
(h
E
S).
Ch. 6. Special classes of function algebras
115
hence CfJI = cpo But CPI -:f= cP, therefore S is simply invariant in H2(dml)' Let FE S with IFI = 1 m1-almost everywhere and S = FH2(dml)' Then, clearly, IFI = I m-almost everywhere and H;,,= FH2(dm). Then from Theorem 5.22. H,; is simply invariant. Consider now H;/ simply invariant and let H;, = ZH2(dm) be its form given by Theorem 5.23. If CfJI E~, according to Theorem 3.12, CfJ1 is bounded in H2(dm) and from Proposition 6.3 we get ICfJ1(Z) I < 1. We define on A a mapping 8, by
'''e have e(.1.)CD. If e(CP1) = e(CP2) for CP1' there results ~ an [CfJ1(z)]n =
=
from Proposition 6.3
IX;
'l:;
CfJI(!)
(j)2 E: ~,
n=O
~ 0n[CfJ2(Z)]" =
cplf)
n=O
for any I E A, and since A separates S'lL we obtain CfJI = CfJ2' Therefore, 8 is an injection of ~ into D. Let us prove that 8 maps A onto D. Let I, g E A and
We have seen in the proof of Proposition 6.3 that
f --
00
- b0 g-
+ 01 Z ---L
---L I On zn TI
+ b 1Z
T'b nZ" -lI
I'"
---L i'"
1 zn+l III ~
h2
zn +
1
where hI = Tn+1/, h2 = Tn+1g and T is the operator defined on H2(dm) with values in H2(dm). given by Til
Hence hI' h2 Let
E
=
Z(h -
Jhdm)
(h
E
H2(dm)).
H2(dm) and are both bounded. (n = 0, 1, 2, ... ).
Function algebras
116
Using the above expressions off and g we find by a simple calculation that
etc.
Let now AE D. For f
E
A we define 00
CPt(f) = ~ anAn , n=O
where
Since an is a bounded sequence, CP1 is well defined on A, linear and, according to the above considerations, also multiplicative. As CPI(l) = 1 we have CPI :F O. Therefore CPI E :v1L LetfEA with
Then lanl <
E
for any
11,
which implies
Hence CP1 is bounded in H2(dm). The representing measure of CPt is, according to Proposition 6.3,
Therefore m and m l are mutually absolutely continuous, with bounded derivatives and, from Theorem 3.12, CPI EA. Since, obviously, O( CPl) = A, 0 is an injection of A onto D, hence A does not reduce to only one point.
Ch. 6. SpeciaJ classes of function algebras
117
Let now 't = 0- 1 be the inverse map of lJ. 't is a one-to-one correspondence between D and A and, for any f E A, we have 00
F(l) = f['t(l)] =
~ allAn, n=O
where all is the bounded sequence
.
Therefore F is an analytic function in D and 't is coniinuous. The theorem is thus proved. 0 To conclude this paragraph we give a variant to F. and M. Riesz theorem in which the Lebesgue decomposition with respect to a representing measure is replaced with a decomposition with respect to all representing measures; the sense of this decomposition will be specified further. , We first prove the following ;completion to Theorem 3.12. Proposition 6.5. Let ({Jl, ({J2
such that ({Jl and CP2 do not belong 10 the same Gleason part of allL Then, the representing measures /1It /12 of CPIt respectively CP2 are mutually singular. Proof. Let dJL2 = hdpl + dps be the Lebesgue decomposition of E j)[
P2 with respect to Ill> Since PI' P2 are positive measures of norm 1 ~ there results 0 ~ h ~ 1. We then have
2 = 11({J1 - ({J211 ~ IIPI - JL211 =
11(1 - h)JLlll
+ IIPsll ~ J (1
=
11(1 - h)JLI
- h) dpi
+ 1=
+ Psil = 2-
I hdpl'
Hence
and, since h is positive, we obtain h = 0 PI-almost everywhere, which implies P2 = Ps' 0 Theorem 6.6. Let A be a logmodular algebra on X and J1 E A ~ .
There exist: an at most countable set {cp,} of distinct elements in ollL any two elements belonging to two distinct Gleason parts, the func-
Function algebras
JI8
tions h; E Ht where A; are the representing measures, for lfJ;, and the measure (J E A.L, singular with respect to all representing measures for the elements of .)lll, such that 00
P =
~
+
h),;
(J,
;= 1
the series being convergent in norm. Proof. If P and A are two' measures on X, we denote by Pi. the absolutely continuous part of P relative to )., and by P;, the singular part of If with respect to l. Let ({J1."" ({In be a finite system of elements in j)1L, each two elements belonging to two distinct Gleason parts and )'1" .. , An their representing measures. According to Proposition 6.5, A; and )'j are mutually singular for i ¥= j. Let P E A.L and
P= P -
"
~ Pi-;' ;=
1
We have p
=
p;'j -
~ Pi.;'
(j
= 1, 2, ... , n)
;t::.j
hence p is singular with respect to every).;. Since P
" = p+ ~
Pi.;
;=1
there results
IIpll = Ilpll
+
t1
~
lip;)!
i= 1
which implies n
~ i=1
Ilpi.;1I
~
IlplI.
Ch. 6. Special classes of function algebras
119
This inequality holds for any system At, ... , An with the above properties, therefore there exists at most a countable set of Gleason parts which contain an clement lfJ with Pi, #- 0, A being the representing measure of lfJ. Let {~j} be this set. We fix cp; E ~j and let ;'j be the representing measure of lfJj. Since if;
~
Illli,; I
~
111l1I
i~l
J:;
'E
the senes i
=
Ji;.i
is convergent in norm. Let
1 if;
a
=
P-
~
p),;'
i = 1
Let A be the representing measure of an element cp E . .mL If there is a j such that lfJ E ~, then A and Aj are mutually absolutely continuous and, for i -=F j, A and Ai are mutually singular. There results (Ii).j);' = Ji). and (p;)). = 0 for i -=F j, hence a i. = Iii, - P). = O. If q> E ~i for any i, then, from the properties of the set {~i}' we get Ii;. = 0 and, according to Proposition 6.5, A and A; are mutually singular for any i, therefore (P;.);. = 0 which implies a). = O. Thus a is singular with respect to any representing measure of the elements of J)1L Following Theorem 5.6 we have Ii;'j E A.L for any i, hence a E Al.. Let h; E Ll(dA;) with dp).j = hidA;. Since Pi.j E Al., there results (fE A)
and, from Corollary 5.18, we get hi E Ht. This completes the proof of the theorem. (, The importance of this theore.m consists in the following: there are function algebras A with the property that the only measure orthogonal to A and singular with respect to any multiplicative measure on A, is the null measure; at the same time, there are orthogonal measures to A which are absolutely continuous with respect to no representing measure. In this sense give an example in § 6.4.
Function algebras
120
6.2. Algebras generated by inner functions Let A be a function algebra on X, its Shilov boundary. Let SI
lsI
= {s E A, s
#- 0: II Is II
m its maximal ideal space and r
=
11/11 for any I
We obviously have IIsll = 1 for any s E SI' Proposition 6.7. The function s of A belongs to =- 1 on the Shilov boundary, i.e.
SI = {s E A: Is(x) I = 1, X
Proof. If Is(x)1 = 1 for any x
II sIll
=
E
r, then
E
IIsll
E
A}.
if and only if
SI
r}. =
1 and
sup Is(x)1 (x) I = sup Is(x) I If(x) I = sup II/(x)1I xEr
xEr
=
11/11
xEr
for any I E A, that is s E SI' Conversely, assume there exist s E SI and Xo E r such that Is(xo)1 < 1. Let e > 0 and U be the neighbourhood of Xo on which
Is(x) I < 1 -
8
(x
E
U).
According to Theorem 2.16 there exists If I < e off U. We then have
Is(x)f(x) I = Is(x) I II(x) I <
I
E
1- e
A with 11/11 = 1 and
(x
E
U)
and
Is(x)f (x)1
=
Is(x)11 f(x) I <
8,
that is IIsfll < 1, which contradicts the relation Ilsfll = 11/11 = 1. 0 Corollary 6.8. If r = dJIt then any element of SI is invertible. Corollary 6.9. If s E SI and is invertible then lsi = 1 on ~.
Ch. 6. Special classes of function algebras
Proof. Since IlslI any ep
E ~
lis-III = IIss- 1 11
=
1,
we get
lis-III =
121
1. Then for
we have 1
Is(ep)1 =
Is-l(ep)1
-~
I
and, since IlslI = 1, there results Is(cp) I = 1. Let now S be a multiplicatively closed system of functions in Sl> Assume 1 E S. The C(r)-closure A s of the set {g E C(r): g = sf, s E S, f E A} is a function algebra on Indeed, since S is multiplicatively closed and lsi = 1 on r for any s E S, there results, on r,
r.
hence As is a uniformly closed subalgebra of C(r). Obviously A CA., therefore A s separates the points of r and contains the constants. As SEAs, any s E S is invertible in As. We call As the algebra of quotients of A with numerators in S. Let ~ s be the maximal ideal space of A sand r sits Shilov boundary . The embeddings defined in paragraph 3.1 allow us to write r s e r e ems em; indeed, we easily verify that A, as subalgebra in A s' separates the points of 81Jts. Proposition 6.10. We have
and
rs = r. Proof. Since any s E S is invertible in A. and lsi = 1 on
r
then,
according to Proposition 6.7 and Corollary 6.9, for any cp E ~., Icp(s)1 = I, i.e. ms C {ep: lep(s)1 = I}. Let cP E @Jt with Icp(s) I = 1 for any s E S. We define the functional CPl on As, by
(s E S,
f
E
A)
and extending continuously to A s' which is possible as lepl( Sf)' ~ II stll.
122
Function algebras
If Sl!l = SJ2 on f, then S2 !I = SI!2 on f, hence also on ,J)lL. Therefore lfJ(S2)CP(!I) = lfJ(sl)cp(/2) and, since IlfJ(s)1 = 1 for any s E S, --there results lfJ(sl)cp(/l) = lfJ(s2)lfJ(/2) which shows that lfJI is a well defined functional. We also have
i.e. lfJl is linear. At the same time lfJl is obviously multiplicative and <{>1(1) = I, therefore lfJl E cill s • On the other hand for any f E A we have lfJI(f) = lfJ(f) and, since A separates the points of ~11I, we get q> = lfJl E ~lIl s' But f s is the Shilov boundary of As, therefore it is determined for A and, since f s C f, there results r s = r. The proof is complete. <)
Proposition 6.11. If S separates the points of f, then As = C(r). Therefore we hal'e
Proof. Since S is multiplicatively closed and separates the points of f, the set of all elements of the form g = s1: CiSi, Ci constants and S, Si E S, is a subalgebra of C(r) which separates the points of f. This algebra is symmetric, as g = a1:ciO'; with a = SI"'Sn E S, O'i = ass; E S. Then, according to the Stone-Weierstrass theorem, there results As = C(f). 0
Corollary 6.12. If S separates r then for any g E CRCf), g ~ 0 and G > 0, there exists f E A such that Ig - f I I < G. Proof. As As = C(f), there exist S E Sand f E A with Ig - sfl < G. Then Ig - III I = Ig - I sf/ / ~ Ig - sfl < G. 0 Proposition 6.13. If S separates the points of f then for any closed subset F of f, with AIF = C(F), and any Ii E A.l. we hal'e /iF = O.
Ch. 6. Special classes of function algebras
123
Proof. Let tl E Al., Iltlll = l. According to Corollary 4.3. there exists k < I such that IltlF11 ~ k Iltlll. If we denote by F' the complement of F then, clearly, there is a c such that IltlFIi ~ c IItlpll. Let 8 > 0 be sufficiently small and a compact KeF' such that 1IJ1.~' -KII < 8. Let g E C(r), g ~ 0, with g = 8 on K. g = I - 8 on F and g ~ I - e on r. Following Corollary 6.12 there exists f E A such that g - 8 < < I fl < g + e. Hence, we have Ilfll ~ 1, If I ~ 1 - 28 on F and If I < 28 on K. Since fJ1 E Al., we obtain
But e is arbitrary small, therefore
IIJ1.FII =
0, i.e. IlF
=
o.
0
Corollary 6.14. If S separates the points of rand F is a closed subset of r for which AIF = C(F), then F is an intersection of peak sets. Proof. The proof follows from Proposition 6.13 and Theorem
4.12.0 Corollary 6.15. If S separates the points of r then any point of r is an intersection of peak sets. Therefore the Choquet boundary of A is identical with r. Now, let A be a function algebra on X, J111 its maximal ideal space, r its Shilov boundary and l: its Choquet boundary. A function SEA will be called (X)-inner if lsi = 1 on X. Let S be a closed multiplicative system of inner function. We assume 1 E S. The algebra A is called S-generated if the closed subalgebra generated by S in A is identical with A. Since S is multiplicative closed, A is S-generated if and only if the closed linear subspace generated by S in A is identical with A. As S separates the points of X, from Proposition 6.11- and Corollary 6.15, there results Proposition 6.16. Let A be an S-generated algebra on X, s)1l its maximal ideal space, r its Shilol' boundary and l: its Choquet boundary. We have ~ =
r
=
X = {cp
E 511l:
IqJ(s)1 = 1, s E S.}.
Function algebras
124
We now give an example of an S-generated algebra. Let G be an abelian group and X the dual of the discrete group G. Let S be a subsemigroup of G such that G = SS-l and S S-1 = {I}. If we consider G as the character group of X then G C C(X). Let p. be the normalized Haar measure on X. We write
n
A(S) = {f E C(X):
Jfgdp. =
0, g
E
G, g ¢ S}.
One easily verifies that A (S) is a uniformly closed algebra. As S C A(S) and G = SS-I, S separates the points of X, which is also true for A(S). Thus, A(S) is a function algebra on X. S is, clearly, a system of inner functions in A(G). We now show that A( S) is S generated. Let f E A( S), f '# O. It is known that for any 8 > 0 there exists u E C(X) such that Ilf*u - fll < 8, where f*u denotes the convolution of f with u. Since the trigonometric polynomials are uniformly dense in C(X), there exists a trigonometric polynomial v = I:.Cjgi such that IIu - vII < 8. Since f*v
= ~Ci(J fgidp.)gi = ~ c"g" gkES
and II!*v - III < 28, the linear subspace generated by Sin A(S), is dense in A(S). Since S S--1 = {I} and
n
JsdJI =
0
(s E S, s ¥= 1),
the measure JI is multiplicative on A(S), hence it represents a point in the maximal ideal space of A( S).
(fJo
Theorem 6.17. The following assertions are equivalent: (a) G ~ SU S-1, (b) A(S) is a Dirichlet algebra on X, (c) A( S) is a logmodular algebra on X. (d) for any cp E ~ there exists a unique representing measure for
with support in x, (e) p. is the only representing measure for lPo (with support in X), (f) if JIl is a representing measure for ({Jo. absolutely continuous with respect 10 p., then JIl = p.
({J
Ch. 6. Special classes of function algebras
125
Proof. If G = S U S-l then A( S) + A( S) contains all trigonometric polynomials, hence it is dense in C(X). Thus, A(S) is clearly a Dirichlet algebra on X and therefore (a) -+ (b). The implications (b) -+ (c) -+ (d) -+ (e) -+ (f) are already known; (f) -+ (a) follows immediately from Corollary 5.19. Consider again an S-generated algebra A on X. The system S will be called analytically free if for any multiplicative map (X from S into the complex plane, with III(S) I = 1, s E S, for any linear combination rCiS; of elements in S, we have
_
A
Let G the abelian group G = SS and G the dual of the discrete group G. Theorem 6.18. Let A be an S-generated algebra X with S analytically A
free. The space X is homeomorphic to the dual G of the discrete group G and A is isometrically isomorphic to A(S). Proof. For x E X let (X x be the character on G, defined by (Xx(S l S2) = Sl(X) S2(X). A
Obviously (Xx E G. If (XXI = (Xx 2, then s(xl )=S(X2 ) for any S E Sand, since S separates the points of X, there results Xl = X 2 • Therefore the A
map
X -+
(Xx
is an injection of X into G. A
A
Let us now prove that this map applies X onto G. If (X E G we put
But (X is a character of the group G, therefore (X is a multiplicative map onto Sand I(X(s) I = 1 for any s E S. Since S is analytic free, there results
hence i.e. ({Ja
({J(I can be E ~. Since
extended to a multiplicative linear functional on A, for any S E S we have
Function algebras
126
from Proposition 6.15 we get cp(l = x
E
X. But, obviously (Xx
=
X,
~
(Xx
A
therefore the map x
~
(Lx
applies X on G. A
According to the defined topologies on 011l and G, the map x A
is continuous, hence it is an isomorphism between X and G. This homeomophism maps isometrically isomorphic the algebra A A
in the algebra of functions on G, generated by the characters of S which, as we have seen, is exactly A(S). . This completes the proof of the theorem. 0 In the following we give a sufficient condition for a system S of inner function to be analytic free. . For a finite system r - [51,52 , ... , sn] of inn~r fum;tions, we denoie by 7r t the map defined on X with values in the n-dimensional torus n
Tn =
II {Izl =
I}, by
1
Proposition 6.19. Let So be a system of inner functions such that for any finite system T of distinct elements of SO' the map of X into Tn is surjective. Then the multiplicatil'e closed system S generated by So is analytic free: Proof. Let C( be a multiplicative map of S in Tl and SI,'''' Sn a finite system of elements in S. Let t = [O'b" .,an ] be a finite system of distinct elements of So such that i i i
- vn IP1 Vn P22 ... Si-
n v
P
n"
(i = 1,2, ... , n).
Since the map 7r t is surjective, there exists x E X such that O'i(X) = = C't(aj) , i = J, 2~ ... , n. Hence, for any finite system of constants c l , c2 , ... , Cm we have
i i i
=
ILci;t(a:l)(X(a~2) ... :x (a:") I =
and, therefore, S is analytic free. ()
Ch. 6. Special classes of function algebras
127
11
Corollary 6.20. Let X=
II {z: Izl =
I}, I ~ n ~
00,
and A the
1
algebra generated by the coordinate functions ZI' Z2' .•. , Zn' Then the multiplicatil'e closed system generated by ZI, Z2,.", zn is analytic free.
6.3. Maximal algebras Let A be a function algebra on X. A is said maximal (in C(X» if for any function algebra B on X, A C B C C( X), we have either A = B or B = C(X). The algebra A is called pervasive on X if for any closed set Fe X, F ¥= X, the algebra AIF is dense in C(F). We call A analytic on X if any function in A which vanishes on a nonvoid open subset of X is identically zero. We recall here that A is said to be an antisymmetric algebra on X if any f E A with f E A is constant on X. Finally, A is called essential on X if the only closed ideal of C(X) in A is the ideal {OJ.
Theorem 6.21. Let A be a function algebra on X, A i= C( X). We have a) If A is perl'asil'e then it is analytic. b) If A is analytic then it is an integrity domain. c) If A is an integrity domain then it is antisymmetric. d) If A is antisymmetric then it is essential. e) If A is essential and maximal then it is pervasive. Proof. a) Let f E A, which vanishes on the non-void open subset U of X and let K = X - U. If g E C(X), since A is pervasive, there exists a sequence gn E A such that gn converges uniformly to g on K. Then, clearly, fgn converges uniformly to fg on all X. Therefore A contains the (closed) ideal generated by f in C(X). Let F = {x E X:f(x) = OJ. Suppose F i= X. It is known that the ideal generated by f in C(X) is equal to the set of all functions in C(X) which vanish on F. Let g E C(X) and fn a sequence of functions in A which converges uniformly to g on F. Using eventually a subsequence, we can find a sequence I1n of functions in C(X), converging uniformly on X to the function h and verifying hn = gn on F. Then gn - hn vanishes on F, hence it belongs to the ideal generated by f in C(X), which is included in A. There results gn - hn E A and, as gn E A, we get hn E A and therefore hE A. But h = g on F, hence h - g E A, that is g E A.
Function algebras
]28
Since g was arbitrarily chosen in C(X) , there results A = C(X). This contradicts the hypothesis, therefore F = X and I is identically zero. b) We assume Ig = 0, I E A, g E A and let K = {x E X:/(x) = OJ. If U = X - K is non-void then g = 0 on U and since A is analytic, there follows g = 0 on X. c) Let B = {I E A:} E A}. We define on X the equivalence relay if I(x) = l(y) for any IE B. tion x Let Y be the compact space obtained by the factorisation of X by this equivalence relation. ,.." " Consider B = {I E C(Y):/(x) = I(x) , I E B}. One can easily verify that B" is a symmetric function algebra on Y. From the Stone-Weiersrass ,.. theorem there follows B = C( Y). If Y does not reduce to one point I'-.;
"""
",..
",..
then there are 11,/2 E B such that 11 ::1= 0, 12 ::1= 0 and 11/2 = O. Then, clearly, 11 ::1= 0, 12 ::1= 0 and IJ2 = 0, where 11,/2 are those functions A " in B which satisfy 11(X) = 11(X), 12(.i) = 12(X). But this contradicts the fact that A is an integrity domain. Therefore V reduces to one point, hence B reduces to constant functions. d) Let us assume that A contains a closed ideall of C(X). These ideals, as it is known, are of the form I = {IE C(X):I = 0 on F}
with F a closed subset of X. If I does not reduce to 0, then F ¥= X. Hence there exists a real continuous function, which is not identically zero but vanishes on F. Yet this function belongs to l, therefore to A, which contradicts the antisymmetry of A. e) Let Fe X, F::I= X, F closed. Since A is essential there exists g E C(X) with g = 0 on F and g ¢ A. Since A is maximal, the closed algebra generated by A and g is identical with C(X). Consider hE C(X) of the form
wherel/ nl EA. Then, clearly, we have
and therefore A is pervasive.
Ch. 6. Special classes of function algebras
129
The theorem is proved. <) Theorem 6.22. Let A be a pervasil'e algebra on X, cl1t its maximal ideal space, r its Shilov boundary and r its Choquet boundary. For any cP E &lR there exists a unique representing measure with support in X. Therefore r = r = X. If cP E j)[ - X then the representing measure of cp has its support identical with X. Proof. Two representing measure Jl and Jl' for the same cP E ~ satisfy Ji.F = Ji.~ for any closed subset Fe X, F i= X. As Ji.{X) = = p'(X) = 1, there result p = p'. Now let cP E S'R - X, !1 its representing measure and F its support. If F:f. X, since PF = Ji., It results multiplicative on C(F); hence there exists x E F such that
(fE A)
lP{/) = JfdJL = f(x)
which is impossible as A separates jJlt and cP
E
j)1t - X. 0
Theorem 6.23. Let A be pervasive on X and lP EA. If there exists CPo E ~)R - X such that
I f(cpo) I = 11111, then f is constant on M. Proof. Let f(lPo) = eiOtll/1l and g =
e- iOt f,
g = u
+ iv.
Since
there results u(CPo) = 11/11. Since IIgll = 11/11 there follows u(cp) ~ u(lPo) for any cp E j)1Z. Let U = {x E X: u(x) < u(lPo)} and P be the representing measure of CPo. We then have
which is impossible. We used the fact that X is the support of JL. Therefore u is constant and, since A pervasive is also antisymmetric, g is constant, hence f is constant. <> 9 -
c. 437
Function algebras
130
6.4. Function algebras on compact sets of the complex plane Let X be a compact set of the complex plane. We denote by P(X) the subalgebra of C(X) obtained by closing in C(X) the set of polynomials on X. By R(X) we denote the algebra of all functions in C(X) which are uniform limits on X of rational functions with poles outside X; A(X) will denote the algebra of all functions in C(X), analytic in any interior point of X. One easily verifies that P(X), R(X), A(X) are function algebras on X and P(X) C R(X) C A(X).
Proposition 6.24. The maximal ideal space of P(X) is homeomorphic to the polynomially convex hull X of x. Proof. It is known that A
A
X
=
{z
E
c: IP(z)1
~ sup IP(x) I for any polynomial Pl. xeX A
A
Then X is a determining set for P(X) and, since P(X) = P(X) I X, " and P(X) are isomorphic, hence from Theorem 2.17 it follows that P(X) their maximal ideal spaces are homeomorphic. Let 8nL be the maximal A
ideal space of P(X), which, by the natural homeomorphism, is equal A
to the maximal ideal space of P(X). We know that X is homeomorphically embedded in 8nL by means of the mapping x -+ e:lt' where A
eif) = f(x)
(fE P(X». A
We now show that the map x -+ ex applies X on rn.. Let cp E ~ and let z be the identity map of the complex plane, which obviously A
belongs to P(X). Let Xo = cp(z}. Since for any polynomial P we have lfJ(P) IP(xo) I
= P(cp(z»
= IP(cp(z» I = IlfJ(P) I ~ xeX S:IP IP(x)l·
there results
Ch. 6. Special classes of function algebras
131
/\
Therefore x 0
E
X and
" (fE P(X» /\
that is exo = q>. Then x -+ ex applies X on m. The proof is complete. 0 We notice that, according to the maximum modulus principle X" is the union of X with the bounded connected components of the complement of X. Let us suppose that X belongs to the boundary of the unbounded connected component of its complement. In this case we shall prove that P(X) is a Dirichlet algebra on X. We first prove two helping lemmas (see CARLESON, [2]). Let dA be the Lebesgue measure in the plane. Lemma 6.25. Let X be a compact set of the complex plane and n the unbounded connected component of the complement of X. Let P be a real measure on X. Then u(z)
= ( log
J
x
1
Iz- xl
dp(x)
converges absolutely almost everywhere with respect to the Lebesgue measure dA. If u(z) = 0 in n, then u(z) = 0 in any point ZEn in which we have absolute convergence. Proof. Since the function log _1_ is locally integrable with respect
Izl
to dA and the measure d
Ipi is finite, there results
(
{ ( log
;zi
x
J
J
1
Iz-xi
dill/(X) } dA(Z) <
00
and therefore ( log
J
x
1
Iz-xl
dllll(x) <
00,
A-almost everywhere. Hence, the absolute convergence A-almost everywhere of u(z) follows.
Function algebras
132
Now assume u(z) = 0 in n. Let Zo E on (the topological boundary of n) in which u(z) converges absolutely. To simplify the notation, we assume Zo = 0, which does not affect the generality of the proof. Let b > 0 and 0 < '1 < '2 ~ b. As 0 E an we can find a positive measure (J with support in n such that
and (J(K) = 0 for any compact K situated outside the circle Izi Since u(z) = 0 in G we have
~
b.
~
p
-1 Ju(z) d(J(z) = o. b
We fix a p > O. Then (
J
{~( log bJ
1 d(J(z) } dJl(x) Iz-xi
+
[xi
+
({~ ( log J ~J
1 d(J(z) } dJl(x) = ~ Ju(z) d(J(z) = O. Iz- xl b
Ix!
By the construction of the measure d(J, there results that for Ixl we have lim 6-+0
~ ( log ~
)
1 d(J(z) = log _1_ . Iz - xl Ixl
If Ixl < p we have: -1 ~
~ log -1- d(J(z) Iz - xl
6
=
_1 (log _ _1_ dr = log _1_ ~ J r - Ixl Ixi o
~
-1 b
~ log
1
d(J(z)
=
IIzl - Ixll
6
+ (log _ _1_ _ dr ~ log _I + C. J 0
i1_
I
2
Izi
Ixl
Ch. 6. Special classes of function algebras
133
where T
C
=
~ log
sup -1 T>O T
o
1
11 -
dt <
tl
00.
Hence (
J
( log I - - -} dC1(Z) dJL(x) ~ '(log - 1 + C )dlpl(x) I; {-8I J Iz - xl ) Ixl I 'I'
!xl>p
and therefore, since b
~
0, we obtain
~
, ( log _1_ dJL(x):
:J
jx ~p
(
Ixl
I
(log _1_
(
J
Ixi
:x,~p
+
c)
dipi (x).
Since 0 is assumed to be an absolute convergence point for u(z) the inequality's right handside tends to 0 when p ~ 0, hence u(O) = lim ( P-'O
J
log _1_ dJL(x) = O.
Ixl
:x:~p
This completes the proof of Lemma 6.25. 0
Lemma 6.26. Let X, JL and u be as in the preceding lemma. If u(z) = 0 A-almost everywhere in all the plane, then Jl = O. Proof. For any function g of class C2 and compact support we know that (see STOILOW, S., [1]): g(x)
= - _1_ ( Ag(z) log
J
2n
.
1
Iz-xl
dA,(z).
By Fubini theorem we get
Jg(x)dJl(x) = _I (Ag(z) { ( log
x
2n
J
J
x
1 21t
= - -
1
Iz - xl
Sg(z)u(z)d).(z) =
dJL(x) } dA(z)
O.
=
Function algebras
134
Since any function in CR(X) is a uniform limit on X of functions with compact support and of class C2, there results Jl = O. 0 Theorem 6.27. Let X be a compact set in the plane, contained in the boundary of the unbounded connected component n of its complement. The algebra P( X) is a Dirichlet algebra on x. Proof. Let Jl be a real measure on X, orthogonal on Re P( X). The logarithmic potential u(z)
= ( log .
J
x
1
Iz-xl
dp(x)
is known as an harmonic function outside X (see BRELOT, M., [I]). For any z with the property Izl > sup lxi, the function xeX
f(x)
1 log--
=
belongs to P(X), since the series
z-x
00
~ n=l
(_
n
l)n
(x)n -
converges uniformly
z
I - = Ref(x), we get u(z) = 0 for any z with on X. As log ___
Iz-xl
Izl > sup Ixl. But since u(z) is harmonic in nand {z: Izi > sup Ix/}cn xeX
xeX
there follows u(z) = 0 for any ZEn. From Lemma 6.25 there results u(z) = 0 in any point of n in which u(z) converges absolutely. Let D be a bounded connected component of the complement of X and a ED. Let Y = DUX. Since D is bounded and aD C X, Y is a compact set in the complex plane. We easily see that n is the unbounded connected component of the co mplement of Y. Let A( Y) be the algebra of continu ous functions on Y, which are analytic in any interior point of Y. According to the maximum modulus theorem X is a determining set for A( Y). Hence there exists a positive measure Aa on Y such that f(a) =
Let
Va
=
ea
-
A.a· Va
Jfd}'a
x
(f E A(Y».
is a real measure on Y.
Ch. 6. Special classes of function algebras
135
I If ZEn, then the function f(y) = log - - obviously belongs
z-y
to A ( Y), therefore
uiz) =
1 ~ log Iz-YI
dviy)
1
= log
Iz-al
x If
Zo E
an,
- ~ log
x
1
d).a(x)
Iz-xi
=
o.
since the potential is lower semicontinuous, there follows
Iuizo) I ~ ~ log
1
Izo - yl
; dlyai ~ ilog
I
~l og I dlL'(x) Z-+Zo Iz - xl Zen x
. + hm
I
.
I
Iz - al I
+
1 :s:;; 2 ijlog - -
Izo - al
!
.
hence u(zo) converges absolutely in any point Zo of an. From Lemma 6.25 there results uiz) = 0 on an. Thus, for any z E X we have log
1
Iz-al
=
~ log
1
Iz-xl
dlix)
and then ( { ( log
J J
x
x
= ( { ( log J J
x =
x
log ( log
J
x
1
Iz-xi
dllll(x) } dA.a(z) =
1
dA.a(z) } dllll(x)
1
dllll(x)
Iz-xl
Iz-al
<
=
00.
Therefore the set of points in X in which u(z) is divergent has the measure zero with respect to A.a.
Function algebras
136
Since u(z) = 0 in any point z E X in which u(z) converges absolutely, we obtain
o = ~ u(Z)dA'(Z) = ~ {~IOg Iz ~ xl x
= ({ ( log
J J
x I
Iz - xl
dll(X) }dA.(Z) =
x
dAa
J
I
Iz - al
dJL(x)
=
u(a).
x x Hence u vanishes in any point belonging to the bounded connected components of the complement of X. Since u(z) = 0 A-almost everywhere on 0, u(z) = 0 A-almost everywhere on the entire plane and therefore, using lemma 6.26, Jl = O. 0 Proposition 6.28. The maximal ideal space of R( X) is homeomorphic to x. Proof. Let ~ be the maximal ideal space of R(X). We know that X is homeomorphically embedded in &m, by means of the map x ~ ex where ex is defined by
exCf) =f(x)
(fE R(X».
We shall prove that this map applied X on jl1L Let z be the identity map of the complex plane. Obviously z E R(X). Let q> E 51R and IX = q>(z). If ex ¢ X then (z - ex)-l E R(X), hence I = q>(l)
=
q>[(z - a) (z - a)-I] = (q>(z) - ex)q>(z - ex)-I = 0
which is impossible. Therefore a
e«(f)
=
f(ex)
X. It is then clear that
E =
f(q>(z) = q>(f)
for any rational function of R(X) , hence for any function of R(X). There results q> = ea and therefore the map x ~ ex is surjective. Lemma 6.29. Let Jl be a measure on X. Let
N(y)
dllli . J Ix-YI
= (
x
Ch. 6. Special classes of function algebras
137
Then N(y) is finite almost everywhere with respect to 1. If
F(y)
= (
dJl(x)
Jx-y
x
vanishes almost everywhere with respect to 1. then Jl = o. If Jl E R( X) 1. , then U = {y E C: N(y) < 00, F(Y)::J= O} C X and for any y E U, the measure Jl y defined by dJl y = f(y)
=
1 (z -y) -1 dJl(z), verifies the relation F(y)
Jfdpy
(f E R(X».
Proof. The first part of the lemma follows from the fact that
1
is an integrable function with respect to 1 on any compact in
Izl
plane, and IJlI is a finite measure on X. Assume now F(y) = 0 A-almost everywhere. Let g be a continuous function with compact support and indefinitely differentiable. It is then known that g(y) = _ _ I C(y _ 2n
J,
W)-l(~+ i~)g(W)dA(W). (}X
(}y
By integrating relative to dJl and applying Fubini's theorem we get
~ g(y)dp(y) = x
-
:
~ H(Y -
W)-l
~ ( o~ + i
:y)
g(w)dA(w) } x
x
= -
_I F(y) ( (~+ i ~)g(W)dA(W) = O. 2n J (}x (}y
Since any continuous function on X may be uniformly approximated on X with continuous functions with compact support and indefinitely differentiable, Jl = O.
Function algebras
138
Suppose now Jl
=
E
R(X)l. and let y
E
U.
If y; X then f(x) =
(x - y)-l E R(X), hence
F(y) =
J(x -
x
y)-ldJl = 0
which is impossible. Therefore U C X. Let y E U. Since F(y) =F 0 and (z- y)-l is integrable with respect to Jl on X, then Jl y defined by 1 F(y)
= - - (z - y)-ldJl
dp Y
is a measure on X. Since for any rational function of R(X), the function
hex) = f(x) - fey) x-y is also a rational function in R(X), there results
fey) - JfdJly x =
=
1
fey) - - - Jf(x) (x - y)-ldJl(x) F(y)
(( f(x) - fey) dll(X) ] F(y) J x - y
____ I
=
=
0
for any rational function in R(X) and, by continuity, we get
fey)
=
Jfdpy,
The lemma is completely proved. 0 Let r be the Choquet boundary of R(X). Since X is metrisable, any point of 1: is a peak point for R(X).
Proposition 6.30. Let x E X - r and let A. be the Gleason part of X which contains x. Then leA) > O. Proof. According to Theorem 2.23 there exists Il ~ ex such that Jl({x}} = o. Let v = p - ex. Then clearly v E R(X)l., v:l= O. Let F
Ch. 6. Special classes of function algebras
139
and U be as in Lemma 6.29. From Lemma 6.29, there results U C X and that F does not vanish everywhere with respect to l. Then l( U) = l{y E C: F(y) '1= O} > O.
We now show that UCA. Let y E U. According to Lemma 6.29, the measure 1
dv y =
Vy
defined by
(z - y)-ldv
F(y)
satisfies the relation
f(y) = Jfdvy
(f E R(X».
On the other hand, from Theorem 3.15 there follows a representing measure for y, absolutely continuous with respect to Vy , hence absolutely continuous with respect to v. This measure must obviously be of the form hdp, + C8y with hE Ll(dJL). Then JL y = (h + c)JL is a representing measure for y, absolutely continuous with respect to p,. Then, from Proposition 6.5 we get YEA. Hence UC A and therefore l(A) ~ l (U) >0. 0 Corollary 6.31. There exists at most a countable set of Gleason parts of X (with respect to R(X)), which do not reduce to one point. Proof. Since the points of ~ form point Gleason parts (Corollary 3.14), the proof follows immediately. 0 Corollary 6.32. A point x E X forms a point Gleason part of X with respect to R(X) if and only if it is a peak point for R(X). Hence the set P of all points x E X which form point Gleason parts is identical to 1:. Theorem 6.33. Let X be a compact set of the complex plane. 1: its Choquet boundary with respect to R( X) and P the set of all the points of X which form point Gleason parts of X (with respect to R( X)). The following assertions are equivalent. (a) R(X) = C(X) (b) X = P (c) X = 1:.
Function algebras
140
Proof. We already know that (a) ~ (c) and that (b) and (c) are equivalent. It remains to prove (c) ~ (a). Let I' E R(X)~. If I' =F 0 then, according to Lemma 6.29, there exists y E X such that F(y) =F 0 with F constructed as in the mentioned lemma, and the measure Jl. y defined by
satisfies the relation fey) =
Jfdp,y
(f E R(X».
Now, according to Theorem 3.15, there exists a representing measure for )" absolutely continuous with respect to /ly, hence absolutely continuous with respect to 1'. Since )' E 1:, By is the only representing measure for )" hence Gy is absolutely continuous with respect to p, that is p( {y}) =F O. Since y is a peak point for R(X), there exists f E R(X) such that fey) = I and If I < 1 on X - {y}. It is then clear that for sufficiently large n
which contradicts the fact that Jl E R(X).l. Then p = 0, i.e. R(X) = C(X). The theorem is proved. <) Let now X be a compact set of the complex plane, contained in the boundary of the unbounded connected component of its complement. In this case, according to Theorem 6.27, P(X) is a Dirichlet algebra on X. As P(X) C R(X), R(X) is also a Dirichlet algebra on X and, following Corollary 6.2, the Choquet boundary of R(X) is identical to X. Then, from Theorem 6.33 there results: Corollary 6.34. If X is contained in the boundary of the umbounded connected component of its complement, then R(X) = C(X). Proposition 6.35. Let X be a compact set of the complex plane, contained in the boundary of the unbounded connected component of
Ch. 6. Special classes of function algebras
141
its complement. If Jl E P( X) 1. and it is singular with respect to any positive measure on X and multiplicative on P( X) then J.I. = Proof. We have seen that the maximal ideal space of P(X) is the
o.
A
A
polynomially convex hull X of X. Hence, if z ¢ X, then the functionf(x}= = _1_
x-z
e P(X), that is
=0
{_I-dJl J(x-z)ft
A
(z¢X, n=I,2, ... ).
A
Let now Z E .X, Z ¢ X and pz be the representing measure of z with support in X. We put dv
=
z-x
- cdJ.l.z
where c = C~J.I.(x) .
J x-z
For n
~
1 we have
since Jl E P(X)l. and Jlz is a representing measure for z. As obviously v{l) = 0, v E P(X)l.. Let dv = dVa dv s be the Lebesgue decomposition of v with respect to J.l.z. Since P(X) is a Dirichlet algebra on X, J.I.= is the only representing measure for z with support in X, hence according to the M. and F. Riesz Theorem (Theorem 5.6), v E P(X)l.. But J.I. is singular with respect to Jlz, therefore dv s = (x - Z)-l dJ.l.. We then have
+
o.
( dJ.l.(x) =
Jx-z Now, if we repeat the procedure, taking (x - Z)-l dJl instead of j.l, we obtain (x - z)-2dJ.l. e P(X)T, and so on. Hence ( dJ.l.(x) = 0
J(x-z)ft
A
(zeX- X, n = 1,2".,).
E
P(X)
Function algebras
142
Then J.l results orthogonal on any rational function with poles outside X and, from Corollary 4.33, we get J.l = O. The proof is therefore complete. 0 We are now able to give an example of a Dirichlet algebra A with the following properties: there exist measures J1 E A.l which are absolutely continuous with respect to no representing measure of A and ~ any measure of A.L, singular with respect to any representing measure of A, is O. LetX= {z: Izl = I}
U {z: Iz - 21 = I}
and A = P(X).
Since. clearlY. . X beloOfzs - to the boundary of the unbounded connected component of its complement, according to Theorem 6.27, P(X) is a Dirichlet algebra on X and following Proposition 6.35, any measure of P(X).L, singular with respect to all representing measures for A, is O. On the other hand, the complex measure dz is obviously orthogonal on P(X) and, since any representing measure for P(X) has its support either in {z: Izl = I} or in {z: Iz - 21 = I}, dz is absolutely continuous with respect to none of these representing measures. ~
'
Theorem 6.36. (Mergelyan). Let X be a compact set of the complex plane with a connected complement. Then any continuous function on X, analytic in any interior point of X, is a uniform limit, on X, of polynomials, i.e. P(X) = A(X). Proof. Let Y = ax (the topological boundary of X). Since X has a connected complement, Y belongs to the unbounded connected component of its complement. According to the maximum modulus theorem, Y is a determining set for P(X), hence P(X) is isometrically isomorphic to P(X) 1 Y = P(Y). At the same time Y is also a determining set for A(X), hence A(X) and A(X)I Yare isometrically isomorphic. Now, let J.l be a measure on Y, orthogonal on P(Y). Since P(Y) is a Dirichlet algebra on Y (Theorem 6.27) from Theorem 6.6 there results 00
dJ.l
=
~ hidAi
+U
; =1
where Ai are representing measures for P( Y), hiE HI i and singular with respect to any representing measure of P( Y). From Theorem 6.35 there follows (j = O.
(j
E
P( y).l is
Ch. 6. Special classes of fDnction algebras
143
Since X has a connected complement, X is polyomially convex, hence (Proposition 6.24) the maximal ideal space of P(X) is homeomorphic to X and also homeomorphic to the maximal ideal space of P(Y). Assume Ai represents the point Xi E X. Since Y is determining for A(X) too, there exists a representing measure A; for Xi (with respect to A(X», with support in Y. But P(X) C A(X), therefore A~ is a representing measure for Xi with respect to P(X), hence also with respect to P(Y); as P(Y) is a Dirichlet algebra on y there results A~ = Ai Thus, 1i is a representing measure for Xi with respect to A(X), hence also with respect to A (X) I Y. Since P(X) C A(X) I Y, A(X)I Y is a Dirichlet algebra on Y. It is clear that HI, relative to P( Y) is included in Ht re1ative to A(X) I Y~ hence hi E Ht relative to A(X) I Y. Then for any f E A(X) I Y, fh i E Ht (relative to A(X) I Y), and therefore (f E A(X)I Y). Hence 00
Jfdp = ~ JfhidAj = 0
(fE A(X)IY)
;=1
that is J.l E (A(X) I y).l. There results (A(X) I Y) = P(Y), hence A(X) = P(X). The proof is therefore complete. 0 Corollary 6.37. If X has a connected complement then the maximal ideal space of A(X) is homeomorphic to X.
6.5. The standard algebra and HOO algebra Let D = {z E C: /zl < I} and T = aD = {z E C: /zl = I}. From the previous paragraph we know that A(D) = P(D) is isometrically isomorphic to A(D)IT = P(T). We denote this algebra by A and consider it is a function algebra on T. As we have already mentioned, A will be called the standard algebra. We have seen that the maximal ideal space of A is homeomorphic to D. At the same time A is a Dirichlet algebra on T, hence T is identical to Choquet and Shilov boundaries of A.
144
Function algebras
The multiplicative closed system of inner functions S = {I, z, Z2, ... } is analytic free and generates A, hence A is identical to A(S). Therefore, a function f E C(T) belongs to A if and only if
(n = 1,2, ... ) where Jl is the normalized Lebesgue measure on T (which, as is already known, is also the Haar measure of T considered as the dual of the discrete group of the additive group of integers). Proposition 6.38. A is a maximal subalgebra of C (T) . Proof. Let B be a function algebra on T such that A C Band Jl is the normalized Lebesgue measure on T. If Jl is multiplicative on B, then for any fEB we have (n
= 1,2, ... )
hence f EA. There results B = A. Let A. be a representing measure for B. If
Jzd). =
0,
then clearly A. is a representing measure of the point 0 (with respect to A) and, as A is Dirichlet, A. = Jl. Thus, if Jl is not multiplicative on B, there results
J zdA. #= 0 for any representing measure of B, hence z does not vanish, on the maximal ideal space of B. Therefore z = Z-l belongs to B and from the Stone-Weierstrass theorem we obtain B = C(T). The proposition is proved. 0 Since the algebra A is anti symmetric, it is also essential and from maximality, there results it is pervasive (Theorem 6.21). Therefore A has all the restrictive properties required until now in the frame of this theory to function algebras, hence all the established results hold for A too.
Ch. 6. Special classes of function algebras
145
If Jl is the normalized Lebesgue measure on X, we denote by HP the space HP(dJl). According to Fatou theorem, HP is the set of all functions in LP which may be analytically. extended in D. We shall complete the theory of HP-space given in chapter 5 by the following results valid in the case of the standard algebra. Proposition 6.39. Let S be an invariant subspace of LP. S is simply invariant if and only if z S '# S. The proof is immediate. If an invariant subspace S is not simply invariant, then z S = S, hence 'is C S. Proposition 6.40. The algebra H~ ;s a weakly maximal subalgebra of the algebra L 00 • Proof. Let B be a weakly closed subalgebra of the algebra Lex: such that H C B. Then B is obviously an invariant subspace of L 00. If B is not simply invariant then zB C B, hence Z E B. But B is a weakly closed subalgebra, then B = L 00 • If B is simply invariant then, according to the theorem of invariant subspaces, there exists a function FEB, IFI = 1, such that B = FH oo • Since FEB, there exists hE H oo such that £2 = Fh and therefore F = he Hoc, that is B = H oo . In § 5.6 we have seen that H oo may be considered as a function algebra on the maximal ideal space of L 00. From Theorem 5.24, HCIJ is even a logmodular algebra on this space. In the following, X will denote the maximal ideal space of L 00 • We know that it is a totally nonconnected compact space. From this remark and from the following proposition it results that H'J:. is not a Dirichlet algebra on X. Proposition 6.41. If A ;s a Dirichlet algebra on X and X is totally nonconnected, then A = C(X). 1 Proof. Let F be a closed subset of X andfeA such that 11- Refl < --2 on F and IRefl <
~ on
X-F. Such anf does exist in A as XFECR(X) 2 and A is Dirichlet on X. Let Y1 = f(K) and Y 2 = f(X - K). Y1 and Y 2 are compact sets 1 1 of the complex plane and we have IRezl < - for z E Y1 and /Rez/ > -
2
10 - c. 437
2
146
Function algebras
for Z E Y2' Then, there exist two disjoint rectangles RI and R2 such that Y1 C RI , Y2 C R2. Therefore, R ~ RI U R2 is a compact set of the plane, with a connected complement, hence there exists a sequence of polynomials Pn which converges uniformly on R to XR ,. Then, clearly, 1 pi!) converges uniformly on X to Xp, hence Xp EA. Therefore A contains the function Xp for any closed subset F of X, hence it is clear that A = C(X). The proof is complete. 0 So HrI) is a logmodular algebra on X which is not a Dirichlet algebra. Following Proposition 6.16, HrI) can not be generated by a free analytic system of inner functions. It remains an open problem if H aJ is generated by the set of its inner functions or if these functions at least separate the points of X. Notes Dirichlet algebras were introduced by J. WERMER [2] and logmodular algebras by K. HOFFMAN [1]. Theorem 6.1 was proved by K. HOFFMAN [1] while proof of Theorem 6.4 is due to J. WERMER [2] for Dirichlet algebras and to K. HOFFMAN [1] for logmodular algebras. Theorem 6.6 was given by I. GUCKSBERG, J. WERMER [1] but its most general form belongs to I. GLICKSBERG [3]. The results of paragraph 6.2 are mainly due to I. SUCIU [1], [2], [3], [5]. Proposition 6.13 belongs to I. GLICKSBERG [1] and Theorem 6.17 to N. MOCHIZUKI [1]. In connection with these results we also mention E. A. GORIN [2]. Theorem 6.20 appears in a first form in H. HELSON, F. QUIGLEY [1]. The form under which it is presented here can be found in K. HOFFMAN, I. M. SINGER [1]. Theorems 6.20 and 6.21 are proved by H. S. BEAR [I]. Concerning the maximality property see E. A. GoRIN [1]. The use of potential theory in approximation problems on compact sets of the complex plane is quite old (cf. M. BRELOT [I]). The treatment of paragraph 6.23 follows E. BISHOP [3], L. CARLESON [2] and I. GLICKSBERG, J. WERMER [1]. Proposition 6.29 and Theorem 6.39 belong to D. R. WILKEN [1]. The proof of Proposition 6.34 belongs to I. GUCKSBERG, J. WERMER [1] as well as the proof of MERGELYAN theorem [1] (Theorem 6.35). Proposition 6.37 was given by J. WERMER [l], Proposition 6.39 by H. HELSON [1] and Proposition 6.40 by K. HOFFMAN [3]. In connection with these problems we also mention L. CARLESON [1], D. GA~ PAR [1], S. MERRILL [1], N. MOCHIZUKI [2], [3].
CHAPTER 7
Operator representations of function algebras 7.1. Positive definite maps on C(X). Spectral and semi spectral measures Let X be a compact Hausdorff space and H a Hilbert space. We denote by L(H) the algebra of all bounded linear operators on H. The linear map qJ -+ Ttp of C(X) in L(H) is called positive definite if for any qJ E C(X), qJ ~ 0, we have Ttp ~ o. One easily verifies that if qJ -+ Ttp is positive definite then it is symmetric (Tep = T:, where qJ is the complex conjugate of qJ) and continuous (II Ttpll ~ K IlqJll). For h, k E H we put (7.1. 1)
JIh,k (qJ)
= (Ttph, k)
(qJ E
C(X)}.
Thus, we obtain a family (JIh, k}h, k EH of measures on X, with the following properties:
(7.1.2)
JIh , k ~ K
IIhll IIkll·
Function algebras
148
A family (Ph, k)h, k E H of measures on X is called semispectrai if it satisfies relations (7.1.2). We shall write Ph = Ph,h' It is easy to see that (7.1.3)
4Ph,k = Ph+k - /ih-Ic
+ iPh+ik -
iph-ik
(h, k
E
H)
(h, k
E
H).
and (7.1.4)
Theorem 7.1. Relation (7.1.1) gives a one-to-one correspondence between the set of all positive definite maps cp -+ TqJ of C( X) in L(H) and the set of all semispectralJamilies (Ph,k)h,kEH on X". Proof. If qJ -+ TqJ is a positive definite map then, as we have already seen, (7.1.1) defines a semi spectral family on X with values in H. If (Ph,k)h,kEH is a semispectral family on X, then using (7.1.1) we may construct the operator TqJ for any qJ E C(X); by a simple calculation we obtain that qJ -+ T", is a positive definite map of C(X) in L(H). 0 Let us denote by R(X) the set of Borel parts of X. A map u -+ F(u) of R(X) in L(H) is called a semispectrai measure (on X with values in L(H» if for any h E H the map (J -+ (F(u)h, h) is a positive Borel measure on X. A semi spectral measure on X with values in L(H) will be denoted by (F(u»/T E BeX). Theorem 7.2. There exists a one-to-one correspondence between the
set of all senlispectrai families (Jih,k)h,k EH and the set of all semispectral measures (F( u)) /T E B(XI' given by Ph(a)
(7.1.5)
=
(F(a)h, h).
Proof. If (Ph ' k)h ,k EH is a semispectral family on X then, it is clear . that, If we put (F(u)h, k)
=
Ph,lu)
(u
E
R(X); h, k
E
H)
we can construct the semispectral measure (F(u»/T E B(X) which satisfies (7.1.5). If (F(u»/T EB(X) is a semispectral measure on X, then for any hE H, Ph defined by (7.1.5) is a Borel measure on X. (7.1.4) is verified by a simple calculus. Now, defining Ph,k for any h, k E H by (7.1.3), one easily obtains that (Ph, k)h, Ie EH is a semi spectral family on X.
Ch. 7. Operator representations of function
aJgebra~
149
The 3-tuple cp -+ TIP - a positive definite map of C(X) in L(H), (PIt"J", k EH -a semispectral family, (F(O'»/1 EB(X) -the semispectral measure, is therefore well defined in the sense of Theorems 7.1 and 7.2, by one of its terms. The semi spectral measure (E(O'»/1 EB(X) is called spectral if for any 0'1,0'2 E B(X) we have E(0'1 0'2) = E(al) E(0'2)'
n
Theorem 7.3. Let cp -+ T" be" a positil'e definite map and ( F( 0') ) E B(X) its semispectra/ measure. The measure (F( 0')) EB(X) is spectral if and only if cp -+ T" is a multiplicative map. Proof. Let cp -+ TIP be a positive definite map of C(X) in L(H), (J1",k)",k EH be the semispectral family attached to cp-+ TIP' and (F(O'»/1 EB(X) its semispectral measure. Suppose that cp -+ TIP is multiplicative. For any f, g E C(X) and h, k E H we have /1
/1
JfgdJ1".k
= (Tfgh, k) = (TfTg h, k) = JfdpTgh, k
JfdJlTgh,k = (TfTg h, k) = (TgTf h, k) = (Tfh, Tgk) = Jfdph. Tgk'
We therefore have the following measure equalities
(g E C(X); h, k
(7.1.6)
Using (7.1.6). for any 0' E B(X), g
(7.1. 7)
JgdJ1F(a)",k =
E
C(X) and h, k
(TgF(O')h, k) = (F(O')/z, Tgk)
=
E
E
H).
H, we have
Function algebras
150
We have the measure equality
(0' E B(X); h, k
(7.1.8) If we use (7.1.8) for any (/,
(F(O'
(jJ
E
E
H)
B(X), we get
n w)h, k) = I'll, k(O' n w) = JlaIC) dph,k = =
(F(O')F(w)h, k)
that is
F(O'
n w) = F(O')F(ro)
(0',
(jJ
E
B(X»
and therefore (F(O'»a EB(X) is a spectral measure. Conversely, assume (F(O'»a EB(X) a spectral measure on X. As for any 0' E B(X), F(O') is a projection (selfadjoint) of L(H), then for any (/, WE B(X) we have
JXC)dJlF(a)h ,k = (F(w.)F(u)h, k) = (F(w n u)h, k) = =
JXwdPF(a)h,k =
JXwX a dph, k
(F(ru) F(u)h, k) = (F(w.)h, F(u)k) =
We therefore obtain the following measure equalities
(7.1.9)
(0' E B(X); h, kEH).
Ch. 7. Operator representations of function algebras
According to (7.1.9), for any U
E
= (F(u)k, Tgh) =
J
B(X) and g
E
C(X) we have
= (Tgh, F(u)k)
gdJlh,F(a)k
=
lSI
=
J
gdJlF(a)h,k'
Hence there results (7.1.10)
From (7.1.10) we obtain (TfTgh, k)
= JfdJlTgh,k = JfgdJlh,k = (Tfgh, k)
which proves that qJ --. TqJ is multiplicative. 0 In general, we denote by qJ --. UqJ the multiplicative maps and by (£(o)a EB(X) the spectral measures. A semispectral family attached to a multiplicative map (or to a spectral measure) will be called a spectral family.
The notion of positive definite map introduced above is justified by the following Proposition 7.4. Let qJ --. TqJ be a positil'e definite map of C(X) in L(H). For any finite system gh ... ,gn offunctions in C(X) and any finite system hI,"" hn of elements in H, we have (7.1.11)
~ i,i
(Tg;gjhj, hi) ~ O.
Proof. Let (Ph , k)h , k EH be the semispectral family attached to the map qJ --. TqJ . For the finite system hI, ... ,hn of elements in H we define the positive measure Jl on X by
Function algebras
152
Following (7.1.3) and (7.1.4), for any i, j, the measure Ph, .hJ is absolutely continuous with respect to J1. Hence dJ1hi ,hJ = lijdp
with lij E LI(dJ1). For any cp E C(X) and any finite system AI' ... A.n of complex numbers, we have
Jcp
~ A.;A.jl;jdp
=
i,j
=
~
i,j
I
~ A.;A.j cpdphj ,hi
=
i,j
A.jA.;{T.hj, hi)
=
(TIP ~ A.}h} , ~ AihJ. j
Therefore, for any cp
~
;
0
I cp
~ A.;Aj/ijdJ1 ~ O. i,j
Thus, for any countable family IF of finite systems A.h ... , A.n of complex numbers, we find a Borel set OJ such that J1(X - co) = 0 and ~ A;A.j/ij(x) ~ 0 i, j
for x E OJ and A.1 , ••• , A.j in IF. If, as IF, we take the set of all systems A.h .•. , An for which A;, 1 ~ i ~ n, has rational real and imaginary parts and we consider the corresponding OJ then, for any x E OJ the numerical matrix (fij(x) is positive definite. Let now gh ... ' gn be a system of functions in C(X). Since for any system AI,. .. , An of complex numbers and any x E X we have ~ i,j
A;A.jg;gj =
I ~ Ajgj 12
~
0
j
then the matrix (gig/X» is positive definite for any x EX. Therefore we clearly have (7.1.12)
~ g,(x)gix)fij(x) ~ i,j
0
(x E OJ).
Ch. 7. Operator representations of function algebras
153
Using (7.1.12) we get ~ (Tg;g/tj,h j )
= ~ JKigjdlLltj,lti= ~ JKigjfijdjl i,j
i,}
=
i, j
= J ~ gigjfijdp.
=
i,;
J.
~ Kigjfijdp. ~
W
i,j
o.
The proposition is proved. 0 Theorem 7.5. (The dilatation theqrem of M. A. Naimark) Let cp --+ Ttp be a positive definite map of C(X) in L( H). There exists a Hilbert space K, a bounded linear operator V: H --+ K and a multiplicative linear map cp --+ Utp of C(X) inL(K), which satisfy VI = I - the identity operator of K, I Vtpll ~ IlcplI, cp E C(X) and Ttp =
(7.1.13)
v*utp V
(cp
E
C(X)).
Proof. Consider the tensor product C(X)®H, with elements of the form u = Ifi ®hi' where fi E C(X), hi E H, and the sum having j
a finite number of non-zero terms. We define the sesquilinear form (7.1.14)
if u = ~ fi ® hi' v = ~ Since cp results
gi
(8) k i•
i
j
--+
Ttp is positive definite, from Proposition 7.4 there (u, u)
=
~ (TJ./ Jhj,hi) ~ 0 i,j
hence (u, v) is a positive definite form. Let N = {u E C(X)(8)H: (u, u) = OJ. The form (u, v) induces an inner product on the quotient space C(X)®HjN. K will be the Hilbert space obtained by completing C(X)®HjN with respect to this inner product. We now define the operator V on H, with values in K, by Vh = I ® h
+ N.
Function algebras
154
It is clear that V is linear and
Hence V is bounded. Let us define the map and
lp E
UqJ. We first define it for u = 'f-gj ® hj
lp -+
;
C(X). U~(~ gj ® hj) = ~ lpgj ® hj. j
j
It is immediate that (U~u, u)
=
lp -+ U~
is linear and multiplicative. Moreover
(~ lpgj ® hi' ~ gi ® hi)
=
i
j
~ (TgiqJgJhj, hi) = i, j
:E (T~gigJ hj, hi) = (:E gj ® hj, ~ lp ~j
gi
® hi) =
i
j
(u, U~u).
=
Hence (U~u,
(7.1.15)
u)
=
(u,
ugu)
(lp E
C(X».
(qJ E
C(X».
For a fixed u we define, on C(X), the functional p(lp)
= (U~u,
u)
p is clearly a linear functional on C(X). Since for any qJ
using (7.1.5), we have p(q;lp)
=
(U~~u, u)
=
IIpll
~
p is a positive functional and
(U~u, U~u) ~
0
p(I) = (u, u). Then
E
C(X),
Ch. 7. Operator representations of function algebras
155
Now, if we put U,,(u
+ N) =
U~u
+N
and extended continuously on all K, then we clearly obtain a multiplicative linear map qJ -+ Uf/' of C(X) in L(K) , which satisfies U I = I,
II U"II
~
Ilcpll,
qJ E
C(X).
Moreover, for any h, k
E
H, cp
E
C(X), we have
(V*U"Vh, k) = (U"Vh, Vk) = (U"l ® h, I ® k)
=
--::-- (qJ ® h, 1 ® k) - (T"h, k).
Hence T "
V* UV lP
=
(cp
E
C (X».
The theorem is proved. <..) Let us note that if Tl = IH - the identity operator on H, then V* V = I K - the identity operator on K, that is V is an isometry. At the same time P = VV* is the orthogonal projection of K on VH. Then, identifying H with VH, we may consider H as a subspace of K, hence T"h = PUlPh
(qJ E
C(X); h
E
H).
To complete the dilation theorem of M. A. Naimark we now give the following uniqueness theorem. Theorem 7.6. Let qJ -+ T" be a positive definite map of C(X) in L( H) and K;, Vi' cp -+ U~, i = 1,2, be as in Theorem 7.5. Moreover, we assume that the elements U~ Vi h, qJ E C (X), h E H, span the space K i • Then there exists a unitary operator U: K2 -+ Kl such that
(cp Proof. We define, on K2 , the operator U by U( ~ U;.V 2h;) i
=
~ Ui i Vlh i . i
E
C(X».
Function algebras
156
Then IIU(~ U;jV2hj112 = (U(~ Ui J V2h), U(~ UiiV2.hi» = j
j
=
(~ U:JV1hj , ~ U:iV1h i) j
;
=
~-(Vl*UgigJVlhj' h) = ~j
i
~ (Tgjgfii,h) =
~ (V2*UjigjV2h i ,hj ) = ...
i, j
i, j
=
= II ~ UiJV2 hjll2. j
Now it is clear that U l11ay be defined on all K2 and that it is a unitary operator. It follows immediately from the definition of U that it satisfies the conditions required by the theorem.
7.2. Representations of function algebras Let X be a compact Hausdorff space, A a- function algebra on X, H a complex Hilbert space and L(H) the algebra of all bounded linear operators on H. An algebra homomorphism j -+ T f of A in L(H), which satisfies (7.2.1) and (7.2.2) is called a representation of A on H . .!ropositioD 7.7. Let j -+ T f be a representation of A on H. If f andf E A then Tj = Tl. Proof. It is sufficient to prove that if f is a real function in A I then T, = 1f*. Indeed, if j, f E A, then the real functions u = - (J +f), 2 -i v = - (f - f) belong to A and, since f = u + iv, we have 2
Ch. 7. Operator representations of function algebras
But, as T"
=
T:, T
=
157
T:, there follows
Tj * -- T*U
-
,'T* - T l'
-
,,-
,'Tt' -- Tf·
Thus, let f be a real function in A. Let 1 be a real number, 1 > IIfll. Then the functions (f + il)-1 belong to A. Write gl = (f + it) (J - il)-I; g2 = (J - it) (J
+ it)-I.
We have gl, g2 E A, glg2 = 1, Igil = Ig21 = 1. Hence Ti"/ exists and Tg~l= Tg1. As I TglH ~ Hgl11 = 1, II Tit 1 11 = II Tgtll ~ Ilg211 =-1, T..~1 results a unitary operator. But (7.2.3) and therefore we have 4Reit(Tf h, h) =
!I(Tf
=
II(Tf - it/)hI1 2 -!l(Tf
it/)hI1 2
-
-
II Tg1(Tf
-
+ it/)hIl
it/)hI1 2
2
=
=0
for any h E H. Here we have a well-known relation in the geometry of Hilbert spaces, relation (7.2.3) and the fact that Tg1 is unitary. Then (Tfh, h) is real for any h E H, hence 1j* = T f . 0 Proposition 7.8. Let f ~ T f be a representation of A on H. If Ref ~ 0 then ReTf ~ O. Proof. If Ref ~ 0, then (J 1)-1 E A. Let G = (f - 1)(f + 1)-1; we have g E A, !lglI ~ 1, hence II Tgli ~ 1. Since
+
for any h E H we get 4Re(Tf h, h) =
II(Tf
=
!I(Tf
+ /)hI1
+ /)hI1 2 -
2 -
II TiTf
II Tf
- /)hI!2 =
+ /)hli2
~ O.
Function algebras
158
Therefore
Corollary 7.9. For any f
E
A we have
IIReT,1I
~
IIRefll.
Corollary 7.10. A representation cp -+ UqJ of C(X) on H is a pusitive definite multiplicative map of C (X) in L ( H).
7.3. Representations of the algebra C(X) We have seen that any representation of the algebra C(X) on a Hilbert space H is a positive definite multiplicative map cp -+ UqJ of C(X) in L(H). Hence there exists a spectral measure (E(U»aEB(X) on X with values in L(H), such that
(Utph, k) =
Jcp(x)d(E(x)h, k)
x
(h, k
E
H; cP
E
C(X».
In this paragraph we shall describe completely the form of the representations of the algebra C(X) on a separable Hilbert space H. Let cp -+ UqJ be a representation of C(X) on a Hilbert space H. A closed subspace M of H will be called cyclic (for the representation cp -+ Utp) if there exists hEM such that the closure Mh of the set {UqJh: cp E C(X)} be equal to M. For any hE H, Mh is obviously a cyclic subspace, called a cyclic subspace generated by h. Two non-zero elements h I,"2 E H are said to be cyclic free if Mhl -.l Mhz. A non-void system S of non-zero elements of H is called cyclic free if it reduces to one element or, if any two elements of S are cyclic free. One easily verifies that the set of cyclic free systems of H is inductively ordered, hence any element of H belongs to a maximal cyclic free system.
Ch. 7. Operator representations of function algebras
159
Proposition 7.11. Let S be a maximal cyclic free system of H. Then
Proof. According to the definition of a cyclic free system, Mhl~Mha for any hh h2 E S, hi '=1= h20 Let now hi E H be orthogonal on any M h , h E S. We then have
There results Mh1.i Mh for any h E S. If hi ¥= 0 then S U {hi} is a cyclic free system which is in contradiction with the maximality of S. Therefore hI = 0, hence
'\
Let Jl be a positive measure on X and N be a Hilbert space. It is already known how to construct the Hilbert space L2(N; dJl) of all functions h defined on X, with values in N, for which
SIIh(x)11 2dJl(x) < The representation
qJ -+-
00.
U'I' of C(X) on L2(N; dJl), where (h
E
L2(N; dJl))
will be called the natural representation of C( X} on L2( N; dJl}. Theorem 7.12. Let qJ -+- U'I' be a representation of C(X) on a separable Hilbert space H. There exists a positive measure J1. on X, a partition X = U Wk of X, of disjoint Borel sets and, for any k = 1, ... , 00 k = I, 2, ... ,
00
a k-dimensional Hilbert space Nk. such that H is isometrically isomorphic (±) L 2 ( Nk.; Xw dJl) and the representation qJ -+ U'I' is unitary equito k=1,2, ... , ":JJ
k
valent to the direct sum of the natural representation of C(X) on L2(Nk.; Xw k dp,}.
Function algebras
160
Proof. Let S be a maximal cyclic free system of elements in H. Since
and H is separable, the set S is at most countable. Let S = {hh h2 , ••• }.
We have <X)
H = (±) M hj • ;=1
We can assume IIhill = 1. For i = 1,2, ... let
(cp
E
C(X»).
Pi is known to be a positive measure on X and, since Il;(l) = IIh i l1 2 = 1, there results I Jl ill = 1. Consider 00
~ i
p =
;=1
1
2
Pi'
Then p is a positive measure on X and p(1) = l. The measures Pi are absolutely continuous with respect to It. Let dpi = fidJl with fi E Ll(dp), Ii ~ O. Let us denote by Sj = = {x E X:/;(x) '# O}. For any set of natural numbers" we put E" = Si) - (U Sj). We have £"1 E"2 = 0 if '# f2' For any =
(n
f E,f
k = 1,2, ... ,
00
"1
n
;f1:."
we write Wk
U
=
card" =k
E".
It is clear that a point x E X belongs to Wk, k < 00, if there are exactly k sets S;I"'" Sik defined as above, which contain x. We have Wi Wj = 0, i,j = 1,2, ... ,00; i '# j and
n
X=
k
U
=
1, 2, ... ,
Wk' IX)
Ch. 7. Operator representations of function algebras
161
Since Si are countable sets and, since by all operations from (S;) to (Wk) for any finite k, we still remain in the Borel clan of measurable· sets, Wk are measurable. Since
is measurable too. As we shall see, the sets W k may be chosen to be Borel sets. For any k = 1,2, ... , 00, let Nk be a Hilbert space of dimension k (lVoc = /2).
(J) ex>
00
Now let h E H have the form h
=
~ Ucp/li' ;=1
In
F or any k Nb by
=
1, 2, ... we define the function g~ on X, with values
if there exists J such that x
g~(X) =
E
E,?, where
o for the other x For k = 00, J = (j1,j2,···),A < j2 < .... We now show that g~ E L2(Nk; Xwkdll). We have
J (!h(x)lqJ jl(x)1 2 + ih(x)lqJ jlx) I2 + ... + !ik(X) IqJik(X) I2d.u(x) ~
~
card .1 = k E,? oc
~ ~
JIqJi(X)12!i(X)d/l(x) =
;=1
~ (U: fPi 2h;, hi) ;=1
11 - c, 437
~
JIqJ;(x)1 2d /l
i
=
;=1 X
oc
=
00
oc
=
oc
~ (UfPjlzi' U fPi hj ) = ~ ;=1
;=1
II UfPj hi ll 2 =
IIh!12.
Function algebras
J62
ffi
Let gh be the element of
L2(Nk' XcolcdJl) defined by
k = 1, 2, .. ,00
:xl
For hE H of form h
~ U~ihj, ;=1
=
we put
One easily verifies that U is a linear map from the subspace of elect)
ments ~ U~ihj of H, which is dense in H, into i
=
ffi k = I, 2, ... ,
1
L2(Nh XcokdJl). 'J.)
Moreover
I Uhl1 2 = IIghl1 2=
Ilg~112 =
~ k = 1, 2, ... ,
k
~
~
k = 1, 2, ... ,
card ,I = k
:xl
~
JIlg~(x)112d,u =
~ k
:xl
=
1, 2, ... ,
00
Wk
J1q>j;(x)12fj;d,u =
i = 1 E,I ct)
~ k
=
~
1,2, ... ,
card ,I = k
ct)
~ i
=
J lq>i(X)12fi(x)dp =
1 E,I
00
=~ i
=
:xl
=
~ ; =
~
1 k = 1,2, ... ,
~
~
00
Jlq>j(x)12fi(X)dp =
1 k = 1, 2, ..• , x
~ i
=
Jlq>lx)1 2dJLi =
;=1
.'Y.)
~ (Uq;;q;jh j , hi)
=
;=1
~ i =1
~ (U~ih;, U~ih;) ;=1
"1)
=
J1q>;(x)12f;(x)dJL =
1
"1)
.'Y.)
=
'J.)
IIh;ll2 = Ilh11 2 •
=
Ch. 7. Operator representations of function algebras
163
At a certain step we used the fact that for x E E" we have fi(X) = 0 for any i ¢ f. The map U can therefore be extended to an isometric operator ® L2(Nk; XWkdJL). from H into k = 1.2, .. .• 'Xl
We now prove that the image of H by U is dense in
®
k= 1,2, ... ,00
®
L2(Nk; XWk dJl). LetgE
L2(Nk; lWk dJl) of the form g=(gl, g2, ... , gOO)
k=1.2, •..• 'Xl
with gk = (g}, ... , gt), gk
E
L2(Nk; lWk dp), be orthogonal to any Uh co
with h E H. For h E H of form h =
o= ~
=
k=l.~ •.
(UIz, g)
(g~, gk) = ..• Xl
UfPi hi, we have
= (gh, g) ~
k=1.2 ..... 'Xl
~ k ~=1.2 ..... Xl
~
~
=
J(gk(x), g~(x)) dp =
k
~ JJ fjlx)
qJii(X)
g7(x) dJl.
card.1 =k i = 1
Let now
=
~
1,2 ..... '1)
~
JJ fn(x) cp(x) g!(x) dJl =
0,
card .f=k E J
where we used the fact that fn = 0 on E.f if n ¢ E". Note that, jf 11 Ef then E" =E" Sn and if n ¢ f then E"nSn= 0. Hence
n
and therefore
U
U
k=1.2 ..... 0') card ,f=k
E" =
U
(OJ k
k=1.2 •... ,00
n Sn) =
Sn·
Let F be a function on Sn such that F = g~ on E", if card f = k. From the above considerations there results
s J flx) qJ(x) F(x) dJl =
Sn
O.
11 E
f and
Function algebras
164
Since this equality holds for any qJ E C(X), there follows fll(x) F(x) = 0 Jl-almost everywhere on SlI and, as fix) ¥= 0 for x E Sn' we get F(x) = 0 for x E SlI' hence g:(x) = 0 for x E E.f with n E J and card J = k. Since n has been arbitrarily taken, there results gk = 0 on any E" with card; = k, that is gk = 0 on Wk' Therefore
J
for any k, i.e. gk = 0, hence g = O. Then, the operator U is an isometric isomorphism between H and Ef) L2(Nk' XWk dJl). k=1.2 .... ,~
Let
qJ -+
U; be the representation of C(X) on k
L2(Nk' XWkdjl),
Ef) =
1,2 •... ,00
defined by
00
If g has the form g = gh, with hE H, h =
~
Ulp hi, we have
\=1
00
U;g = UUfPU-lg" = UUlph = U ~ Ulplpj hi =
qJg".
i=1
As we have seen, the elements of this form are dense In Ef) L2(Nk;XeJk dJl); then
k=1.2 •.. .. ~
for any g E
Ef) k
=
L2(Nk; XWk dJl).
1.2 •.. .• X
The proof is complete. <) From the proof of this theorem there also results the following
Ch. 7. Operator representations of function algebras
Proposition 7.13. Let q>
165
~
Utp be a representation of C(X) on H. For any hEH, the representation q>~ Utp/Mh of C(X) on Mh is unitarily equivalent to the natural representation of C( X) on L2( d/1h)' To conclude this paragraph we shall establish the form of invariant subspaces of natural representations of C (X). Theorem 7.14. Let /1 be a positive measure on X and q>
Utp the natural representation of C(X) on L2(d/1). The closed subspace M of L2(dp.J is invariant to q> ~ Utp if and only if M has the form -+
where XE is the characteristic function of a Borel set E in B(X). E is Jl-essentially uniquely determined by M. Proof. Let M be a closed subspace of L2(d/l) such that UtpM C M for any q> E C(X). Let F be the projection of 1 on M. We have (q> E C(X)).
Then F = IFI2 Jl-almost everywhere, hence F = XE with E a Borel set in X. It is clear that XEL2(d/1) C M. Now let h be a function in M, orthogonal to XEL2(dJl). We have
and
Therefore, h
=
XEh /1-almost everywhere and
for any q> E C(X). There results h = 0 /l-almost everywhere, so M = = XEL2(dJl). If M = XE I L2(d/1) = XE2L2(d/l) with £1 and £2 Borel sets in X then, clearly, XEI = XEz Jl-almost everywhere.
166
Function algebras
Notes A unitary treatment of positive definite maps on C( X), spectral and semispectral families and spectral and semispectral measures can be found in C. FOIA~ [4] and C. T. IONEScu-TuLcEA [1] as well as in M. A. NAIMARK [3]. The first direct proof to Proposition 7.4 has been given by W. F. STlNESPRING [1]. Naimark's dilation theorem first appears in M. A. NAIMARK [2]. A new proof belongs to C. FOIA~ [4] and it has also been presented in the Romanian edition of this book. In the present chapter the proof follows W. F. STlNESPRING [1] where the theorem is proved in the case of C·-algebras. See also W. B. ARVESON [2]. A first study of operator representations of arbitrary function algebras has been carried out by C. FOIA~ and I. SUCIU [1]. The same work contains the results of paragraph 7.2. The origin of Theorem 7.12 is to be found in the theory of spectral multiplicity (cf. P. R. HALMOS [2]); the present proof follows M. ZERNER [1]. Another proof of Theorem 7.14 in the case of the unit circle of the complex plane and the Lebesgue measure appears in H. HELSON [1].
CHAPTER 8
Elements of spectral theory of representations of function algebras 8.1. The canonical decomposition Let A be a function algebra on X and H a Hilbert space. A representation I-+- T J of A on H will be called X-spectral, or simply spectral, if it is the restriction to A of a representation qJ -+- Utp of C(X) on H. The representation qJ -+- Utp of C(X) on H, which satisfies
= UJ
TJ
is called the extension to C(X) of
I
-+-
TJ .
Proposition 8.1. A spectral representation I -+- TJ 01 A on H has a unique extension to C(X). Prool. Indeed, let qJ -+- Utp and qJ -+- Vtp be two extensions to C(X) of the representationj -+- T J . Since U J = T J = V J forjEA, there results Uf = 1j"f = Vl, hence U,£cdjgj =
for
Ib gi EA.
~
ciUfiU:i
= ~ CiVJjVg~ =
V,£cdjgj
Therefore Utp = Vtp for any function of the form qJ = I.ciligi with Ii' gi E A. Since the set of all functions with such a form is dense in C(X) and the representations qJ -+- Utp' qJ -+- Vrp are continuous, there follows Utp = Vtp for any qJ E C(X). A closed subspace AI of H is said to be inrariant (to I-+- T J ) if TJM C M for any I EA. The subspace M is called doubly inl'ariant (to 1-+ T J) if TJM CAt, Tj*M C kf for any f EA.
Function algebras
168
Proposition 8.2. Let f
Tf be a spectral representation of A on II, qJ -+ U its extension to C( X) and M a doubly invariant subspace. Then UqJM C M for any qJ E C(X). Proof. Obviously UqJM C M for any
Corollary 8.3. If f -+ Tf is a spectral representation of A on H,
(8.1.1)
T*Th f 9 = T9 T*h j
(h
E
M;/, g E A).
For finite systems fl"",/n, gl"'" gn we define the operator (8.1.2) From (8.1.1) T is a normal operator on M. We shall prove that (8.1.3) Let k = IIrfigdl and ET(G) be the spectral measure of T. To prove (8.1.3) it is sufficient to show that ET({Z
E
C: I Z I ~ k}) M
=
M,
hence it is sufficient to show that (p
=
1,2).
Ch. 8. Elements of spectral theory
169
Let hEMp and assume h ¢ ET({Z E C: Izl ~ k}) M. That means ET({Z
E
C:
Izi >
k})h '# 0
hence there exists e > 0 such that h'
= ET({Z E C: Izi
~ k
+ e}) h
#- O.
\Ve have (k ~
+ e)2" Ilh'1I 2 =
J
{z;iz'~k+
(k
+ e)2" Jd(ET(z) h, 11)
IzI2"d(ET(z) h, 11) ~
~
JIzI2"d(ET(z) h, h) =
II T2"hI1 2 •
f}
Then (8. 1.4)
Let
!
-+
-+ U~
be the extension to C(X) of the spectral representation Tfl Mp. As h E M h , we have
Hence
and therefore (8.1.5) Using (8.1.5) in (8.1.4) we obtain (k
+ 8)2" IIh'1I 2
~
k2"
IIhll 2
for any naturai n, which is impossible. Hence (8.1.3) is true. Let cP E C(X) be of the form
Function algebras
170
Proposition 8.S. Let {MiL E.1 be an increasingly directed family of spectral subspaces. Then, the closure of U M j is a spectral subspace. iE"
Proof. Since {MJiEf is an increasingly directed family,
U Mi is ; E,?
a vector subspace. Let M be its closure in H. Then, clearly, M is doubly invariant. Let lfJ -+ U~ be the extension to C(X) of the spectral representation f -+ Tfl M j • If MI and M2 are spectral then, according to Corollary 8.3, Ml M2 is also spectral and U;h = U:h for h E MI M 2 • Let h E U Mi and define Uq1h = U~h for h E Mi' Following the iE,? above considerations, the operators Uq1 are well defined on U .~f,
n
and, for any lfJ
n
E
iE,f
U M i , we have
C(X) and h E i
Eo?
I Ulphll
~
IllfJl! Ilhll·
Therefore, Ulp may be extended, by continuity, to M and it is easily verified that lfJ -+ Ulp is an extension to C(X) of the representation f -+ TfIM. (> Theorem 8.6. Let f -+ Tf be a representation of A H has a unique decomposition of the form H
=
011
H. The space
Hs EB He
such that Hs and He are doubly invariant subspaces, the representation f -+ TflHs is spectral and the representation f -+ TflHe is completely non-spectral. Hs is the largest spectral subspace of H. Proof. Let {M j }iE.1 be the set of all spectral subspaces of H. The subspace {OJ is, obviously, spectral, hence this set is non-void. {Mih E,? is increasingly directed, according to Proposition 8.4. Let Hs be the closure of U A(. Then, from Proposition 8.5, Hs is spectral and, iE.1
clearly, it is the largest spectral subspace of H. We write H = Hs EB He· B s is doubly invariant, hence He is doubly invariant too. Hs being spectral, f -+ TflHs is also spectral and, as Hs is maxin1al, f -+ TflHc
is completely non-spectral.
171
Ch. 8. Elements of spectral theory
Consider another decomposition of H
with HI, H2 doubly invariant, / -+ TjlHl spectral and / -+ TjlH2 completely non-spectral. Hence, HI is a spectral subspace and, as Hs is maximal, HI C Hs. Let
Since Hs and HI are doubly invariant, M is also doubly invariant and, according to Corollary 8.3, M is spectral. But M is orthogonal to Hh hence M C H2 and, as f -+ TjlH2 is completely non-spectral, there results M = {o}. Therefore
and the proof is complete. 0 The representation / -+ 1jIHs will be called the spectral part of the representation f -+ 1j, and f -+ TjlHc the completely non-spectral part.
8.2. The spectral dilation and attached spectral measures A representation q> -+ U
(/ E
A;
hE
H)
where P is the orthogonal projection of K on H. \Ve obviously have (8.2.2)
Tjh
=
PUjh
=
PUrh
(/ E
A .. Iz
E
H)
Function algebras
172
The spectral dilation qJ --+ Utp of I --+ Tf is said to be minimal if the space generated by the elements of the form Utph, with qJ E C(X) and h E H, is dense in K. If the representation I --+ Tf admits a spectral dilation, then it clearly also admits a minimal one. In the following all the considered spectral dilations will be assumed minimal. Let qJ --+ Utp be a spectral dilation of I --+ Tf and (Ilk}, k2)kl, k2 E K the spectral family attached to the representation qJ --+ Utp of C(X) on K. For hE H we put Ilh = Ilh,h' For any I, g E A we have (T;h. h) ' J ;
=
(PU,h. h) J
=
(U,I1, 11) = J
ffdph
..
and
II(Tf
+ Ti)hll
2
=
IIPUf+:ehIl2 ~ IIUf+g1z112 = (Uf+gh, Uf+gh) =
= (Ulf+gj2
11, h) = SI/~: gl2 dll h •
A positive measure Ilh on X, which satisfies (jEA)
(8.2.3) and
(I, g E A)
(8.2.4)
will be called a spectral measure attached to h by I --+ ~. The measure Ph = Ph,h, hE H, where (Pk b k 2 )k 1 ,k 2 EK is the spectral family attached to a spectral dilation qJ --+ Utp of I --+ Tf , is, as we have already seen, a spectral measure attached to h by f --+ Tf . In the following we shall study the existence of attached spectral measures and spectral dilations. Theorem 8.7. Let A be a Dirichlet algebra on X and I --+ Tf a representation of A on H. There exists a unique spectral dilation qJ --+ Utp off --+ Tf · Proof. From Proposition 7.8 there results (8.2.5)
I ReTf11
~
II Re/ll·
Ch. 8. Elements of spectral theory
173
Now using (8.2.5) we obtain
+ i (ReT_ if + ig)II + I Re ( -
~ IIReTf +g
II
+ IIReT_if+igll
if + ig) II ~ II Re (f + g)
II
+ II Re ( -
~
I Re(f + g)11
if + ig) II ~ 211 f
+ + g II
which yields liT Tl!c" /' "'''f -0 II f T ill ~"'Il TglI·
(8.2.6)
I
I
For hb h2 E H we define on A
+ A the functional
According to (8.2.6), Il hl,h2 is well defined on A + A and bounded; therefore, since A is a Dirichlet algebra on X, Ilhl,h2 may be extended to some Radon measure Ilhl,h2 on X. One easily verifies that the family (llhl,h2)hl,h2 E H satisfies (7.1.2), hence it is a semi-spectral family. Applying Theorem 7.5, we obtain a representation qJ -+ Utp of C(X) on a Hilbert space K which is a spectral dilation of f -+ Tf . Let qJ -+ Utp and qJ -+ U~ be two spectral dilations of f -+ Tf and {J1kl,k2)kl,k2EK' {J1~"k')k"k'EK' be the corresponding spectral families. I
For j, g
E
2
I
2
A and hlh2 E H, we have
Hence, the measures Il h l,h2 and ll~l,h2 are equal on A + A and, since A is a Dirichlet algebra, they are equal. Since the spectral families
Function algebras
174
(Ilkl. k2)klo k2EK and (Il~i. k~)ki.k~EK' are minimal dilations of the semi-
spectral families (p ht. h 2)ht ,h2 E H' (J1~1 ,h2)ht ,h2 E H' respectively, then according to Theorem 7.6, they are equal (in the sense precised in Theorem 7.6). This proves the uniqueness of the spectral dilation in the case of a Dirichlet algebra. The theorem is completely proved. 0 According to formula (8.2.5), which holds for any representation f -+ 1f on H of a function algebra A, if for any q> E A + A of the form q> = f + g,f, g E A, we put
then the map qJ -+ TqJ of A + A in L(H) is well defined, linear and bounded. We now prove the existence of the spectral measure attached to h, for the representations f -+ Tf of A which satisfy the condition (8.2.7) for any finite system
qJb""
q>n
of elements in A
+ A, with l:lqJd
2
~
1.
Proposition 8.8. Let f -+ Tf be a representation of A on H sllch that (8.2.7) holds for h E H. Then there exists a positive measure Ph on X such that Ilh(J) = IIhll 2 and (8.2.8) Proof. For g
E
C(X), g
~
0, we put
(8.2.9) where the supremum is taken for all finite systems ments in A + A, with I:lqJ;l2 ~ g. We have (8.2.10)
for
qJl"'"
qJn
of ele-
Ch. 8. Elements of spectral theory
and from (8.2.7) there results (8.2.11 )
that is (8.2.12) Let us note that (8.2.13) For some g (8.2.14)
E
CR(X) we write J1*(g) = inf (cllhll 2 c>
-
:g!
Jl*(c - Igl))·
From (8.2.12) and (8.2.13) we find that
for any c >
Igi. Hence
for any c >
Ig/. Therefore, for any g E CR(X),
(8.2.15)
J1*(g)
~
O.
175
Function algebras
176
It is clear that for
> 0, we have p,*(rtg)
(8.2.16)
C2
~
Let now gI' g2 > Ig21, and
E
=
rtp,*(g).
I , C2 be two constants, with CI > IgII,
CR(X) and
C
We used the fact that p,* is a positive functional, and (8.2.12). Hence (8.2.17)
for any gI' g2 E CR(X). From (8.2.14) there follows (8.2.18) and therefore (8.2.19)
JI*(g) ~ IIgll IIhl1 2
-
p,*(llgll - Igl) ~ Ilgll IIhl1 2
Ch. 8. Elements of spectral theory
177
From (8.2.15), (8.2.16), (8.2.17) and (8.2.18), p* is a seminorm on CR(X) and, according to Hahn-Banach theorem and (8.2.19), there exists a real measure Ph on X such that Ph(1) = IIhl1 2 and (8.2.20) Then, obviously
hence Ph is a positive measure on X. We have
for g
~
0 and c
~ g,
and therefore
for any g ~ 0 and c ~ g. Since any positive function in CR(X) may be written under the form c - g with g ~ 0, c ~ g, then for any g E CR(X), g ~ 0, we have (8.2.21 ) Then
that is
and the proposition is completely proved. 0 12 - c. 437
178
Function a1gebras
Proposition 8.9. Let ilia be a positive measure on X such that PIa(I) = IIhl1 2 and
(8.2.22) Then Ilh is a spectral measure attached to h by f
(/, g E A). -+
Tf .
Proof. It remains to prove only that (8.2.23) Let A be an arbitrary complex number and / there results
E
A. From (8.2.22)
hence
or (8.2.24) Since (8.2.24) holds for any complex number A, we get
and the proof is complete. 0 From Proposition 8.8 and 8.9 there results the following Theorem 8.10. Let / -+ Tf be a representation of the algebra A on Hand h E H for which (8.2.7) holds. Then there exists a spectral measure attached to h by f -+ Tf . In the case of logmodular algebras we shall prove the uniqueness of attached spectral measures together with the existence and unique-
Ch. 8. Elements of spectral theory
179
ness of the spectral dilation when there are spectral measures attached to every point h of H. Theorem 8.11. If A is logmodular on X and f --+ Tf is a representation of A on H, then the spectral measure attached to 11 E H by f --+ Tf is unique. Proof. Let Ilb /12 be two spectral measures attached to h E H by f --+ Tf · For f, f- 1 E A we have
Since A is a logmodular algebra, we have
for any u E CR(X). Hence, the 'eal function
has an extreme point in 0 for any u E CR(X) and therefore
o=
qJ'(O)
for any u E CR(X), i.e. J11
= 2( Jud III -
= J12.
Jud /12)
0
Theorem 8.12. Let A be a logmodular algebra on X and f --+ Tf a representation of A on H such that for any hE H there exists a spectral measure attached to It by f --+ Tf . Theil there exists a unique spectral dilation of f --+ Tf . Proof. Let Ph be the spectral measure attached to h by f --+ Tf . We first prove the following relation (8.2.25)
180
Function algebras
Let /,j-I EA. We have 2(lIh 1 1l 2 = (1f(h l
~ I Tf(h l ~
+h
2 ),
+ IIh2112) = Ilhl + h2112 + IIhl Tj-l(h l
+ h » + (Tf(h 2
l -
1z2112 =
h2 ), TJ-l(hl - h2» ~
+ h2) I I Tj-l(hl + h2)11 + II Tj~-1(hl-h2)11 I Tf (h l -h 2) I
[II Tf(h l
+ h2)11 2 + 111/(h l
-
h2)112]1I2[11 Tj-l(h l
~
+ h2)11 2 +
Since A is a logmodular algebra we get
for any U E CR(X). Then, the standard argument used also in the proof of the preceding theorem, yields (8.2.25). If we put
we get a semispectral family (/1h 1 • h2)h1 • h2 eH on X such that
Using Theorem 7.5 we obtain a representation ({J --+ U", of C(X) on a Hilbert space K~ which is a spectral dilation of j --+ Tf .
Ch. 8. Elements of spectral theory
Let cp
--+
U; be another spectral dilation of f
181
--+
Tf . Writing
for hI, h2 E H, then, as we have seen, the measure mil = mh,h is a spectral measure attached to h by f --+ Tf . From Theorem 8.11 there follows mh= Ilh hence, the semispectral families (mhl> hz)h1 , h2 EH and {J1h1 • h2 )h1 , h2 EH coincide. Since the spectral dilation for semi spectral families is unique (Theorem 7.6), the spectral dilations qJ --+ UqJ and cp --+ U; of f --+ Tf coincide (in the sense given above). The theorem is proved. 0 Theorem 8.13. Let A be a logmodular algebra on X and f --+ Tf a representation of A on H. The representation f --+ Tf has a spectral dilation if and only if (8.2.7) holds for any h E H. If f --+ Tf has a spectral dilation then it is unique. Proof. If (8.2.7) holds for any h E H then, from Theorem 8.10 there results that f --+ Tf has a spectral dilation. The uniqueness of this spectral dilation follows from Theorem 8.12. Let qJ --+ UqJ be a spectral dilation off --+ Tf and qJI,"" qJn a finite system of elements in A + A, of the form cp = fi + gi' with l: IqJil 2 ~ 1 and h E H. We have
~IITlpih112
=
~11(Tfi
+ TiJ hll
~ ~ I Ulpjhll2
<
2
=~IIP(Ufi+g) hll 2 ~
~ IIqJill 2 11hl1 2 ~ 1111112.
The proof is therefore complete. 0 The condition (8.2.7) in Theorem 8.13 can be reduced to a much simpler one. Indeed we have the following. Proposition 8.14. Let A be a logmodular algebra on X and f --+ Tf a representation of A on H. Then the following two conditions are equivalent. (i) f --+ Tf satisfies (8.2.7) jor all h E H. (ii) For all h E H andj, g E A we have (8.2.26)
Function algebras
182
Prool. The implication (i) => (ii) is obvious. So we can suppose that condition (ii) holds. Firstly, we shall prove (by induction) that n
n
~
(8.2.27)
I/il
2
~ 1 => ~
;=1
I Tf ,hll 2
~
IIhl1 2
;= 1
for any finite system {/b ... ,I,,} of elements in A and hE H. For n = 2 (8.2.27) is contained in (8.2.26). Let thus n > 2, e > 0 and n
lb'" ,I" E A such that ~ I/il 2 < I. Since A is logmodular algebra on ;=1
X, we can find g
E
A such that g-1 n-l
~
(8.2.28)
E
A and
I/il 2 < Igl2 < 1 -1/,,\2 + e
;=1 n-l
Let gi =/jg-l, i
=
1,2, ... , n - 1. Then, by (8.2.28) ~ Igil2~ 1 ;=1
thus by the induction hypothesis n-l
~
II Tg ,kll 2
~
IIkl1 2
;= 1
for all k
E
H. Particularly, for k
= Tgh we obtain
n-l
~ II Tfi h II 2 ~
II TJzll 2
;=1
But, from (8.2.28) we have also
//,,/2 + Igl2
~
1+e
so that using (8.2.26) we infer
that is n
~
II Tfihll2
~ (I
+ e) IIhl1 2
;= 1
Letting e -+ 0, we obtain finally (8.2.27).
Ch. 8. Elements of spectral theory
183
In a similar way we infer that n
n
~ Ifil2 ~ 1 => ~ II TJjhl12 ~
(8.2.29)
;=1
IIhl1 2
;=1
for any finite system {/h" .,I,.} in A and h E H. Now define for gEeR(X), g ~ 0 Il*(g)
sup ~ 1\ T/ hll 2
=
j
where the supremum is taken for all finite systems {/h ... ,f,.} C A such that 1:/fiI2 ~ g. Then reproducing the proof of Proposition 8.8 we obtain (using (8.2.2i) instead of (8.2.i») a positive measure ilion X such that 111(1) = IIhl\2 and 1\ Tf hl1 2
(8.2.30)
~
J1/1 2d1l
(f E A)
1,
Analogously, (8.2.29) will lead (again by reproducing the proof of Proposition 8.8) to the existence of a positive measure 112 on X such that 112(1) = IIhll 2 and
(fEA)
(8.2.31 ) In virtue of the proof of the Proposition 8.9 we will have
(fEA)
(8.2.32) Actually III
J.l2 since for f,/-l
=
IIhl14
E
A we have
~ (Tfh, T'f-lh)2 ~
~
IITfh!\21IT7-1hI12
~
J1/1 2dll J1/1- 2dIl2• 1
The fact that III = 112 is easily obtained as in the proof of Theorem 8.11. Let us put Ilh = 111 = 112' Finally if qJ; = fi + Ki' with 1:lqJ il 2 ~ 1 and f1,'" ,f,., gh"" g,.EA then n
~ IIT"jh112 ;=1
n
=
~ IIT/jh
+ T~jhll2
n
~ ~ (II Tfih II 2
;=1
+ 2Re (Tfigjh, h) + "TgihI12) =
;=1 n
~ ;=1
(II Tfihll2
+
+
Function algebras
184
+ 2Re Sfigi d J1h + II T~/1112) ~
n
~
(S IJ;\2dJ1h +
;=1
+ 2Re Jfigidp. + JIgiI 2dp.) = J( ~1
Irp.!2 d P.) .;;; 111111 2,
The proof of the proposition is complete. 0
8.3. Szego measures and natural representations Let fl be a positive measure on X. We recall that H2(dJ1) is the Hilbert space obtained by closing A in L2(dJ1). The measure J1 is called a Szego measure (relative to A) if, for any Borel set E of X for which
XEL2(dJ1) C H2(dJ1), we have J1(E) = 0 (1.E = the characteristic function of E). The following theorem asserts the existence of Szego measure for any algebra A '# C(X): Theorem 8.15. (i) Any representing measure of A is either a Szego measure or a point measure. (ii) if v is a non-zero complex measure on X, orthogonal to A, then Ivl is a Szego measure. (iii) If A i= C(X) then there exist Szego measures on X (relative to A). Proof. (i) Let J1 be a representing measure of A. One easily verifies that J1 is multiplicative on H2(dJ1.). Assume that for a Borel set E we have (8.3.1) Then XE
E
H2( dJ1) and
S XEdJ1 =
J X1dJ1 = [J XE dll]2
Hence J1(E) = 0 or /1(E) = 1. If /1(E) = 1 then from (8.3.1), there results L2(dJ1) = H2(dll), hence /1 is multiplicative on L2(dJ1) and, in particular, on C(X). It is known that such measures are point measures. (ii) Let v be a complex measure, orthogonal to A. Then v is orthogonal to H2(dll) and, writing J1 = lvi, we have (8.3.2)
dv = FdJ1,
IFI
=
1
Ch. 8. Elements of spectral theory
Suppose XEL2(d/1) C H2(d/1). following (8.3.2), we have
/1(E) =
There results
185 --
XEF E H2(d/1) and
JX d/1 = JXEPFd/1 = JXEFdv = O. E
(iii) is a consequence of (ii) and of the Hahn-Banach theorem. The proof is therefore complete. 0 A positive measure /1 on X be called Szego - singular (relative to A) if L2(d/1) = H2(dJi.). We shall now establish the relation between the Szeg6 measure and a completely nonspectral representation. Theorem 8.16. Let f -+ Tf be a representation of A on Hand Ji.h a spectral measure attached to the element h E H. Let M be a closed subspace of H such that hEM and TfM C M for any f E A. If the representation f -+ TflM of A on M is completely non-spectral, then Ji.h is a Szego measure. Proof. Let V, = TfIM; we define the operator V on A + A, with values in M, by (f, g
Since Ji.h is a spectral measure attached to h by f
I V(f + g)112
=
II Vfh ~
+ V:h1l 2 <:
II T,h
-+
+ T:h1l
2
E
A).
T/, we have ~
JIf + gI2dJi.h·
Hence V may be extended to a contraction (denoted also by V) from the space K, the closure of the set spanned by A + A in L2(dp), into M. For qJ E C(X) we denote by Uip the multiplication by qJ in L2(dp). Then, for f, g E A,
and, by continuity (8.3.3)
Function algebras
186
Similarly, we find (8.3.4) where
Let E be a Borel set of X, such that
Then obviously,
as XEL2(dph) is doubly invariant to qJ -+ Uq>' then, according to (8.3.3) and (8.3.4), the M-c1osure of VXEL2(dJlh) is a doubly invariant subspace Nand (8.3.5)
V(Uf IXEL2(d'Lh»
=
Vf (Vlx EL2(dJlh»
V(UjIXEL2(dJlh» = vj(VIXE L2(d,Lh».
Following (8.3.5), we have V:VfVe = V:VUfe = VU:Ufe = VUgfe = VUfge = = VUfU:e
= VgV:Ve
for any e E X EL2(dJlh)' and by continuity (8.3.6) For cP
(/, g E
C(X) of the form
(8.3.7) with/i' gi (8.3.8)
E
A, and n EN, we put
E
A; n EN).
Ch. 8. Elements of spectral theory
187
Then, using (8.3.5), we obtain V~ Ve
(8.3.9)
=
~
vj;vgi Ve = V(lpe)
for e E XEL2(dp,,). Hence, if lp = 0 then V~Ve = 0 for any e E XEL2(dp,,) and, by continuity, V; = o. Therefore lp --+ V~ is a well-defined linear map on the elements lp of the form (8.3.7). Also from (8.3.9) we get (8.3.10) and, following (8.3.9), (8.3.10), there results
!! v;mVe!! = !! v;mVell = II v(~/)me)!! ~ JIlpl21el 2mdph = Illpllmllell
~
where IIlpll resp. lIell are the norms in C(X) resp. L2(dp,,). Therefore
II v;mVell
(8.3.11) Since
~
IIlpllmli e ll.
V; are normal operators, we have II V; Veil = (V; Ve, V; Ve)1/2 = {V;*V; Ve, Ve)1/2
~
II V;*V;VeIl 1/2 Veil 1/2 = (V;V;Ve, 11
=
~
V;*V~Ve)1/2111 Veil 112 =
(V:'2 V;2 Ve, Ve) 1/21 11 Vell1!2 ~ 1
1
~ II V;*2 V;2 Veil II Veil 2 + 22. Taking into account (8.3.11) we obtain 111
(8.3.12) and, for m
I V; Veil -+ 00,
~
-+-+ ... +--
II Veil 2
1
2mlllplillell2m
21
therefore results
hence (8.3.13)
I V~II
~
IIlpll.
Since the set of all functions of the form (8.3.7) is dense in C(X) then, according to (8.3.13), we may extend the map lp -+ V; to a representation of C(X) on N. As VI = V; for f E A, the subspace N results spectral for the representation f --+ VI'
Function algebras
188
But
I
Vf is a completely non spectral representation, hence N = {O}. Therefore Ve = 0 for any e E XEL2(dll h). Since XE L2(dll h ) C C H2(dph), for any e E XEL2(dll h ) there exists a sequence In of functions in A, such that e = limL2 In. We have --+
o=
(Ve, h) = lim (Vln' h) = lim (Vfnh, 11)
=
= lim (Tfnh, h) = lim SInd/lh = Sedph for any e E XEL2(dll h). In particular
o = SXEdll h =
Illl E).
Therefore Ph is a Szego measure. The theorem is proved. 0 In the" following we shall study a class of representations of A on H for which the converse assertion is also true. The representation I --+ Tf of A on H will be called subspectral if there exists a spectral dilation qJ --+ Utp of I --+ Tf such that Ufh E H for any I E A and h E H. We obviously have
(I E A; hE H)
(8.3.14)
One easily verifies that any other spectral dilation of the subspectral representation I --+ Tf , which satisfies (8.3.14), coincides with q> --+ Ucp. In the following by spectral dilation of the subspectral representation I --+ 1/, we mean the (unique) spectral dilation qJ --+ Ucp of I --+ Tf which satisfies (8.3.14). Proposition 8.17. Let 1--+ Tf be a subspectral representation of A on Hand M C H a doubly invariant subspace to I --+ Tf . The representation I --+ TflM 01 A on M is subspectral. Prool. Let qJ --+ Ucp be the spectral dilation ofI --+ Tf and K the dilation space. If we put K(M)
= clm {Ucpm; mE M,
qJ E
C(X)}
then q> --+ UcpIK(M) is clearly a spectral dilation of I satisfies (8.3.14) for I E A and m E M. 0
--+
TfIM, which
Ch. 8. Elements of spectral theory
189
An important class of subspectral representations is the class of natural representations. Let p, be a positive measure on X, p(X) = 1 and N a Hilbert space. We have already seen in paragraph 7.3 how the natural representation q> ~ UqJ of C(X) on L2(N; dp) is constructed: (q> E C(X); k E L2 (N;
dp»
If we consider the elements of N as functions with constant values from X to N, we can consider N as a subspace of L2(N; dll). Let H't.(N; dp)
=
H't.(N; A; dp,) = elm [{ Ufn, f
It is clear that, for any f
E
E
A, n E N}J.
A, we have
If we write Tf = Uf IH2(N; dp), one easily verifies that f ~ Tf is a subspectral representation of A on H2(N; dp,), q> ~ UqJ being its spectral dilation. We call this representation the natural representation of A on H2(N; dJl). If N is one-dimensional we shall write H2(dJl) instead of H'l(N; dp,), and it is easy to see that this notation is in agreement with that of the previous chapters. N ow let f ~ Tf be a subspectral representation of A on Hand q> ~ UqJ its spectral. dilation. Let (P,k l • k 2 )kl • k2 EK be the spectral family attached to the representation q> ~ UqJ of C(X) on K(K is the dilation space) and Ph = Ph,h the spectral measure attached to h by f ~ Tf , for any hE H. We have
that is (8.3.15) For h E H, we denote by Mh the closed subspace of H spanned by Tfh, f EA. Using (8.3.15) we may uniquely define the operator Von M h , with values in L2(dJlh), such that
Function algebras
190
The operator V is an isometry from M" onto V(Mh). It is clear that V(M h) is invariant to the multiplication by functions of A and (8.3.16) Proposition 8.18. The closed subspace M of M" is spectral with respect to f ~ TfIM" if and only if there exists a Borel subset E of X such that VM = XEL2(dph). E is Ph-essentially unique determinedby M. Proof. Let VI = TIIMh. Assume VM = XEL2(dph). For mE M and qJ E C( X) we define U•tpm --
1.'-1 J'
cP If J' m.
Since XEL2( dph) is an invariant subspace to the multiplication by functions of C(X) , Utp are well defined and linear on M. Obviously
At the same time
Hence qJ ~ Utp is a representation of C(X) on M. From (8.3.16) there results
(fE A; For m
E
M and k
(V;m, k) =
E
=
M).
M" we also have
= (m, Vfk)
(Vm,fVk)
mE
=
(fVm, Vk)
(m, Tfk) =
=
(m, V-IfVk)
(V-I fVm, k)
=
= (U;m, k).
We used (8.3.) 6) and the fact that Vand V-I are isometries. Therefore M is doubly invariant to f ~ Vf and qJ ~ UqJ is an extension to C(X) of f -+ VI' i.e. the subspace M is X-spectral relative to f -+ Vf·
Ch. 8. Elements of spectral theory
191
Conversely, assume M is X-spectral relative to f -+ Vf and let cp -+ Ucp be the representation of C(X) on M, which extends / -+ V,IM. According to (8.3.16), V M is invariant to multiplication by functions of A. Let P be the projection of L2(dflh) on VM. For m, n E M we have
(vjm, n) = (m, Yin) = (m, Tin) = (m, V-I/Vn)
= (Vm,JVn)
=
=
(fVm, Vn) = (P/Vm, Vn) =
Hence (8.3.17)
V;m = V-IP/Vm.
Since V, are normal operators, we have
= I V-l/Vmll = IIJVml1
-
=
II/Vmll
for any mE M. Therefore fVM C VM for any f EA. Then VM is invariant to multiplication by any function of C(X) of the form cp = r.figi and, by continuity, to multiplication by any function cP E C(X). From Theorem 7.14 -there results VM = XEL2(dflh)' with E a Borel set, JLh-essentially unique. The proof is therefore complete. 0 Theorem 8.19. Let f -+ Tf be a subspectral representation of A on Hand flh the spectral measure attached to h by f -+ Ti . flh is a Szego measure if and only if the representation f -+ TflMh oj A on Mh is completely nonspectral. Proof. Since, obviously, V(Mh} C H2(dJLh) then, according toProposition 8. I 8, f -+ TflMh has no X-spectral subspaces different from zero, if and only if for any Borel set E with XEL2(dJLh) C H2(dJLh) we have JLh(E) = 0, that is, if and only if JLh is a Szeg6 measure.
192
Function algebras
Let Jl be a positive measure on X with Jl(X) = 1 and f -+ Tf the natural representation of A on H2(dJl). Taking as h E H2(dlt) the function identically equal to 1 on X, we have
hence Jl is the spectral measure attached, by the natural representation, to the element 1 of H2( dJl). Furthermore MI = H2(dJl) and the operator V defined above is the embedding operator of H2(dJl) in L2(dJl). Theorem 8.20. Let p. be a positil'e measure on X and f -+ Tf the natural representation of A on H2(dp.). (i) The subspace M of H2(dJl) is X-spectral if and only if M = = XEL2( dJl), where E is a Borel set of x. E is Jl-essentially unique determined by M. (ii) The natural representation of A on H2( dJl) is completely nonspectral if and only if Jl is a Szego measure. (iii) The representation f -+ Tf is spectral if and only if p. is Szego-singular. (iv) The measure Jl has a unique decomposition under the form p, = JlI + Jl2 where JlI is a Szego-singular measure and Jl2 a Szego measure. M oreOl'er
,
where the subspaces H2( dJlI) , H2(dJl2) are doubly invariant,f-+ TfIH2( dP.I) is the spectral part of f -+ Tf,f -+ TfIH2( dJl2) is the completely non-spectral part of f -+ Tf . The measure Jl2 is the supremum of the Szego measures v, such that v ~ Jl. Proof. (i), (ii) and (iii) are direct consequence of Proposition 8.18 and Theorem 8.19. It remains to prove (iv). Let
the canonical decomposition of the representation f -+ Tfo Since Hs is an X-spectral subspace, then, according to (i) it has the form
Ch. 8. Elements of spectral theory
193
with E a Borel set in X. Let JlI = XEJl. We have
Since any function of L2(dJl) which belongs to H2(dJl), belongs also to H2(dJ1l)' there results L2(dJll) = Hs = H2(dJll)' hence Jl is a Szego·singular meas ure. Let J1.2 = (l - xd Ji. Since XE E H2(dJl), there follows
Then
The representation f -+ TjlHc is therefore the natural representation of A on H2( dJ1.2); as it is completely non-spectral, Jl2 results a Szego measure. Hence (8.3.18)
Jl = J1.l
+ 112
with 111 a Szego-singular measure and 112 a Szego measure. Let v be a Szego measure, v ~ J1.. We have
and therefore veE) = 0, i.e. v ~ Jl2. Thus, Jl2 is the supremum of the Szego measures v, v ~ Ji. This proves also the uniqueness of the decomposition (8.3.18). The Theorem is proved. 0 The measure Jll is called the Szego singular part of Jl, and Jl2 the Szego part of Jl.
8.4. The Wold decomposition· Let A be a function algebra on X and Jl a representing measure for A. Recall we denoted A Jl 13 - c. 437
=
J
{f E A : f dl1 = O}.
Function algebras
194
In the following we suppose that Jl has the uniqueness property, i.e. any representing measure which coincides with J1. on A is equal to J1.. The subspectral representation f ~ Tf of A on the Hilbert space H is said to be J1.-spectral if elm [U TfH] I
= H.
f::. A,l
Proposition 8.21. Let f ~ Tf be a J1.-spectral representation of A on Hand M C H a doubly invariant subspace. The representation f - 4 Tjl-I\{ is a It-Spectral representation of A on M. Proof. According to Proposition 8.16, f ~ TflM is subspectral. Let m E M, orthogonal to [U TJ-A-/]. For any f E A JL and h E H of the form h = hI
+ h2'
lEA"
with hI E M, h2 E MJ., we have
Hence m is orthogonal to
which is equal to H, therefore m = O. 0 The subspectral representation f ~ Tf is called /l-completely non spectral if for any doubly invariant subspace M, for which f -+ TflM is jl-spectral, we have M = {OJ. Theorem 8.22. Let f ~ Tf be a subspectral representation of A on H and Jl a representing measure of A with the uniqueness property. The space H has a unique decomposition of the form
such that HI' H2 are doubly invariant subspaces for f ~ Tf , f ~ TflHI is a J1.-spectral representation andf ~ TflH2 is a J1.-completely Ilonspectral representation. HI is the largest doubly im'ariant subspace for which f~ TflM is a J1.-spectral representation.
Ch. 8. Elements of spectral theory
195
Proof. Let us write N
= [U
TfA]-L.
ftAp.
Let qJ -+ Utp be the spectral dilation off -+ Tf and (J1k h kzh 1 • k2 E K the spectral family attached to qJ -+ Utp (K is the space of spectral dilation). For any n E N andf E All we have:
according to the definition of N. Since
using the uniqueness property of j1, there results (n EN).
8.4. 1) Let us put HI
elm [U TfN].
=
lEA
It is clear that TfHI CHI for any fE A. If n E Nand g for any hE H we have:
E
All' then
hence
T*n g
(8.4.2)
=
0
(n EN, g E All)'
It results that Tt N C N for any g E A. Let now h E HI be of the form h = Tfn with n E Nand f EA. For any g E A we can choose a gk with fk' gk E A such that sequence fk
+
lim Slfk k -+ x"
+ Kk -
fil 2dJl
=
0,
Function algebras
196
since A
+ A is dense in L2(dJl) (Corollary 5.19). Since
we obtain lim (Tfk n
k .... oo
+ T:kn) =
T:Tfn.
Hence T:h
=
Tg*1jn = lim (Tfkn k .... oo
+ Tg~n) E HI
thus the subspace HI is doubly invariant. Let us write
where HI and H2 are doubly invariant subspace. We have
Indeed if m E HI is orthogonal to
then for any f E All and h2 E H2 we have
H of the form h
hE
therefore m is orthogonal to elm
[U
TfH]
fEAt~
which contains HI_ i.e. m = O.
= hI
+ h2
with
hI E
HI'
Ch. 8. Elements of spectra] theory
197
It results that I -+ T,IHI is Jl-spectral. Let M be a doubly invariant subspace of H such that
clm
[U
T,M]
=
M.
fEA/.1
I
E
Then it is easy to see that M is orthogonal to N. Then for any A, m E M and n E N we have (m, T,n)
= (T,*m, n) = O.
Hence M is orthogonal to H2 i.e. M C HI. Therefore HI is the maximal doubly invariant subspace M for which I -+ T,IM is Jl-spectral. It is now clear that I -+ T,IH2 is Jl-completely non-spectral. Let H= H{ €a H~
be another decomposition of H as required by the theorem. By the maximality of H2 we have H; C HI. Let us put
I t is then clear that M is a doubly invariant subspace and, since M C HI, from Proposition 8.21 there results thatl -+ T,IM is Jl-spectral. Since M C H~ we conclude that M={O}. Thus H; = HI, H~ = H 2 • 0
PropositioD 8.23. II Jl is not a point measure and M is a spectral subspace of I -+ T, then f -+ Til M is Jl-spectral. Proof. Let
n
and assume there is an hEM N, II h II = 1. Let Mil be the closed subspace generated by the elements of the form Tfh, f EA. As in the case of H2 from the previous theorem, we can prove that Mil is doubly invariant to f -+ Tj • As Mil C M and M is spectral, Mil is also spectral
Function algebras
198
and, according to Proposition 8.18, there exists a Borel set E, J1.h = = J1. - essentially unique, such that
where V is the operator defined on M h , with values in L2(dJ1.), by
But, clearly, VMh C H2(dJ1.), and therefore XE L2(dJ1.) C H2(dJ1.). According to Theorem 8.14, J1 is a Szeg6 measure, hence J1.(E) = O. There results Mh = 0 which contradicts the fact that hE Mh and IIhll = 1. Therefore
Now, let m
E
M be orthogonal to
For any h E H of the form h andl E AJl we have
This means that
mE
[U
=
hI
+ h2'
TfH~.l = N,
with hI
hence
mE
E
M, h2
M
nN=
E
Ml.
{O}.
fEAl1
I.e. m = O. 0
Corollary 8.24. Let 1 -+ Tf be a subspecfral representation 01 A on Hand J1. a measure which is not a point measure. If 1 -+ Tf is spectral, then it is also Ii-spectral. If.f -+ Tf is Ii-completely non-spectral, then it is also completely nonspectral. A subspectral representation 1 -+ Tf of A on H, J1.-spectral and completely non-spectral, will be called J1.-singular. Theorem 8.25. (Wold-Helson-Lawdenslager). Let f -+ Tf be a subspectral representation 01 A all Hand Jl a representing measure (l1,'hich
Ch. 8. Elements of spectral theory
199
is not a point measure) for A with the uniqueness property. H admits a unique decomposition under the form
such that H t , H 2 , H3 are doubly inl'ariant subspaces to f -+ ~,f --+ TflHt is spectral, f -+ TflH2 is p-singular and f --+ TflH3 is p-completely non-spectral. Proof. The proof follows immediately by applying successively Theorem 8.6, Corollary 8.24 and Theorem 8.22. 0 The following theorem furnishes some characterizations of natural representations and gives the reason for which Theorem 8.24 was called the Wold Theorem.
Theorem 8.26. Let f --+ Tf be a subspectral representation of A on Hand p a non-point mass representing measure of A with the uniqueness property. The following assertions are equivalent. a) There exists a Hi/bert space N such that f -+ Tf is unitarily equivalent to the natural representation of A on H2(N; dp). b) The representation f -+ Tf is p-completely non-spectral c) If M is a closed subspace of H such that Tt M C M for any fEA and -
M C elm [ U ~H], lEAp
then M = {OJ. d) If M is a doubly invariant subspace of H with M C elm
[U TfH] lEAp
then M = {OJ. e) If M is a doubly invariant subspace such that M
n [U
fEA/1
then M = {OJ.
TfH]J. = {OJ
Function algebras
200
Proof. Let us prove the implications a) -+ c). For this, let f -+ T, be the natural representation of A on H2(N; dJl). First we shall show that
[U
N =
(8.4.3)
T,H2(N; dJl)]l.
JEAJl
Indeed if n E N,f E AI" g
(T,gm, n)
E
A and mEN, we have:
= (fgm, n) = I (f(x) g(x) m, n) dJl(x) =
=
(m, n) Ijg dp = O.
Conversely, if hE H2(N; dp) is orthogonal to Nand
[U
T,H2(N; dJl)],
fEAJl
then for any n EN and g
E
A, if we put
f
= g -
I gdp,
we obtain (h, gn) = (h,jn) =
(h, 1Jn)
+ (h, I gdf.l n) =
+ I gdp(h, n) =
O.
Let now M be a closed subspace of H2(N; dJl) as in c). From (8.4.3) M results orthogonal to N. Then for any m E M, f E A and n EN we have
(m, fn) = (m, T,n) = (T/m, n) = 0 i.e. m = O. Hence M = {O} and implication a) -+ c) is proved. c) -+ d) and d) -+ b) are obvious. b) -+ e). Let M be a doubly invariant subspace as in e). Then we have (8.4.4)
elm
[U 1jM] = fEAJl
M.
Ch. 8. Elements of spectral theory
201
Indeed, if m E M is orthogonal to
then for any hE H of the form h = hI andfE A we have
+ h2'
with hI EM, h2 E Ml,.
Thus m
E
[U
TfH]l
f.:. A Il
nM
i.e. m = O. Hence M verifies (8.4.4) and from b) we obtain M = {O}. This completes the proof b) -+ e). e) -+ d) is obvious. To complete the proof of the theorem it remains to prove b) -+ a). ]f b) is true then from Theoreln 8.22 there results that
where
Let (Ilk l • k2 )k l • k2 eX be the spectral family attached to the spectral dilation qJ -+ U" off -+ Tf . Using (7.1.3) and (8.4.1), for any n, mEN, We obtain
Function algebras
202
Then for any
hE
H of the form ~ Tfin j • with
Ii E A
and ni
E
N,
i=1
we have:
p
I;
-
p
IliljdPninj =
p
-
~ Ililini' nj) d/l = I ~ lilini' n) dp = i, j=l
i,j=l
-
i,j=l
Now it is clear that the map TIn ~ In may be extended to an isometric operator U from H on H2(N; dp) such that ({EA, hEH).
This completes the proof of the theorem. <) The following theorem completes Theorem 8.26 in the case of onedimensional N.
Theorem 8.27. The subspectral representation f ~ Tf of A on H is unitarily equivalent to the natural representation of A on H2 (N; d/l) with N-one-dimensional, if and only if it is not p-spectral and has no proper doubly invariant subspaces. Proof. Letl ~ Tf be the natural representation of A on H2( N; dJl) with N one-dimensional, and M a doubly invariant subspace. If M N = 0 then from Theorem 8.2.6 there results M = {a}. If M N ¥= 0 then M N = N since N is one-dimensional, hence M = = H2( N; dJI).
n n
n
Ch. 8. Elements of spectral theory
203
Now, let I -+ Tf be a subspectral representation which is not ,u-spectral and has no doubly invariant subspaces. Since H
n [U
TfH:l. ¥= 0,
JEA'I
according to Theorem 8.26 point e), there exists a Hilbert space N such that I -+ Tf is unitarily equivalent to the natural representation of H2(N; d,u). Let NoC N be one-dimensional and M be the closed subspace spanned by the elements of the form In, I E A, n E N. Repeating the arguments used in Theorem 8.22 for proving that H2 i~ doubly invariant, we can show that M is a doubly invariant subspace. Since M :1= 0 we have H = M = H2(N; d,u). 0
8.5. Decomposition with respect to Gleason parts LetA be a DirichletalgebraonX and {Ga}aEJ the family of Gleason parts of the maximal ideal space of A. For any (l E J let ma be a representing measure for an element qJa of Ga. Since A is a Dirichlet algebra on X, m(1. is the unique representing measure, with support in X, for CPa' According to Proposition 6.5, for (l :1= p, m(1.' mp are mutually singular. Let H be a Hilbert space and I -+ Tf a representation of A on H. As A is a Dirichlet algebra, there exists a unique spectral dilation cP ~ UqJ of I -+ Tf , hence there is a unique semispcctral family (J1h, k)h, k EH on X, such that (Til, k) = Sld,uh,k
(I E A; h, k
E
H).
If P is the orthogonal projection of the dilation space on H then, ,uh,k are defined by
(g E C(X); h, k
E
H).
We write, as usual, ,uh = Ph. h' The representation 1-+ Tf is called Ga-continuous if for any h E H, I1h is absolutely continuous with respect to ma' The representation f ~ Tr will be called singular if for any (l E J and h E H, ma and I!h are mutually singular measures. Following Theorem 3.12, if CPI' qJ2 E Ga and m l , m 2 are their representing measures, with support in X, then m l and m 2 are mutually absolutely continuous. Therefore, the notions of G(1.-continuity and singularity do not depend on the particular choice of the elements qJa in Ga·
Function algebras
204
Theorem 8.28. Let A be a Dirichlet algebra on X and {G«}« EJ the family of Gleason parts of the maximal ideal space of A. Let H be a Hilbert space and f ~ Tf a representation of A on H. Then, H has a unique decomposition under the form
where H«, ~ E J, and Ho are doubly invariant subspaces, f ~ TfIH«, ex E J, is G«-continuous and f ~ TflHo is singular. Proof. Let (/lh, Ie)", Ie E H be the semi spectral family attached to the representation f E T r . For any ex E J we fix the representing measure m« of an element qJ« in G«. Let (h, k
E
H)
be the Lebesgue decomposition of the measure /l", Ie with respect to m«; m'h.k is the absolutely continuous part of /lh, k relative to mer and U".k the singular part. Since (/l", ,J", k E H is a semispectral family on H, one easily verifies that (m".k)",kEH and (O'''.k)",leEH are also semispectral families on H, for any ex E J. Let qJ ~ r; be the positive definite map of C(X) in L(H) attached to the semi spectral family (m".k)" , Ie EH
(T:h, k)
=
JqJ dm".k
(qJ
E
C(X); h, k
E
We have
+ Jfg dUh.k
(Tfgh, k)
=
Jfg d/l h, Ie = JIg dm".k
(Tfgh, k)
=
(TfTgh, k) = J/d/lTgh,k = J/dmTg",k
+
H).
Ch. 8. Elements of spectral theory
205
for f, g E A. Therefore, the measures
g d mh, k + g d (J h,a k -d mra gh, k - d (JTa g h, k (l
are orthogonal to A. According to F. and M. Riesz Theorem (Theorem 5.6) the absolutely continuous and singular parts of these measures, relative to ma , are orthogonal to A too. Then, we have
I fg dm~ k ,
=
I g dm: T* k ,
I
(8.5.1 )
(f, g E A; h, k
E
H)
and
(8.5.2)
(f, g
E
A; h, k
E
From (8.5.1) we get
If dm;ah k + If d(J;ah k - Ifg dm hu, k g , g ,
(TfT~h, k) - I fg dm h, k =
=
If dflT;h, k
=
(T;h, Tjk) - Ifgdm:,k = I gdm~. T-k - Ifgdm~,k = 0
-
Ifg dmh,k
=
=
f
for any I, g EA. Hence, the measure
is orthogonal to A and, F. and M. Riesz Theorem yields (8.5.3)
(f, g
E
A; h, k
E
H).
H).
Function algebras
206
Now from (8.5.3) there results
for any f, g
E
A, h, k
E
H, that is
(f, g
(8.5.4)
E
A).
Therefore, the restriction of qJ ~ T; to A is multiplicative. In particular Tf = P':J. is a projection on H. Using (8.5.1) and (8.5.3) we obtain
=
I dmh,k = (T;h, k)
(P(1.Ti1, k) = (T~Tfh, k)
= I dmT1h,k = If dmh,k =
(l~h, k)
which yields (8.5.5)
(fE A)
P(1. TaJ = Tap J
(8.5.6)
(1.
=
TaI
(f E A).
If we· put Ha = PaH, then from (8.5.5) and (8.5.6) Ha results doubly invariant to f ~ Tf and j ~ T;, and 1JIHa. = T~IHa' It is also clear that f ~ T;IHa is a Ga -continuou5 representation of A on H(1.' We now show that P(1. Pp = 0 for tX ::j= p. We have
Sfdm;qh,k -+- Sfda;qh,k - Sfdm!,k
=
S jdllTfh,k - Sjdm!.k
= (TfTfh, k) - Sf dmZ,k = (Tfh, T:k) - Sf dm~, k =
S
dm!. 1;' - Sf dm~,k = o.
=
Ch. 8. Elements of spectral theory
207
From (8.5.1), the measure a
{J
a
mTPh k 1 '
+ (JT/Jh k 1 '
-
mh
,
k
results orthogonal to A. Then, using again F. and M. Riesz Theorem and taking into account that ma and mp are mutually singular, we get
(I E A).
(8.5.7) Hence
for any h, k
E
H, that is
Therefore the subspaces HIZ are pairwise orthogonal. Let
As each HfZ is doubly invariant to f~Tf' Ho results doubly invariant to 1 ~ Tf too. We know that f ~ TflHa, (1 E f is Ga-continuous. Let us prove that f -+ TflHo is singular. First we observe that from (8.5.3) there results
Jdm~, h
=
Jdm;~h, h = J dm~ah,h
=
Jdm~.h =
0
for any h E Ho and, as m h is a positive measure, m h = 0 for any h E Hoo This means that 1 ~ TflHo is singular. There remained to prove that the decomposition is unique. Let H = ffi H~ ffi H~ aE,!
Function algebras
208
be another decomposition of H as required in the theorem, and let p~, p~ be the orthogonal projection of H on H~, H~ respectively. Then I1P'h P'k a'
a
=
m;'h P'k a'
a
(cc
E
f; h. k E H)
(8.5.8) At the same time, since follows
H~
are doubly invariant to f
j fdl1h,k =
(Tfh, k) = 0
Sf dl1k,h =
(Tfk, h)
-.
Tf , there
= 0
for any h E H~, k E H~l andf EA. Therefore I1h.b 11k ,h are orthogonal to A, and applying F. and M. Riesz Theorem we obtain (8.5.9)
Sfdmh,k
=
Sfdm'k ,h =
0 (f E A; h E H~; k
E
H~l.;
Using (8.5.8) we have
=
Sdl1 p a'h' P'k a
= (P~h, P~k) = (P~h, k)
that is (8.5.10) Now, from (8.5.9) and (8.5.10) we obtain
C! E
J).
Ch. 8. Elements of spectral theory
209
and therefore (8.5.1) ) From (8.5.11) we have
Quite similarly, one proves that
which gIves PrJ
= P(I; (P~
+ 8:t=a ~ Pp + P~) =
P2P~ = P~.
The theorem is proved. 0
Notes Theorem 8.6 has been proyed by C. FOIA~, I. SUC(U [1 J. The same work presents for the first time the notion of spectral dilation of a function algebra representation and the existence and uniqueness theorem of a spectral dilation for representation of Dirichlet algebras (Theorem 8.7). Actually Theorem 8.13 is given in C. FOlA~ and I. SUCIU [2] in a form including Proposition 8.14. The present discussion in § 8.2 is self-contained, avoiding the theory of p-summing operators, which was used in the above quoted Note. The Szego measures have been introduced by C. FOIA~ and I. SUCIU [I]; the same paper contains the proofs to Theorems 8.15 and 8.16. The connection between completely non-spectral representations and Szego measures is thoroughly studied in W. MLAK [5]. Subspectral representations have been introduced by I. SUCIU [10]. Theorem 8.20 has been proved for natural representations by C. FOIA~ and I. SUCIU [I J. In I. SUCIU [10] it appears under the same form as presented here. Jl-spectral, Jlsingular and j.l-completeJy non-spectral representations as well as the Wold type decomposition theorem for subspectral representations (Theorem 8.25) can be found 14 - c. 437
210
Function algebras
in I. SUCIU [II]. The characterizations of natural representations contained in Theorem 8.26 have been proved in the same work. Ga-continuous and singular representations have been introduced by W. MLAK [6]. In [6], W. MLAK proves Theorem 8.28 for any function algebra. Important contributions in this direction can also be found in D. GA~PAR [2]. As concerns the problem of the existence of a spectral dilation for function algebras representations, a quite important result has been obtained in W. ARVESON [1], [2]; it is proved (as a particular case of a more general result for C*-algebras) that a representation f ....... Tf of A on H has a spectral dilation if and only if it is completely contractive, i.e. if and only if for any n = 1,2, ... and any 11 >< Il matrix (/;) over A, with 1(/ij):: ~ 1, we have II(Tfij)ir ~ 1. The example given in chapter 10 (Corollary 10.11), based on PARROTT'S example [l J. shows that there exist representations of function algebras which are not completely contractive, hence have no spectral dilation.
CHAPTER 9
Elements of prediction theory on S-generated algebras
9.1. Semigroups of contractions Let G be an abelian group. We write 1 for the unit element of G. A unitary representation of G on the Hilbert space H, is a map g -+ U g of G in L(H) such that UI = I, Ug-l = and Ug1g2 = Ug1Ugs for any g, gI' g2 E G. Obviously, Ug is a unitary operator on H for any g E G. Let g -+ Tg be a map of G in L(H). A unitary dilation of the map g -+ Tg is a unitary representation of the group G on a Hilbert space K such that H C K and
U:
(9.1.1)
(h
E
H, g
E
G)
where P is the orthogonal projection of K on H. The unitary dilation is called minimal if the space K is the closure of the set spanned by the elements Ugh, h E H, g E G. In the following we shall assume every unitary dilation to be minimal. If g -+ Tg admits a unitary dilation, then it is clear that Tl = I, Tg-l= Tg*, II Tgil ~ 1 for any gEG. Theorem 9.1. (M. A. Naimark-B. Sz.-Nagy). Let g -+ Tg be a map of G in L( H), which satisfies Tl = I, Tg-l = T;, for any g E G. The map g -+ Tg admits a unitary dilation if and only if for any finite
Function algebras
212
system gh"" gn of elements in G and any finite system "1"'" hn of elements in H, we hare
(9.1.2) Proof. Let g gl,"" gn E G and
~
Ug be a unitary dilation of g hn E H, we have
"1' ... ,
~ (Tgj-1 g I h·J' h.)= ~ (PUg,-1 g j h·J' h.) ~ I ~ I
'.J
-+
T g • For any
=
I. }
Therefore (9.1.2) is a necessary condition for the existence of the unitary dilation of g -+ T g • Now assume that g -+ Tg satisfies (9.1.2). For h E H, we define on G the real function Ph' by (g E G).
From (9.1.2) there
re~ults
for any finite system c1 , .... en of complex numbers. This means that Ph is a positive definite function on G and then, according to HerglotzBochner-Weil theorem. there exists a positive measure I1h on the dual X of the discrete group G, such that (gE G)
where g is considered as a character on X.
Ch. 9. Elements of prediction theory
One easily verifies that
213
Ilphll = Ilhl! and
Writing (h, k
E
H),
by usual arguments one shows that (p" "J".k EH is a semi spectral family on X. Let qJ -+ TfP be the positive definite map attached to this family (Theorem 7.1) and q, -+ UI{J the spectral dilation of qJ -+ T., (Theorem 7.5). It is ea~ily verified that g -+ Ug , is a unitary dilation of g -+ T g • The proof is complete. 0 The map g -+ Tg of G in L(H), which verifies (9.1.2) will be called a positive definite map. From the proof of the preceding theorem there also results the following: Proposition 9.2. Let G be an abelian group and X the dual of the discrete group G. There exists a one-to-one correspondence between the set of positive definite maps g -+ Tg of G in L(H), and the family of semispectral measures (F(u)) a EB(X) on X, given by the relation (Tg h. k)
= Jg(x) d(F(x)h, k)
The map g-+ Tg is a unitary representation is a spectral measure.
(h, k
E
H).
if and only if (F(u)) a EB(X)
Corollary 9.3. Let G be an abelian group, X the dual of the discrete group G and g -+ Tg a positive definite map of G in L( H). Then for any finite system of constants c1 , .. ·, Cn and allY finite system gh"" gn oj elements in G, thefol/owing inequality (Von Neumannformula)
I LCi T ig I ~ sup I Lcigi(x) I xeX
holds. The umqueness theorem for minimal dilations can be given as Theorem 9.4. Let g -+ Tg be a positive definite map of G in L( H) and g -+ Ug' g -+ U; two minimal unitary dilations acting on K and K',
Function algebras
214
respectlively. Let V, V' be the embedding operators of H in K and K', respectively. There exists a unitary operator U: K ~ K' such that UV = V' UU g
=
(g
U'U g
E
G).
Proof. The proof follows from Proposition 9.2 and Theorem 7.6 0 We establish now a theorenl which gives a sufficient condition for (9.1.2) to hold.
Theorem 9.5. Let G be an abelian group. S eGa subsemigroup of G with snS-1 = {I}. Let g ~ Tg be a map of G in L(H) such that Tl = I, Tg-l = Ti, IITgll ~ 1 and (9.1.3) Then g ~ Tg satisfies (9.12). Proof. Let gl,"" gn be a finite system of elements in G, with gi ¥= gj for i '# j. If we choose minimal elements relative to the order induced by S in G, we may rearrange this system, such that
i < j,
(9.1.4) Let
T=
(Tij) be the operator matrix given by Tij = Tg~l , gj'
T is
n
an operator acting on EE> Hk(with H" = H for any k). k=l .......
n
If h = (h k ) is a vector in EE> Hk, then k=l
""-
hence (9.1.2) is equivalent to the positivity of the operator T. We show that T = W*DW, where and 15 are operators on
tv
n
""-
EE> Hk and D is positive. k=l
Ch. 9. Elements of prediction theory
215
We put Wij
={
T gj-1 gj.. if i
o
~
j
if i > j
and
Du = I D·· 'J
then
W=
= 0 if i:l=}'
(Wij),
D=
(Dij).
Since Tg are contractions, jj is, clearly, a positive operator. We now show that T = As T and W* DWare self-adjoint operators, it is sufficient to prove that Tij = (W* D W)ij for i ~ j. At the same time (W*)ij = ~~ = 0 for j > i, hence
w*nw.
........,
-. . . . . .
(w*nW);j
=
i.................. .........
~ (W*)iiDW)kj. k=1
For i
=
1 we have
and for i > 1 ........ . . . . . --..
i
.....
(W*nW)ij = ~ (W*)UDllWIj 1=1
=
Function algebras
216
But from (9.1.3) and (9.1.4) there results
T*
~-1
,
g i
T* g-1
- T* 8-1 g
g -
1-1'
1-1
i
and therefore
This is nothing else than
which yields
(t'h, h) = (w*n wh, h) = (DWh, Wh) ~
o.
Thus, T is a positive operator, hence g --+ Tg satisfies (9.1.2). The theorem is completely proved. <> Let G, S, H be as above. A semigroup of contractions on H is a map s --+ Ts of S in L(H), such that Tl = I, TSI TS2 = T S1S2 ' for any SI' S2 E S and II Tsil ~ 1 for any s E S. A unitary representation g --+ Ug of G on a Hilbert space K, IS called a unitary dilation of the semigroup {Tsh eS, if H C K and (s E S, hE H),
where P is the orthogonal projection of K on H. Corollary 9.6. If G is totally ordered by S, i.e. if G = SU S-l , then allY semigroup {Ts}s eS of contractions on H has a unitary dilation.
Ch. 9. Elements of prediction theory
217
Proof. Indeed, jf we write
g=
SE
S
then we obtain a map g -+ Tg of G in L(H) which, obviously, satisfies the conditions of Theorem 9.5. Therefore g -+ Tg verifies (9.1.2), hence admits a unitary dilation. Theorem 9.7. Let G, S, H be as written above, with G - SS-l. Let {Ts}sES be a semigroup of contractions such that
(9.1.5) and
T*T =STG T* G S
(9.1.6)
for
SO'-14q;; S U S-1 •
Then there exists a unitary dilation of the semigroup {Ts} S ES' Proof. Let us write
T:TS { T= g
T* G
if g =
0'-1, O'E
S.
According to (9.1.5), the mapping g -+ Tg is well defined and Tl = I, Tg-l=T:, IITgll ~ 1. We now show that it satisfies also (9.1.3). Letgb g2 EG be such that gl' g2, glg~ ~ S-I. We put gl = slO'l\ g2 = SO'il. Since gh g2' g1g2 ¢ S-1 we have
If gh g2
E
S, then (9.1.3) is clearly satisfied. Suppose g2 ~ S. Then
Function algebras
218
and, using (9.1.6), we obtain (9.1. 7) Now taking into account (9.1.7) we get
Similarly, we have
Therefore (9.1.3) is verified by the map g 4> Tg • Now by Theorem 9.5, here exists a unitary dilation for the semigroup {Ts} s ES' 0 Let G, S, H be as written above and {Ts}sES a semigroup of contractions on H. A closed subspace M of H is called invariant (to {Ts}s ES) if TsMCM for any s E S. The subspace M is said to be doubly invariant if TsMCM, T;MCM for any s E S. If M is invariant, we denote by TslM the restriction of Ts to M and by {TsIM} s ES the corresponding semigroup of contractions. The semigroup of contractions {TsL ES on H is called unitary if the operator Ts is unitary on H for any s E S. The semigroup {Ts}sES is called completely non-unitary if for any doubly invariant subspace M of H for which {TsIMLES is unitary we have M = {OJ. Theorem 9.8. Let {Ts} s ES be a semigroup of contractions on H. The space H has a unique decomposition under the form
such that Hu and He are doubly invariant, {TsIHJsES is unitary and {TsIHcJ s ES is completely-non-unitary. Proof. Let.A be the family of doubly invariant subs paces M of H, with the property that {TsIM}sES is unitary.
Ch. 9. Elements of prediction theory
219
U M.
One easily verifies
Let Hu be the subspace generated by
MEV«
that Hu is doubly invariant and {TsIH,,} s ES is unitary. Hence Hu is the largest doubly invariant subspace M, for which {TsIM}sES is unitary. Let
He results doubly invariant. Let M C He be double invariant, with {TsIM}sES unitary. Then MCH" and, since it is orthogonal to H", we have M = {OJ. Therefore {TsIHe} s ES is a completely non-unitary semlgroup. Let
be another decomposition of H, with HI, H2 doubly invariant {TsIH1}sES unitary and {Ts IH2 } s ES completely non-unitary. Hu being maximal there results HI C H u • Let us write
Then M is doubly invariant and, as MC H u , {TsIM}SES results unitary. But M is orthogonal to HI' hence Me H2 and, as {Ts IH2 }sES is completely non-unitary, M = {O}. Thus Hu = HI and Hs = H 2 , i.e. the decomposition is unique. The theorem is therefore completely proved. <) g
-+
Theorem 9.9. Let {Ts} sES be a semigroup of contractions on Hand Ug one of its unitary dilations. Let
be the canonical decomposition of {Ts} s ES. We have (9.1.8)
H" = {h
H: U;
E
= {h EH: II TsT:hll
=
E
H,
hE
G}
Ilhll, s,
(J
= E
S}.
Function algebras
220
Proof. Let
For hE H2 we have
Hence Ush we get
E
IIPU/l1I
=
H, Uo-lh
E
liT./711
=
Ilhll
IIU)lII
=
H for any s, a
E
S. Then, for g = sa-I,
Therefore U/1 E H for any g E G, which yields H2C HI' HI is obviously a doubly invariant subspace and {TsIHl}sEs is unitary. As Hu is maximal, there results HI C Hu' Since TslHu is unitary for any s E S, there follows
for any h E Hu and
S E
S, hence Hu C H 2 • Then
which proves relation (9.1.8). 0 Remark. From (9.1.8) we obtain immediately HII
= {h E H: IIT:TJzII = IIhll, s, a E S}.
Ch. 9. Elements of prediction theory
221
9.2. The Wold decomposition In the case {Ts}s~s consists of isometries we obtain a decomposition theorem of the Wold type. First we establish that any semigroup of isometries has a unitary dilation. Theorem 9.10. (Ito) Let G be an abelian group, S eGa subsemigroup of G sud that G = SS-1, and H a Hilbert space. E,'ery semigroup of isometries {Ts} s ES has a unitary dilation. Proof. We write, for g E G of the form g = 0-- 1 S, as E S,
If we multiply at right by T~(12' we get
Therefore the map g ~ Tg of G in L(H) is well defined, and TI = I, Tg* = Tg-l. We now show that this map is positive definite. Let gl,"" gn E G, gi = (FilS;, Ui' Si E S, ;=1,2, ... , n, and hI"'" hn E H. Let s = UIU2" 'Un and g; = sgj, ; = I, 2, ... , n. We obviously have g/~ E S for any i, and g;lgj = g;-lg; for any i, j. There results
and then
222
Function algebras
Thus g -+ Tg satisfies (9.1.2). Therefore, according to Theorem 9.1, g -+ Tg has a unitary dilation which, obviously, is a unitary dilation of the semigroup {Ts} sES' 0 The semigroup of isometries {Ts} sES will be called quasiunitary if (9.2.1)
elm[
U
T;TsH]
o-lsEES- 1
= H.
The semigroup of isometries {Ts} sES is said totally-non-unitary if for any doubly invariant subspace M for which {Tsl M} is quasi unitary , we have M = {Ole It is clear that any unitary semigroup is quasiunitary, hence any totally non-unitary semigroup is completely non-unitary. It also happens that a semigroup of isometries be quasiunitary and completely non-unitary at the same time. Such semigroups are called singular. Let {Ts} s ES be a quasiunitary semigroup and M a doubly invariant subspace. Then elm [ U T; Ts M] C M. Let m E M be orthogo-
U
nal to elm[
a-1sEES- 1
T:Ts M] and hE H. If h = hI
a-1stES- 1
+ hz,
with hI E M
and h z E Ml., we have
for
(1-1
S
¢ S-I. Therefore m
orthogonal to elm [
IS
U
T:TsH] =
o-ISG'S-l
= H, i.e. m
=
O. Hence M = elm [
U
T:TsM],
o-lsEl:S- 1
which means {TsIM}sES quasiunitary. If M ic: a doubly invariant subspace, then {Tsli\1LES is unitary (quasiunitary, completely nonunitary, totally nonunitary, singular) if {Ts} s ES has the same property. Theorem 9.11. Let {TS}SES be a semigroup of isometries 011 H. The space H admits a unique decomposition under the form
Ch. 9. Elements of prediction theory
223
where Hq, H, are doubly inl'ariant subspaces, {TsIHq} s ES is quasiunitary and {TsIHtL ES is totally nonunitary. Proof. We put
(9.2.2)
N
=
U T;TsH]J..
[
so-lEES- 1
For
1l E
N,
hE
H and a-I
S
E£ S, we have
hence
(9.2.3)
Sn
T*T (] Sn
=
The subspace TsN S-1 = {I} we get
for any m, n EN. s, a E S, Let us write
(n
0
are
S
E
N, u- 1 S E£ S).
pairwise
orthogonal. Indeed,
SInce
i= u.
(9.2.4) and
Hq and Ht are doubly invariant subspaces. Indeed, Ht IS obviously invariant. If m E Hq and s E S, then from (9.2.3)
for any U E S for which a-Is E£ S and for any n then S = US 1 and
E
N. If u-1s =
Sl E
S,
Function algebras
224
for any n E N, following the definition of H q • Hence Tam is orthogonal to Ht' that is Tam E H q. Therefore Hq is also invariant, which implies that H t and Hq are doubly invariant. The semigroup {TslHqlses is quasi unitary. Indeed, since Hq is doubly invariant we have: clm [
U T:TsHq] C Hq.
o-lsEES- 1
Let
mE
form h = hI Then
Hq be orthogonal to [
+ h2o,
U T:TsHq]
and hE H of the
o-lsEES- 1
hI E H q , h2
E
Ht •
for any s, a E S, a-Is EE S-I. Hence m is orthogonal to elm [ U T:TsH] =:) Hq and therefore m = O. There results o-lsEES- 1
i. e. {TsIHqJs es is quasiunitary. Let now M be a doubly invariant subspace included in H t , such that {TsIM} s es is quasiunitary. Since M = elm [ U T:Ts M), M o-lsEES- 1
results orthogonal to N. We then have (m, T /1)
=
(T:m, n) = 0
for any In EM, n E Nand S E S. Hence M is orthogonal to He:J M, i.e. M = {OJ. Therefore {TsIH,}ses is totally nonunitary. Now let
be a new decomposition of H. where HI, H2 are doubly invariant. {TsIH}}sEs is quasiunitary and {TsIH2 }sES is totally nonunitary. As H}
Ch. 9. Elements of prediction theory
is doubly invariant and HI = elm [
U
o-lsEES- 1
T: TsHIl
225
it is orthogonal
to H t (as above), that is HI C H q • Write
Then M is clearly doubly invariant and, since Me Hq , {TsIM}sES is quasiunitary. But M is orthogonal to H, hence M C H2 and since, {Ts IH2 }sES is totally nonunitary, we have M = {Ole Therefore Hq = HI, Ht = H 2 • The proof is con1plete. 0 Now, if we combine Theorem 9.8 with Theorem 9.10 we ohtain the following generalization of the Wold decomposition theorem. Theorem 9.12. Let {Ts} s ES be a semigroup of isometries on H. The space H has a unique decomposition of the form
where Hu, H a, H t are doubly invariant subspaces, {TslHulsEs is unitary, {TsIHaLES is singular and {TsIHt}sEs is totally nonunitary. Proof. Applying successively Theorem 9.8 and Theorem 9.10 we get the decomposition. To prove the uniqueness it is sufficient to observe, for instance, that H a(£) H t is the completely nonunitary part of {Ts} s ES •
9.3. The semigroup of unilateral translations Let N be a Hilbert space; we denote by L2(N, G) the Hilbert space of families Ii = (h g ), g E G, hg E N, with the property
Ilhll ~ ~ I hg ll 2 <
00,
gE:G
with the usual operations and the inner product
(I;: {) =
~ (h g, kg), gEG
15 - c. 437
226
Function a]gebras
H2(N; S) will denote the closed subspace of L2(N; G) consisting of elements h = (h g ) E L2(N; G) with 17g = 0 for g E£ S. If P is the orthogonal projection of L2(N; G) on H2(N; S), then P(h g ) = (h;), where hg
if g
o
if g E£ S.
E
S
/7'g = {
The map n ~ (n g) defined on N with values in H2(N; S) by n g = 0 if g :j:. 1 and nl .- ti, is an isometric embedding of lv in H2(1V; S). For s E S we defined on H2(N; S) the operator (9.3.1) As we can easily verify, {Ts}sES is a semigroup of isometries on H2(N; S). We call this semigroup, the semigroup of unilateral translat ions in H2( N; S). If {Ts} sES is the semigroup of unilateral translations in H2(N; S) then for any s E S, we have (9.3.2) and
H2(N; S)
(9.3.3)
=
TsN. sES
Theorem 9.13. Let {Ts} s ES be a semigroup of isometries on H. The following assertions are equivalent. a) There exists a Hilbert space N such that the semigroup {Ts}sES is unitary equivalent to the semigroup of unilateral translations in H2 (N; S). b) The semigroup {Ts}sES is totally nonunitary. c) If M is a doubly invariant subspace of H. with M C
C clm [
U
c;-ls EES-l
Ta*TsH], then M
=
{OJ.
d) If M is a doubly invariant subspace of H, with M
n[
U
a- 1sEES-l
T:TsH]l. = {O}, then M = {OJ.
Ch. 9. Elements of prediction theory
227
Proof. The implications d) -4 c), c) -4 b) are obvious. b) -4 a). Let {TsLES be totally nonunitary. Then, from Theorem 3.5, there results
H=H=t.:pTN t ~ s
(9.3.4)
sES
where
N
U
= [
T:TsH]J..
o-lsEES- 1
Tht mappiIlg II
-4
(h g ), with hs = ns if h =
~
Tsns and hg = 0 for
sES
g e S, is a unitary equivalence between Hand H2 (N; S) ; it transforms the semigroup {Ts} s ES in the semigroup of unilateral translations in H2(N; S). Therefore b) -4 a) is proved. a) -4 c). Let N be a Hilbert space and {TsLes the semigroup of unilateral translations in H2(N; S). We first prove the relation
N
(9.3.5)
=
[
U
T:TsH2(N; S)]l
o-lsEES- 1
where N is embedded in H2 (N; S) as seen before. = (n g), ng = 0 for g:f= I. Using (9.3.2) we have Let If E N,
n
where
, I1g
=
{ns-1Ug if gE S
o
if g
e S.
Hence ng = 0 for g :f= I and therefore n; = 0 for g ::j= u- 1s. Then, for n EN and u- 1 s e S we get (9.3.6) This yields
Function algebras
228
for s,
(J E
S,
S(J-l
e S.
Hence
NC [ Conversely, if Ii
=
U
T:TsH2(n; S»).L.
a-1sa;:S-1
(ng)
U
E [
T:TsH2(N; S)].L then we have
a-1Sa:S- 1
for any hE H2(N; S),
0'-1 S
e S.
Therefore
T:Tsli = 0 for any s, (J E S, S(J-l e S. In particular T:fi = 0 for any (J E S, (J'1= 1. If T:;Z = (n;), then n; = 0 for any g E G. But n(1 = n~ = 0; hence n(l =0 for any (J t/= 1, which yields n EN. The relation (9.3.5) is proved. a doubly invariant subspace included in Let now M be clm [ U T:TsH2(N; S)]. From (9.3.5) there results M C N.L and, a-1sQ;S-1
since M is doubly invariant, we obtain
Therefore M is orthogonal to TsN for any there follows M = {O}. b)
~
S E
S and, from (9.3.2),
d). Let M be a doubly invariant subspace of H, such that
(9.3.7)
M
n[
U
T:TsH].L
a-1sa:S- 1
=
{Ole
We have (9.3.8)
M
=
U
elm [
T:TsM].
a-'sEES- 1
Indeed, since M is doubly invariant, there results elm [
U
a-1SQ;S-1
T:TsM]
eM.
Ch. 9. Elements of prediction theory
Let m
E
h = hI
U
M be orthogonal to [
+ h2'
for any s, (I
E
T: TsM], and h
229
E
H be of the form
0-l s EtS-1
with hI
S, (I-IS
E
EE
M and h2
S-I.
E
Mi. Then
Hence m
U
E[
T*TsH]l·, and;from(9.3.7),
0-lsEES-1
there follows m = o. Therefore (9.3.8) holds. From (b) there results M = {OJ and b) -+ d) is proved. Since the following implications are true
a<
.
b
1/1 c(
d
the theorem is proved. 0 The following theorem is a completion to Theorem 9.13 in the case N is one-dimensional. Theorem 9.14. Let {Ts} s ES be a semigroup of isometries on H. The semigroup {Ts} sES is unitary equivalent to the semigroup of unilateral translations on a H2(N; S), with N one dimensional, if and only if {Ts} seS is not quasiunitary and has no proper doubly invariant subspaces. Proof. Let {Ts} s ES be the semi group of unilateral translations on H2(N; S), with N one-dimensional. Let M C H2(N; S) be a doubly invariant subspace of H2(N; S). If M
n[
U
T:TsH2(N; S)]-l = 0
0-ISa:S- 1
then from Theorem 3.7 there follows M = {OJ. Suppose M
n[
U
T:TsH2(N; S)]J. ¥= O.
o-lsEES-l
From (3.4.5) there results MnN ¥= {OJ. As N is one-dimensional, we have MnN = N, therefore H2(N; S) = ® TsN C M. se:S
Let now {Ts} s ES be a semigroup of isometries on H which is not quasiunitary and has no proper doubly invariant subspaces. Then, clearly {Ts} s ES satisfies point d) of Theorem 9.] 3. Hence {Ts} s ES is uni-
230
Function algebras
tarily equivalent to the semigroup of unilateral translations in an H2(N; S). Let 110 E N. no t::/= 0 and let No be the one-dimensional space spanned by no. Let us write
M is a doubly invariant space. Indeed, if m =
T(Jm T:m =
=
~ SES
~ sES
T /1s with nsENo, then
Tsa ns E M
~
TSlnS
E
M.
SI =o-IS
Since M nN = No =1= 0, we have H N = No is one-dimensional. The theorem is proved. 0
= M,
which implies that
9.4. Representations of S-generated algebras Let A be an S-generated algebra on X such that SnS-1 = {I}, H a Hilbert space and f -4 Tf a representation of A on H. Then it is clear that {Ts} s ES is a semigroup of contractions on H. Moreover, for any finite systems Sl,"" Sn E Sand C1 ,., " en E C, we have (9.4,1) hence the representation f -4 Tf is generated by the semigroup {T} s E s' Conversely, if {Ts}sES is a semigroup of contractions on H, which satisfies (9.4.1) then, if we write
and extend by continuity, we clearly obtain a representation on A on H.
f
-4
Tf
Ch. 9. Elements of prediction theory
231
Theorem 9.15. (i) The subspace M of H is spectral for ! ~ Tf if and only if M is doubly invariant to {Ts} s ES and the semigroup {Tsl M} s ES is unitar)'. (ii) The representation f ~ Tf is spectral if and only if {Ts} s ES is unitary. . (iii) The representation ! ~ Tf is completely non-spectral if and only if {Ts} s ES is completely non-unitary. (iv) The representation f ~ Tf and the semigroup {Ts} s ES hOl'e the same canonical decomposition. Proof. (i) Supose M is spectral for f ~ Tf and let cp ~ UqJ be the extension to C(X) of the representation f ~ TfIM. Then, clearly, M is doubly invariant to {Ts} s ES and
for any m E M and s E S. Therefore {TsIM}sES is unitary. Assume now that M is doubly invariant to {Ts}sES and {TsIM}sES is unitary. For f E A of form! = ~CiSi we have TfM C M, TtM C M and by continuity, M results doubly invariant to f ~ Tf . For a function cP E C(X) of the form cp = l:CiS;S and m E M, we write -
hence
for any m E M. We used here the fact that Tsl M is unitary for any s E S. Moreover
I UqJmll
=
IlL cjTsjTs*mll ~
~
IlL c;Tsjll I Tsmll
=
IlL cjTsjll Ilmil
~
IlL CjSj Ilmil = IlcpII Ilmll·
Therefore the map cp ~ UqJ of C(X) in L(M), which is well defined on elements of form cp = l:CjSS, is linear and continuous. We can
232
Function algebras
easily verify that cp -+ VII' is also multiplicative. Since the elements cp of the form cp = I:cjSjS are dense in C(X) , cp -+ VII' may be extended to a representation of C(X) on M which, obviously. is an extension to C(X) of the representation f -+ TfIM. Hence, the subspace M IS spectral for f -+ Tf . (ii), (iii) an (Iv) are immediate consequences of point (i). 0 We shall further assume that S is analytic free. According to Theorem 6.18 we are, in fact, in the following context: X is the dual of the abelian discrete group G = SS-1 and A is the algebra A(S)-generated in C(X) by the elements of S, viewed as characters on X. Theorem 9.16. Let A be an S-generated algebra on X and f -+ Tf a representation of A on H. The representation f -+ Tf has a spectral dilation if and only if the semigroup {TsJs ES has a unitary dilation. Proof. Let cp -+ VII' be a spectral dilation of f -+ Tf . Then the map g -+ Ug is a unitary dilation of the semigroup {Ts} s ES. Let now g -+ Ug be a unitary dilation of the semigroup {Ts}ses. From von Neumann formula (CoroIIary 9.3) there results
for any finite system C1 ,. '.'Cn of constants and any finite system gl,.'.' gn E G. For cp of the form cp = I:CjSjS, s;, S E S, we may define
and, using the same argument as in Theorem 9.15, we obtain the representation cp -+ Utp of C(X) on K (the dilation space), which is a spectral dilation of the representation f -+ 1f. Theorem 9.17. The representationf -+ 1j of the S-generated algebra A is subspectral if and only if the semigroup {Ts} s ES is isometric. Proof. Assume f -+ Tf is subspectral and let cp -+ Utp be its spectral dilation for which Ufh E H for f E A and h E H. We have
for any s E Sand hE H. Hence Ts is an isometry on H for any s E S.
Ch. 9. Elements of prediction theory
233
Conversely, assume Ts isometric. Any semigroup of isometries admits a unitary dilation (Theorem 9.10). Let {Ug}gES be the unitary dilation of {Ts} s ES. Then
hence Ush E H for any s E Sand h E H. Let cp --+ UqJ be the spectral dilation of f -+ Tf , constructed as in Theorem 9.16. For f E A, f = = I:.CjSj, we have (h
E
H)
and, therefore, it is clear that Ufh E H for any f E A and h E H. Thus the representation f --+ Tf is subspectral. Let m be the Haar measure of X. We say that the representation f --+ Tf of A on H is m-spectral (m-completely non-spectral, m-singular) if the semigroup of contractions {Ts} s ES is quasiunitary (totally non-unitary, singular). If G = SUS- 1 then, m has the uniqueness property and, as one easily verifies, these definitions are in agreement with those given in paragraph 8.4. Now, Theorem 9.11 may be transposed in a function language. Theorem 9.18. Let f --+ Tf be a subspectral representation of the S-generated algebra A on H. The space H has a unique decomposition under the form
where Hs, Ha, H t are doubly invariant subspace, f --+ TflHs is a spectral representation, f -+ TflHa is m-singular and f -+ TflHt is m-completely non-singular. In the same sense we write again Theorems 9.13 and 9.14. Theorem 9.19. The subspectral representation f -+ Tf of the S-generated algebra A on H is unitarily equivalent to the natural representation of A on H2 (N; dm) if and only if it is m-completely non-spectral. The representation f --+ Tf is unitary equb'alent to the natural representation of A on H2(dm) = H2 (N; dm), with None-dimensional, if and only if f -+ Tf is not m-spectral and has no proper doubly in-variant subspaces.
Function algebras
234
9.5. Prediction theorems Let A be a function algebra on X and m a representing measure for A. All along this paragraph A and m are supposed to verify one of the following assertions: (1) m is a representing measure for A with the uniqueness property. (2) A is an S-generated algebra, with S analytic free and nl is the Haar measure of X, viewed as the dual group of the discrete abelian group G = SS-l. Moreover, ifin (2) we assume G = SUS-l then, clearly, for A and m satisfying (2), the assertion (1) is also true. But, as we have seen (in the case of the HOC) algebra), A and m may satisfy (1) without satisfying (2), hence the two cases do not indude each other. In these two cases we enounced and proved a decomposition of Wold type for the subspectral representations. In the present paragraph we shall analyse the Wold decomposition for the natural representations of A on H2(dp,), where p, is a positive measure on X. The notions of m-spectral, m-completely nonspectral and m-singular representation will have the sense required by the situation.
Proposition 9.20. Suppose that A and m satisfy (1) (respectively) (2) and let p, be a positive measure on X with p, (X) = 1. If N = [ U Tf H2(dp,)JL (respectively N = [
U
felfm
Ts*TsH2(dp,)]l-) then dim N ~ 1.
a-1sEES- 1
Proof. In both situations one easily verifies that C + Nl. is dense in H2(dp,). Let P be the projection of H2(dp,) on C. We have Pent - n 2)=O for n1 , n2 E N, with Pnl = Pn 2 , hence (n} - n 2, c
+ n1.) =
(n} - n 2, c)
+ (n} -
n 2, nJ.) = 0
for any c E C and n1. E Nl.. Then n1 = n2 and therefore PIN is an InJection of N in C, that is dim N ~ 1. 0
Corollary 9.21. The natural representation f
Tf of A on H2(dll) it is not m-spectral and -+
is m-completely nonspectral if and only if has no proper doubly invariant subspaces. Proof. The proof follows from Theorem 9.19 and the preceding
proposition.
Proposition 9.22. Any doubly invariant subspace M
~
0 of the
natural representation f -+ Tf of A on H2(dll) has the form M = = XEL2( dp,) nH2(dp,), where E is a Borel subset of X. E is Il-essentially unique determined by M.
Ch. 9. Elements of prediction theory
235
Proof. Let L(M) be the minimal subspace of L2(dfl), invariant to the multiplication by functions of C(X) and which contains M. The elements gm with g E A and m E M span L{M). If P is the projection of L2(dfl) on H2(dp), we have Pgm = T:mEM.
Hence PL(M)
eM C
PL(M),
that is M= PL(M).
Since L(M) is an invariant subspace to the multiplication by functions of C(X), it has the form
where E is a Borel set, Jl-essentially unique determined (Theorem7.14). There results
which yields
The proposition is therefore proved. 0 A positive measure Jl on X, Jl{X) = 1, will be called a Szegototal measure if it is not a point measure and if for any Borel subset E of X, with XE L2(dJl)nH2(dJl) #- {O} we have fleE) = I. There immediately results that the non-point measure Jl is Szego-
total if and only if any function of H2( dJl), which vanishes on a set of positive measure, is equal to zero almost everywhere on X.
Proposition 9.23. Any Szego-total measure on X is a Szego measure. Proof. Suppose that a Szego-total measure is not a Szego measure. Then there exists a Borel subset E of X, with Jl(E) > 0, such that XEL2(dJl) C H2(dJl). Therefore lEL2(dJl)nH2(dJl) = ZEL2(dfl) #- O. Since
Function algebras
236
J1 is a Szego-total measure, we have J1(E) = 1, hence L2(dJl) = = XEL(dJl) C H2(dJl) C L2(dJ1), that is L2(dJl) = H2(dp). This yields XEL2(dJl) C H2(dJ1) for any Borel set E, hence peE) = 0 or Jl(E) = I for any Borel set E of X; that means Jl is a point measure which contradicts the fact that J1 is a Szego-total measure. 0 Theorem 9.24. Let J1 be a positive measure on X, Jl (X) = 1, and f -+ Tf be the natural representation of A on H2( dJ1). Then (i) If Jl is a Szegii-singular measure, f -+ Tf is a spectral representation. (ii) If Jl is a Szegii measure and N = [ U T:Ts H2(dJl)]J- = O,f-+ Tf a-1sffS- 1
is m-singular. (iii) If Jl is a Szego-total measure and N :f= 0, f -+ Tf is m-completely nonspectral. Proof. Points (i) and (ii) result from Theorem 8.19. Point (iii) follows from Corollary 9.17 and Proposition 9.18. We know that, for (i) and (ii) the converse assertions are also true. As concerns point (iii), we have the following statement: Proposition 9.25. If f -+ Tf is m-completely nonspectral, then for any Borel set E of X, with XE E H2(dJ1), we have Jl(E) = 0 or J1(E) = 1. Proof. Let E be a Borel set with XE E H2(dJl). Then M = XEH2(dJ1) is a doubly invariant subspace of H2(dJl). Indeed, TgM eM for any g E A. Let hEM and gh = hI + h2 with hI E M and h2 E XE L2(dJl) orthogonal to M. We have
Jkh dJl = JXEkh2 dJl = 2
0
for any k E H2(dJl), hence h2 E H2(dp)l.. There results Tg*h = Pgh = = hi E M, and therefore Tg* M C M. Thus M is a doubly invariant subspace and, since f -+ Tf is totally nonspectral, from Corollary 9.17 we get M = {O} or M = H2(dJl), i.e. Jl(E) = 0 or J1(E) = 1. 0 We now show that Theorem 9.24 may be interpreted as a prediction theorem. Let K be a Hilbert space (K may be, for instance, t~e space L 2(dP) of square integrable functions on a probability space)~-- G an abelian group and S eGa subsemigroup of G such that G = SUS-I, SnS- 1 = {l}. . We say that the map g -+ Fg of G in K is a G-stationary process, if (9.5.1 )
237
Ch. 9. Elements of prediction theory
The prediction problem relative to the future S, for such a process,
is that of approximating an element Fg by linear combinations of elements Fy with g-l yeS. More precisely, let us suppose, without any loss in generality, that the space :f( is the closure of the set spanned by the elements Fg , gE G. For any gE G we put Jig
If gl Jl gt
C
=
clm [{Fy; Y E G, g-l}, e S}].
~ g2'
Jig!,
i.e. if g2gl -1 E S, then gil yeS yields gil yeS, hence Thus, {..II g } is an increasing family of subs paces of .it.
Let us write :fl
=
n .A
g•
KEG
Let Eg be the orthogonal projection of Fg on the orthogonal complement of "lt g. For any g E G, Eg is orthogonal to:f l and {Eg; g E G} is an orthogonal set. It spans the closed subspace :f3 of :ff, orthogonal to :fl' We put (9.5.2) We say that g~ Fg is a deterministic process if % = %1' If % = %3 the process is called innovation process and if :f = :f 2 the process is called evanescent. The prediction problem is, then, nothing else than the characterization of the deterministic, evanescent and innovation parts of a process. Let us define on G the real function p, by (g
E
G).
p is a positive definite function on G. Indeed, according to (9.5.1)
we have ~ CjC;p(g;l gi)
=
~ cjc; (F
-1
gi
• gj
Fl )
=
Function agebras
238
for any finite system cl , ... , Cn of complex numbers and any finite system gl' ... ' g,. of elements in G. Following the Herglotz-Bochner-Weil theorem, there exists a positive measure lIon the dual X of the discrete group G, such that
peg)
=
SgdJ-l
(g E G),
where on the right handside g is viewed as character on X. We now define the operator U on :f{, with values in L2(dJl), by (g E G).
(9.5.3) Since
U may be defined by (9.5.3) and is a unitary operator from :f{ to L2(dJl) U realises an equivalence, from the point of view of the prediction theory, between the G-process g -+ Fg on :f{ and the G-process g -+ F~ = = g, on L2(dJl). Indeed, if we write
then
Let us remark that }Jg = elm [{y: y E G, y = gs, S
E
S, s #- 1}].
Let A be the algebra generated by S in C(X) and f -+ Tf be the natural representation of A on H2(dJl). We denote by m the Haar measure of X, viewed as the dual group of the group G.
Theorem 9.26. The process g -+ Fg is deterministic (evanescent, innovation) if and only if the natural representation f -+ Tf of A on H2(dJl) is m-spectral (m-singular, m-completely non-spectral).
Ch. 9. Elements of prediction theory
Proof. Let K
239
= L2(dJI) and H = H2(dJI). If we denote by Ug the
operator of multiplication by g in L2(dJl), then g ~ Ug is the unitary dilation of the semigroup {Ts} sES which corresponds to the natural representation f --+ Tf of A on H2(dJI). Let and
H
=
HI (±) H2 (±) H3
be the Wold decomposition of the semigroup of isometries {Ts} sESe Since MI
=
elm [{ S:
S
E S, s i= I}]
there results MI C H, hence KI C H. By the definition of M g we get that, for any S E S, S i= 1 and h E H, we have gsh E Mg. Therefore, UgKI CKI for any g E G and, from (9.1.8) we obtain KI CHI. Conversely, since HI reduces any operator U g to a unitary operator, for any hE HI and g E G there exists hI E HI such that h = gShI' with EE sS, s i= 1, i.e. h E Mg. Then HI C K I , hence KI = HI. Since the natural representation f --+ Tf is spectral if and only if HI = H2(dJI) = L2(dJI), then f --+ Tf is spectral if and only if KI = = L2(dJI), that is if and only if the process g --+ Fer is deterministic. Let now e g be the projection of g on the orthogonal complement of Mg in K. As Ug is a unitary operator on K, there results (9.5.4) Assume now that f
--+
Tf is m-singular. Then
1 E elm [ U T s H2(dJI)] = MI. s+=1
Hence el = 0 and, from (9.5.4) we get eg = 0 for any g E G. Therefore Ka = O. At the same time from HI = 0 there results KI = 0, that is g --+ Fg is an evancescent process. Conversely, if the process is evanescent, then KI = K3 = {Ole From K3 = 0 there results el = 0 and therefore 1 E Ml = elm [ U TsH]. S1=1
Function algebras
240
f
-+
On the other hand, as KI = 0 implies HI = 0, we obtain that T, is m-singular. Let M be a subspace of H, doubly invariant to f -+ T" such that elm [ U TsM]
=
M.
s+1
According to a previous remark, M C MI' Then el is orthogonal to M. Since I E H2(dJl) and Ml C H2(,dJ1.), there results el E H2(dJ1.). We have
(m, e s ) = (m, sel )
= (sm, el ) =
(Tsm, e l ) = 0
for any s E S, where P is the projection of L2(dJ1.) on H2(dll). Therefore M is orthogonal to K 3 • If the process is innovation, then M = {OJ, that is f -+ T, is completely non-spectral. If f -+ T, is completely non-spectral then H2(dJ1.)
=
(±) TsN sES
where N = Mr is one-dimensional. Then, clearly, L2(d/l)
=
tt> UgN. gEG
Let k E L2(d/l) be orthogonal to eg for any g E G. As e1 EN, el i= 0 and N is one-dimensional, we have ng = cgel for any ng E N, with cg constant, and (k, gn g)
=
(k, Cggel) = cik, gel)
=
cik, eg} = O.
Hence k is orthogonal to UgN for any g and, from (9.5.4), we obtain k = O. Thus, we have L2(d/l) = K 3 , i.e. g -+ Fg is an innovation process. The theorem is proved. 0 In the case G = Z is the additive group of integers and S = Z+ the semigroup of positive integers, Theorem 9.24 gives exactly the
Ch. 9. Elements of prediction theory
241
classical prediction theorem for stationary discrete processes. The following two theorems are precising this fact. In this case we know that X is the unit circle {Izl = I} of the complex plane, m is the normalized Lebesgue measure on X and A the standard algebra on X. Theorem 9.27. Let A be the standard algebra on X = {Izl = I} and m the normalized Lebesgue measure on X. A positil'e measure Jl. on X is a Szeg5 measure if and only if it is a Szego-total measure. This happens if and only if dJl = hdm with h ELI ( dm) and (9.5.5)
Jlog h dm >
-
00.
Proof. First observe that, in this case, the function 1 belongs to the L2(dJl.)-closure of A~ if and only if L2(dJl). H'I(dJ.l). Let dJ.l = hdm + dJl.s be the Lebesgue decomposition of J.l with respect to m. If E is the support of Jl.s then, from the Szego theorem and the above remark, there results (9.5.6) Suppose Jl. is a Szego measure. Then (9.5.6) yields J.l(E) = 0, that is Jl.s = O. Hence dJ.l = hdm. As L2(dJl) + H2(dJ,l), once more from the Szego theorem and the above remark, we get
Jlog hdm >
-
00.
Now let dJ,l = hdm where h satisfies (9.5.5). The Szego theorem yields H2(dJ.l) ~ L2(dJ,l). Let E be a Borel set and
Then h- 1h1l. E E H2(dm) and we get m(E) = 0 or m(X - £) = 0, hence p, (E) = 0 or p,(X - £) = 0, i.e. J,l is a Szego-total measure. Since a total Szego measure is a Szego measure, the theorem is completely proved. 0 Theorem 9.28. Let f ....... Tf be the natural representation of the standard algebra A on H2(dJ1}. f . . . . Tf is an m-completely non-spectral representation if and only if J,l is a Szego measure. 16 - c. 431
Function algebras
242
Proof. There remains to prove that if Jl is a Szeg6 measure, then N
= [
U
Tf H2(dJI)1l. # O.
lEAm
Indeed, if N = 0 then I belongs to the L2(dp)-closure of Am and therefore the Szeg6 theorem yields H2(dp) = L2(dp); but this contradicts the fact that Jl is a Szeg6 measure. The theorem is proved. 0 Summing up all these results we now obtain the prediction theorem for discrete stationary processes.
Theorem 9.29. Let n -+ Fn be a discrete stationary process given by a positive Ineasure p on the unit circle X = {Izl = 1} of the complex plane, and m the Lebesgue measure on X. Let dp = hdm + dps' hELl(dp) be the Lebesgue decomposition of p with respect to m. 11 -+ Fn is a deterministic process if and only if
Slog It dm n
-+
Fn is an innovation process
=
-
00.
if and only if dp
Slog h dm > -
=
hdm and
00.
In the last case the prediction is made with a non-zero error el given by Szego formula
el = exp
[~
jIoghdm
l
Notes The first proof to Theorem 9.1 has been given by M. A. NAIMARK [1]. Other proofs as well as essential generalizations to non-commutative case and to semigroups are due to B. SZ.-NAGY [1], [2]. Theorem 9.5 has first been proved in the case of the additive group of integers and of the semigroups of positive integers by B. SZ.-NAGY [l], [2], by means of complex functions methods. The proof given here has been first done by I. HALPERIN, also in the case of integers group. W. MLAK [3] has rewritten the same proof for the case when G is totally ordered by S. The general form appears in I. SUCIU [7].
Ch. 9. Elements of prediction theory
243
The theorems of canonical decomposition for semigroups of contractions have their origin in the works of B. SZ.-NAGY and C. FOIA~ [I], [2], [6], where they are proved for the semigroup of positive integers or strongly continuous semigroups of real parameters. Important contributions in this direction have appeared in W. MLAK [1], [2], [3]. The characterization of the unitary space given by Theorem 9.9, in the general case, is given in 1. SUCIU [4 J. Dilation theorem for semigroups of isometries has been given by T. ITO [1 ]. Wold type decomposition theorems for semigroups of isometries have their origin in H. WOLD [1]. A first Wold type decomposition in three terms has been proved by H. HELSON and D. LOWDENSLAGER [2]. Theorems 9.11, 9.12 as well as the characterization of the semigroups of unilateral translations given by Theorems 9.13 and 9.14 are expounded in 1. SUCIU [9]. The results of paragraph 9.4 appear in I. SUCIU llO] together with Theorem 9.24. The formulation of the prediction problem of paragraph 9.5 follows H. HELSON and D. LOWDENSLAGER [1]. Theorem 9.26 which establishes the connection between Helson and Lowdenslager prediction theorem and Theorem 9.24 is unpublished. Theorem 9.7 appears in C. FOIA~ and 1. SUCIU [1]. Theorem 9.29 is the classical prediction theorem for discrete stationary processes (cf. J. L. DOOB [1 Concerning prediction theorems see also G. LUMER [2].
n.
CHAPTER 10
Some examples in the spectral theory of non-normal operators
10.1. The case of a single contraction Let X = {z E c: Izl = I} be the unit circle in the complex plane and A the standard algebra on X, that is the algebra of continuous functions on X which can be analytically extended in the interior of the unit disk D = {z: Iz\ < l}. It is known that A is the uniform closure P(X) of the polynomials in z and that A is a Dirichlet algebra on X. At the same time, S = {l, Z, Z2, ••• }, where z is the function f E A defined by f (x) = x, X E X, is a closed mUltiplicative system of inner functions in A, analytic free, which generates A. If we put G = SS, then X is the dual of the discrete group G, the normalized Lebesgue measure is the Haar measure m of X, and A
= {IE C(X): Jz"fdnl = 0, n
=
I, 2, ... }.
Let H be a Hilbert space and T a contraction on H, i.e. T E L(H), "Til ~ I. We write Ts = Tn for S E S, S = zn. We thus obtain the semigroup of contractions {Ts} sES on H. Since G = SUS we obtain the map g -. Tg of G in L(H) by Ts = T:. According to Proposition 9.5, g -. Tg is a positive definite extension of the semigroup {Ts}sES to G. Then from Corollary 9.3 there results that for any linear combination ~CiSi of elements in S,
Function algebras
246
we have
that is for any polynomial p(z) !!p(T)!1 ~ sup !p(z) I
(10.1.1)
!z: =1
which is von Neumann formula for a contraction. Since A = P(X). f(T) can be constructed for any f E A by using (10.1.1). If we put Tf = J(T) then it is easy to see that J ~ Tf is a representation of A on H such that T z = T. It is clear that any representation J ~ Tf of A on H may be obtained in the same way starting from the contraction T = Tz of L(H). T is called completely non-unitary if the only subspace which reduces T to a unitary operator is {O}. One easily verifies that T is unitary (completely non-unitary) if and only if {TsL ES is a unitary (completely non-unitary) semigroup, hence if and only if f ~ Tf is a spectral (completely non-spectral) representation. From Theorem 8.6 or 9.8 and Theorem 9.9 there results Theorem 10.1. Let T be a contraction a unique decomposition of the form
011
H. The space H has
where Hu reduces T to a unitary operator and He reduces T to a completely non-unitar.v contraction. We have
Hu = {h
E
H: IITnhll = IIT*nhll
=
Ilhll,
11 =
1, 2, ... }.
F olowing Proposition 9.2 there exists a uniquely determined semispectral measure (F(U))aE B(X) such that (n
= 0, 1, ... ; h, k
E
H).
Ch. 10. Examples of spectral theory
247
Let (Ph ,k)h ,k e H be the semispectral family attached to the semispectral measure (F(O'))aEB(X) and J1h = J1h,h' hE H. From Theorem 8.16 and Theorem 9.7 there results Theorem 10.2. Let T be a completely non-unitary contraction on Hand (Ilh ,k) h,k E H the semispectral family on X attached to T. For any h E H the measure Ilh is absolutely continuous with respect to the Lebesgue measure m on X and
According to Corollary 9.5 the semigroup {TsLES generated by T admits a unitary dilation g -+ U g. The unitary operator U = Uz is called the unitary dilation of T. Therefore Tnh = PUnh
(h
E
H, n
=
1,2, ... )
p.
is the orthogonal projection of K (the dilation space) on H. It is immediate that T is an isometry if and only if {TslsES is an isometric semigroup or if and only if f -+ Tf is a subspectral representation. In this case we have:
where
elm [
U T:TsH] = clm [ U TfH] = TH.
so-lEES- 1
fe Am
Therefore {Tslses (f -+ Tf ) is a quasy-unitary semigroup (m-spectral representation) jf and only if it is unitary (spectral). At the same time {Ts} sES (f -+ Tf ) is totally-non unitary (m-completely non-spectral) if and only if it is completely nonunitary (completely non-spectral) and the singular semigroups (m-singular representations) do not appear. The decomposition theorems of Wold Type (Theorem 8.25, Theorem 9.12) reduce in this case to the classical Wold theorem. Theorem 10.3. (Wold). Let T be an isometry on H. Then H admits a unique decomposition under the form
Function a1gebras
248
where Hureduces T to a unitary operator and the restriction of T to H t is unitarily equivalent to the shift (translation) operator in H2( N) with N = [TH).l .
. Then Theorem 8.21 and Theorem 9.19 give Theorem 10.4. The isometry T on H is unitarily equivalent to the shift (translation) operator on H2(N). with N one--dimensional, if Qnd only if T is not unitary and has no proper doubly invariant subspaces.
10.2. Operators having spectral sets with connected complement Let X be a compact set of the complex plane contained in the boundary of the unbounded connected component of its complement. Let A = P(X) be the function algebra on X which is the uniform closure in C(X) of the set of all polynomials in z (z is the function f E E C(X) defined by f(x) = x for x E X). We recall that A is a Dirichlet algebra on X (Theorem 6.26). We also know that the maximal ideal space of X may be identified too with the polynomial convex hull X of X. We have A
A
X =
XU
G1
U
G2 U···
where G1 , G2 ,. •• , are the bounded connected components of the complement of X. One easily verifies that G1 , G2 ,. •• are the non point Gleason parts of X (relative to A) and any x E X forms a point Gleason part. A
Theorem 10.5. Let f
Tf be a representation of A on the Hilbert space H. Then H admits a decomposition under the form -+
where H s ' H", h = 1,2, ... are doubly invariant subspaces, f is spectral and for any k = 1, 2, .. . ,f -+ 1JIH" is G,,-continuous.
-+
TfIH.
Ch. 10. Examples of spectral theory
249
Prool. Let (G(I.)(l.E.1 be the family of all Gleason parts of X" relative to A, and H
= ES HfI. ES Ho (I.E.!
be the decomposition of H relative to I -+ T, as follows from Theorem 8.28. Therefore Ho,H«, (l E J are doubly invariant subspaces, 1-+ .T,IH« is Gtz-continuous, (l E J, andl -+ T,IHo is singular. If G!I={x} we denote Htz by Hx and write
Hs = EB Hx ES Ho· xeX
We now show that I -+ T,IHs is spectral. It is. sufficient, for this, to prove that I -+ TflHo and I -+ T,IHx, x E X, are spectral. Let (Ph ,k)h ,k E H be the semi-spectral family attached to I -+ Tf · If h, k E Hx , then Ph ,k is absolutely continuous with respect to ex and, since Ph,k(1) = (h, k), there results (l 0.2.1)
Ph,k(CP)
=
ip(x) (h, k)
Hence it is immediate that (ilia ,k)1a ,k E H:Ie generates a representation of C(X) on Hx which is an extension of I -+ T,IHx, that is I -+ 1/IHx is spectral. Now let h, k E Ho. We then know that I1h,k is singular with respect to any representing measure for A. For any I, g E A we have
Therefore the measure gdjl" ,k - dl1TgIl ,k is orthogonal to A. Since, obviously, it is singular with respect to any representing measure for A, from Proposition 6.34 we get
(f E C(X), g
(10.2.2)
. Let now g
E
A).
Tg be the positive definite map of C(X) in L(H0) given by the semi spectral family iP",l)",lEHe' Clearly g -+ Tg is an exten-+
Function algebras
250
Slon of I
-+
Tf . From (10.2.2) there follows
for any I E A and g E C(X). Therefore dph.T;k - g dph,k IS an orthogonal measure to A and, as it is singular with respect to any representing measure of A, it is eq ual to zero. Hence (10.2.3)
(h, k E Ho,
I, g E C(X».
From (l0.2.3) there results
for any h, k E Ho, i.e. g -+ Tg is multiplicative. Then I -+ TflHo is a . spectral representation. Since a direct sum of spectral representations is a spectral representation, I -+ TflHs is also spectral. The theorem is pr.oved. 0 Let T be a bounded operator on H. Assume that for any polynomial p in z we have (10.2.4)
/lp(T)/I ~ sup Ip(x)!. XEX
Then it is clear that for any I E A we can construct I(T) and, writing Tf = I(T), I E A, we obtain a representation 1-+ Tf of A on H, with Tz = T. One easily verifies that I -+ Tf is a spectral representation if and only it T is a normal operator and its spectrum is contained in X.
Ch. 10. Examples of spectral theory
In this case the spectral measure attached to f measure of T.
---+
251
Tf is just the spectral
Corollary 10.6. Let T be a bounded linear operator on H which verifies (10.2.4). For any k = 1, 2, ... , let mk be a representing measure with support in X, of a point in Gk • The space H admits a decomposition under the form
where Ho reduces T to a normal operator To with spectrum contained in X and its spectral measure singular with respect to m k , k = 1, 2, ... , H k, k = 1, 2, ... , reduces T to an operator which admits an absolutely continuous semispectral measure with respect to mk'
10.3. Finite system of commuting contractions
n {z n
Let X =
E
e: Izl =
I} be the n-dimensional torus in
en.
1
We denote by x = (Z1, Z2"",Zn) the points of X and by Zi the coordinate functions on X, i.e. Zi(X) = Zi' i = I, 2, ... , n. Let S be the closed multiplicative system of inner functions on X generated by Z1, Z2,"" Zn and G = SSe G is an abelian group, S is a sub-semigroup of G with S S-1 = {I} and X is the dual of the discrete group G. Let A = A(S) be the function algebra on X generated by S. Therefore A is the closure in e(X) of the algebra of polynomials in the variables
n
Z}, Z2,""
Zn'
Let T}, T2 , ••• , Tn be a system of n commuting contractions on a Hilbert space H. If for any s E S, S = Z~l Z~2 ••. z~n, we put Ts = = T~l T;2 ... T!n we obtain a semigroup of contractions {Ts}sES on H. Relative to the semigroup {Ts} sES it is natural to ask the following questions: I) Does a unitary dilation g ---+ Ug of {Ts} sES exist? 2) Does a representation f ---+ Tf of A on H, which extends the . '? semIgroup
Function algebras
252
It is clear that (2) is equivalent to the following inequality (von Neumann inequality) (2') for any polynomial P(Zl"'" z,,) in A we have (10.3.1) From Corollary 9.3 we get (I) ~ (2'). In the following we prove that (I) is true for n = 2 (Ando Theorem). For n = 3 we give an example (Parrott's example) when (2') is true and (1) is false. We obtain, as a consequence, an example of representation of the algebra A which has no spectral dilation. The assertion (2') for n ~ 3 is an open question .). We start with a lifting commutant theorem which is the version we need of a theorem of B. Sz.-Nagy-C. Foia~. Let T be a contraction on Hand U acting on K be its minimal unitary dilation. Therefore H is a subspace of K, U a unitary operator on K and we have
Tnh
= PUnh
(h
E
H, n = I, 2, ... )
00
K
=
elm [
U Un H]
n= -
00
.
0()
where P is the orthogonal projection of K on H. Let K+=elm [U U"H], ,,=0
and P + be the projection of K + on H. K + is obviously invariant to U and V = UIK+ is an isometric operator on K+. We also have (10.3.2)
(h
E
H, n = I, 2, ... ).
This operator V will be called the isometrical dilation of T. Let L = (V - T)H. *) For n ~ 3 and dim H V ARAPOULUS [1 J).
~
5 there are counterexamples for (2') (cr.
Ch. 10. Examples of spectral theory
253
From (10.3.2) there results
« V - Dh, h') = (Vh, h') - (Th, h') = (Th, h') - (Th, h') = 0 for any h, h'
E
H, hence H 1 L. At the same time
(VII(V - T) h, (V -
Dh')
=
(VII+lh, V h') -
- (vnTh, Vh') - (VII+lh, Th')
+ (VIITh, Th') =
=
(Vllh, h') - (vn-l Th, hi) - (vn+lh. Th/)
+ (vnTh, Th') =
=
(Tn+lh, h') - (Tn+lh, h') - (TII+lh, Th/)
+ (Tn+lh. Th') =
0
for any n > 0 and h, h' E H. We used here (10.3.2) and the fact that V is an isometry on K+. Therefore vnL-LL for any n > 0 and, since V is an isometry on K+, there results Vnl L -L V n2 L for any n1 ¥= n 2 From the obvious relation
+ ... -+-
vn -1 (V - T) h
there follows (10.3.3) and from (10.3.3) one has (10.3.4) Theorem 10.7. Let Z be a bounded linear operator on H with ZT = TZ. There exists a bounded linear operator Y on K+ with YV = V Y and such that
=
Function algebras
254
(i)
Tnzm h = P + V"ymh
(ii)
Y(K
e
H) C H +
(iii)
I YII
=
I Zll·
Proof. We may assume
e
IIZII
(/z
E
H,
11,
m
=
0, /,2 .. ) -.
H
=
I and look for an Y of the form;
(10.3.5) where Bo, BI , B 2 , ... are bounded linear operators from K+ into L. We shall determine Bo, BI , B 2 ,. .. such that Y satisfies (i), (ii), and (iii). Let us define the operator B acting on K + with values in L, by
We have IIBkll2 = II(V -
=
«V -
T) ZP +k1l2 = «V -
T) ZP +k, VZP +k) =
T) ZP +k, (V -
I VZP +k1l2
T) ZP +k) =
- (TZP +k, VZP +k)
=
where we used (10.3.2) and (10.3.4). Hence (10.3.6) Since IIZI1 = I, the operator IK + - P +Z*ZP + is positive on K+. Let Do be the positive square root of this operator. Then
255
Ch. 10. Examples of spectral theory
and by (] 0.3.6) we have
that is (10.3.7) The inequality (l0.3.7) allows us to define a contraction Co from K+ inio L such that
B
= CoDoV
as follows: we write CoDoVk = Bk on the subspace DoVK+ of K+, then extend by continuity to the closure of Do VK+ and, for k in the orthogonal of DoVK+ in K+ we write Cok = O. Let
Bo = CoDo·
Bo is an operator defined on K + with values in L. We have Bo V = B and .~~.
that is (I0.3.8) We now construct by induction the sequence B o, B1 , B2 , ••• of operators on K + with values in L such that (l0.3.9)
(n
= ], 2, ... )
and (10.3.10) UZP +kI12+ ~ IIBnkl12 ~ IIkl1 2 O~n
(k
E
K +, P
= ], 2, ... )
Function algebras
256
Starting with Bo defined above, suppose we constructed B,., 0 ~ ~ n < p such that (10.3.9) for n < p, and (10.3.10) are satisfied. We ~ B: B,. is a positive get from (l0.3.10) that Ix + - P +Z*ZP + O~n
operator on K+. Let Dp be its positive square root. Then D;
=
IK+ -
~ B:B"
P +Z*ZP + -
O~n
IIDicll2
= (D;, k, k) = IIkll 2- IIZP +k112 -
~
IIB"kI12, (k E K+).
O~n
We have
IIZP+ Vkl1 2 + ~ IIB"VkI12 = IIZP+ Vkll 2+ IIBkl12 O~n
~ IIZP +k1l2
+
~
IIB,.kIl2 = IIZP +k1l2
O~n
- II Bp_1k 112
~
+ O
n_1
+
~
k11 2 ~
II Bilk I 2 -
o~n,p
II k 112 - II Bp_1k 112 = II Vk 112 - II Bp_1k 112
for any k E K+. We used here relation BoV = B, (10.3.6), (10.3.9) for n < p and (10.3.10). Therefore IIBp_1k1l2 ~
II kl1 2 - IIZP + Vkll 2 -
~
IIBII Vkl1 2 = IIDpVkl12
O~n
thus (10.3.11) A similar argument to that used in constructing Co, allows us to construct, by (10.3.11), a contraction Cp from K+ into L, such that
Let
Then
Ch. 10. Examples of spectral theory
257
and
I\B"k112 = IICp Dl'1I2 for any k
E
~ IID"k1l2 =
~
IIkll 2 - IIZP +k1l2 -
II Bilk I 2
O~II
K+, hence
IIZP +k1l2
+
~
IIB,.k1l2 ~ IIkll 2
O~n
Thus we get the operators Do, Bb "., Bp from K+ into L such that relations (10.3.9) and (10.3.10) are satisfied for n < p + 1. Therefore relation (10.3.5) defines an operator Y on K. We have
II Ykll 2 = IIZP +k1l2 +
00
~
IIBnkll2 ~ IIkll 2
n=O
hence II YII results
~
1. According to the definition of Y and to (l0.3.3) there
ZP+ = P+Y, hence we have
znp +
= zn-lZP +
=
zn-lp + Y = zn-2ZP + y _
-
=
zn-2p + y2 =
_p + yn .
... -
Thus
(h
At the same time
17 -
c. 437
E
H; n, m
= 0, 1, 2, ... ).
Function algebras
258
and, therefore,
II YII = IIZII =
1.
We now calculate
= VZP+
+
~
00
~ n=u
- BoV +
vn+lBn - ZP+V-
~
VnBnV= VZP+ - ZP+V-
;; ....::;0
00
~
Vn(Bn_1
-
BnV) = VZP + - ZTP + - BoV =
n=O
= VZP + - TZP + - BoV = (V - T) ZP + - BoV = B - BoV = 0
so YV = VY.
The theorem is proved. 0 Proposition 10.8. Let Z be a bounded linear operator on H with ZT = TZ, Y a bounded linear operator on K+ with YV = VY and suppose the assertion (i), (ii), (iii) of Theorem 10.7 are satisfied. If Z is an isometry on H then Y is an isometry on K+ and YH C H., YL C L. If Z is unitary on H then Y is unitary on K+. Proof. Suppose that Z is an isometry on H. For any h E H we have
Thus hence
Ch. 10. Examples of spectral theory
259
Thus YHCH; Z= YIH.
Since Y(K+
e
H)
e
C K+
yvn(v - T)h
=
H there results that H reduces Y. Now
Y"Y(V - T)h
=
vn(vy - YT)h
and since
=
YTh
ZTh
=
TZh
we have
YY"(V - T)h = yn(v - T)Zh.
Thus yvnL C V"L. It remains to be proved that
Y/ vn L is isometric.
But
I Y(V - T)h11 2 = II(V - T)Zh112 = IIZhl1 2 - IIZThl1 2 = = IIhll 2
I Thll2 = II(V - T)hIl 2•
-
Hence
IIYIIl
= 11/11
(I E L).
Now
therefore YI v,. L is isometric. Consequently Y is an isometry on K +. If Z is unitary, then Z is an isometry and ZH = H. Therefore Y is an isometry on K+ and Yh = Zh, hE H. Let k E K+ be of the form k
=
p
~ V"h,. 11=0
(h,.
E
H).
Function algebras
260
There exist h~ E H such that h" = Zh~, 0 ~ n ~ p, so p
p
p
k = ~ vnZh~ = ~ VnYh;, = Y ~ vnh~. n=O
,,=0
,,=0
Since the elements of this form are dense in K+ and Y is an isometry on K+, there follows YK+ = K+ that is Y is unitary on K+ . The proposition is proved. 0 Theorem 10.9. (lifting commutants theorem). Let T be a contraction on Hand U its unitary dilation acting on K. For any bounded linear operator Z on H which commutes with T there exists a bounded linear operator Y on K which commutes with U such that
(i)
T"Zmh = PU"ymh
(ii)
YK+
(iii)
II YII
C =
K+, Y(K+
(h
E
H, m, n
=
0, 1, 2, ... )
e H) C K+ e H
II ZII·
If Z is unitary then Y is unitary and Y H C H, Y L C L. Prooj. Let Y + be the operator on K + obtained by the preceding
theorem for operators T and Z on H. Since Y + commutes with V then Y! commutes with V*. If we apply once more the preceding theorem for V*, Y!' and take into account that the isometrical dilation of V* is U* we obtain an operator y* on K which commutes with u* and satisfies V *ny+*mk = PK+u*ny*mk
IIY*II
=
(k
E
K , n, m = I , 2, ... )
IIY!II
where P K + is the projection of K on K+. It is easy to see that Y = (Y*)* is the desired operator. 0 Theorem 10.10. For n = 2 the semigroup {Ts} s ES, constructed at the beginning of this paragraph, has a unitary dilation.
Ch. 10. Examples of spectra] theory
261
Proof. Let TI , T2 be the two commuting contractions on H. Let U~ acting on KI be the unitary dilation of TI • According to Theorem 10.9 there exists a contraction T~ on Kl such that
where PI is the projection of KI onto H. Now let U2 acting on K be the unitary dilation of T~. If we apply once more Theorem 10.9 we get the unitary operator U1 on K which commutes with U2 such that
where P 2 is the projection of K on K t . Let P be the projection of K on H. We have
that is
TiT2'h
=
PU~Urh
(h
E
H, n, m = 0, 1,2, ... ).
If we put Ug = U~U2' for any g E G, g = z~zr, n, m = 0, + 1, + 2, ... , then g..-+ Ug is clearly a unitary representation of G on K which dilates the semigroup {Ts} sES' 0 Coronary 10.11. For n = 2 the inequality (10.3.1) holds, that is for any two commuting contractions T I , T2 on H and any polynomial in two variables p(Zt, Z2) we have IIp(Tl' T 2)11 ~
sup ;Zl
Ip(Zl' z2)1.
=Z2: = 1
.
We now give an example of three commuting contractions T I , T 2 , T3 on H for which the semigroup {Ts} sES generated by them has no unitary dilation but verifies inequality (10.3.1). This example is mainly due to Parrott ([lD. Theorem 10.12. For n = 3 there are semigroups {Ts}sES which satisfy (10.3.1) but do not admit unitary dilations.
Function algebras
262
Let Ho be a Hilbert space on which there exist two non-commuting unitary operators AI, A2
Let us denote by A3 the identity operator on H o• We write H = Ho(j)Ho and consider on H operators T i , i = 1. 2. 3 defined by the matrix
It is obvious that II Till ~ I, TiTj = TjTj = 0 for i, j = 1,2, 3. Hence T I , T 2 , Ts is a system of three commuting contractions on H. Assume there exists a Hilbert space K and the commuting unitary operators U1 , U2 , U3 on K such that He K and (10.3.12)
T·1l I
=
PU·h I
(11
E
H, i = I, 2, 3)
where P is the orthogonal projection of K on H. For h E H, h = rho, 0], ho E H o, we have
and from (10.3.12) we get Tih = UJI. Then
Since U1 , U2 , U 3 are commuting operators, there results
Ch. 10. Examples of spectral theory
263
that is AlA2 = A2Al which is impossible. Therefore, the semigroup {TS}SES has no unitary dilation. Let us now show that T l , T 2 , Ta verify inequality (l0.3.1). Let
be a polynomial in variables = 1, 2, 3, there follows
Since T;Tj
Zl, Z2' Za.
=
0 for any i, j =
that is P(Tl' T 2 , Ta)
=
3
~aiAi
ao
, i =1
Let H' be the subspace generated by {I, z} in L2(d/l) where /l is the Lebesgue measure on the unit circle {z E c: Izi = I} of the complex plane, and A the operator on H' given by the matrix 0\
laol
A=
3
~ lail
10 0 1
i =1
One easily verifies that IIp(Tl' T2 , Ta) I ~ I A II. On the other hand if P' is the projection of L2 on H' and q a one-variable polynomial q(z)
= 10 0 1+
3
~ lalz
+ ...
:=1
then Ah
= P'qh
(Il
E
H).
Hence IIAhll = IIP'qh" ~ Ilqhll =
JIql21hl dJl 2
that is
IIA II
~
sup Iq(z)l. 'z =1
~ sup Iq(z)1 :z' =1
IIhll.
264
Function algebras
Since
with the convention ai
lail sup Ip(zl'
Z2'
=
1 if a;
3
IZ il=l
IZ:=1
sup 'zl=1
100 1+
~
I
+ .... ~
;=1
ZI=Z3=Z3·=Z
=
la;lz;
3
1
;=1
I
10 01 + I; lailzi + ... i =
sup 3
0, there results
100 1+ I;
z3)1 = sup
jZ il=1
~
=
lailz + ...
~
IIAII
~ IIp(Tl' T2 , T 3 )1I
;=1
that is IIp(T!, T2 , T a)" ~ sup Ip(Zl'
Z2,
za)l·
IZil =1
Corollary 10.13. There are representations of a function algebra which admit no spectral dilation.
Notes The results presented in paragraph 10.1 have been mainly taken from the contraction theory on Hilbert spaces derived by B. SZ.-NAGY and C. FOIA~ in a series of common works and unitarily exposed in the book of B. SZ.-NAGY and C. FOIA~ [6]. The second part of Theorem 10.2 has been proved by W. MLAK [2]. Theorem 10.3 is the classical theorem of H. WOLD [1]. Theorems of paragraph 10.2 lead to the principal result obtained by D. SARASON [1]. Theorem 10.7 appears in a more general form in B. SZ.-NAGY and C. FOIA~ [5] while Proposition 10.8 and Theorem 10.9 follow simply from it. Theorem 10.10 has been proved by T. ANDO [1] and the example contained in Theorem 10.2 has been essentially constructed by S. PARROTT [1].
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