PROCEEDINGS OF THE CONFERENCE ON
Banach Algebras and Several Complex Variable
AmE ICAn mATH mATICAL SOCIETY
U0LU E32
PROCEEDINGS OF THE CONFERENCE ON
Banach Algebras and Several Complex Variable
AmE ICAn mATH mATICAL SOCIETY
U0LU E32
COIlTEMPORAn mATHEmATICS Titles in this Series Volume
1 Markov random fields and their applications, Ross Kindermann and J. Laurie Snell
2
3 4 5
6
7
8 9 10
11 12
13 14 15 16 17
18 19 20
Proceedings of the conference on Integration, topology, and geometry in linear spaces, William H. Graves. Editor The closed graph and P-closed graph properties In general topology, T. R. Hamlett and L. L. Herrington Problems of elastic stability and vibrations. Vadim Komkov. Editor Rational constructions of modules for simple Lie allebras, George B. Seligman Umbral calculus and Hopf algebras, Robert Morris. Editor Complex contour Integral representation of cardinal spline functions, Walter Schempp Ordered fields and real algebraic geometry, D. W. Dubois and T. Recio. Editors Papers In algebra, analysis and statistics. R. Lidl. Editor Operator allebras and K-theory, Ronald G. Douglas and Claude Schochet. Editors Plane ellipticity and related problems. Robert P. Gilbert. Editor Symposium on algebraic topology In honor of Jose Adem, Samuel Gitler, Editor Algebraists' homage: Papers in ring theory and related topics, ·S. A. Amitsur. D. J. Saltman and G. B. Seligman, Editors Lectures on Nielsen fixed point theory, Boju Jiang Advanced analytic number theory. Part I: Ramification theoretic methods. Carlos J. Moreno Complex rep.r esentations of GL(2, K) for finite fields K, lIya Piatetski-Shapiro Nonlinear partial differential equations, Joel A. Smaller. Editor Fixed points and nonexpansive mappings, Robert C. Sine. Editor Proceedings of the Northwestern homotopy theory conference, Haynes R. Miller and Stewart B. Priddy. Editors Low dimensional topology, Samuel J. lomonaco. Jr.. Editor
Titles in this Series Volume 21
Topological methods in nonlinear functional analysis, S. P. Singh. S. Thomeier. and B. Watson. Editors 22 Factorizations of b" ± 1. b = 2, 3. 5,6, 7, 10, 11. 12 up to high powers. John Brillhart. D. H. Lehmer. J. L. Selfridge. Bryant Tuckerman. and S. S. Wagstaff. Jr. 23 Chapter 9 of Ramanujan's second notebook-Infinite series identities, transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi 24 Central extensions, Galois groups, and ideal class groups of number fields, A. Frohlich 25 Value distribution theory and its applications. Chung-Chun Yang. Editor 26 Conference in modern analysis and probability, Richard Beals. Anatole Beck. Alexandra Bellow and Arshag Hajian. Editors
27
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28 29
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33
Axiomatic set theory, James E. Baumgartner. Donald A. Martin and Saharon Shelah. Editors Proceedings of the conference on Banach algebras and several complex variables, F. Greenleaf and D. Gulick. Editors Contributions to group theory. Kenneth I. Appel. John G. Ratcliffe and Paul E. Schupp. Editors
conTEMPORARY MATHEMATICS Volume 32
PROCEEDINGS OF THE CONFERENCE ON
Banach Algebras and Several Complex Variables F. Greenleaf and D. Gulick. Editors
AMERICAn MATHEMATICAL SOCIETY providence • RhOda Island
EDITORIAL BOARD Kenneth Kunen James I. Lepowsky Johannes C. C. Nitsche Irving Reiner
R. O. Wells, Jr., managing editor Jeff Cheeger Adriano M. Garsia
PROCEEDINGS OF THE CONFERENCE ON BANACH ALGEBRAS AND SEVERAL COMPLEX VARIABLES HELD AT YALE UNIVERSITY NEW HAVEN, CONNECTICUT JUNE 21-24, 1983
These proceedings were prepared by the American Mathematical Society with partial support from the National Science Foundation Grant MCS 8218075. 1980 Mathematics Subject Classification. Primary 32Axx, 32Bxx, 32Exx, 32Fxx, 46Hxx, 46Jxx.
Library of Cong,.. Cataloging in Publication Data Confere.,ce on Banach Algebras and Several Complex Variables (1983: Vale University) Proceeding. of the Conference on Banach Algebras and Several Complex Varlablesp (Contemporary mathematics; v. 32) Held to honor Prof. Charles E. Rlckart. Bibliography: p.
1. Banach algebras-Congresses. 3. III.
2.
Rlckart, C. E. (Charles Earl), 1913) Alckart, C. E. (Charles Earl), 1913-
Functions of several complex varlables-Congre.ses. .
I. Greenleaf, Frederick P. II. Gulick. Denny. . IV. Title. V. Series: Contemporary mathematics
(American Mathematical Society); v. 32.
0A326.C65
1983
512'.55
84-18443
ISBN 0-8218-5034-2 (alk. paper)
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Copyright © 1984 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government This volume was printed directly from author prepared copy. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.
A conference in honor of CHARLES E. RICKART
upon his retirement from Yale University.
CONTENTS
xi
Introduction
xiii
Signatures of Participants H. Alexander
Capacities in
¢n
G. Allan
Holomorphic Left-Inverse Functions
B. Aupetit
Geometry of Pseudoconvex Open Sets and Distribution of Values of Analytic Multivalued Functions
15
J. Bachar
Some Results on Range Transformations Between Function Spaces
35
w.
Bade
Recent Results in the Ideal Theory of Radical Convolution Algebras
63
w.
Bade & P. Curtis
Module Derivations from Commutative Banach Algebras
71
F. Bonsall
Criteria for Boundedness and Compactness of Hankel Opera tors
83
H. Dales
Algebra and Topology in Banach Algebras
97
J. Esterle
Mittag-Leffler Methods in the Theory of Banach Algebras and a New Approach to Mlchael's Problem
107
I. Glicksberg
Orthogonal and Representing Measures
131
B. Kramm.
Nuclearity (resp. Schwartzity) Helps to Embed Holomorphic Structure into Spectra
143
D. Kumagai
Maximum Modulus Algebr~s and Multi-Dimensional Analytic Structure
163
K. Laursen
Central Factorization in C*-Algebras and its Use in Automatic Continuity
169
J. McClure
Nonstandard Ideals and Approximations in Primary Weighted tl-Algebras
177
A. O'Farrell, K. Preskenis & D. Walsh
Holomorphic Approximation in Lipschitz Norms
187
1
7
x
CONTENTS
M. Rajagopalan & P. Ramakrishnan
Uses of ~S in Invariant Means and Extremely Left Amenable Semigroups
195
R. Rochberg
Deformation Theory for Uniform Algebras: Introduction
An
209
W. Rudin
Nevanlinna's Interpolation Theorem Revisited
217
S. Sakai
Unbounded Derivations in C*-Algebras and Statistical Mechanics
223
D. Sarason
Remotely Almost Periodic Functions
237
Z. Slodkowski
Analytic Hultifunctions, q-Plurisubharmonic Functions and Uniform Algebras
243
E. Stout
Algebraic Domains in Stein Manifolds
259
J. Wada
Sets of Best Approximations to Elements in Certain Function Spaces
267
J. Wenner
Green's Functions and Polynomial Hulls
B. Yood
Continuous Homomorphisms and Derivations on Banach Algebras
273 279
w.
The Maximal Ideal Space of a Commutative Banach Algebra
Zame
285
INTRODUCTION A Conference on Banach Algebras and Several Complex Variables was held on June 21-24, 1983, to honor Professor Charles E. Rickart as he retired after 40 years at Yale University. *
These Proceedings contain articles submitted
for the Conference. The topics, both at the Conference and in these Proceedings, represent recent advances in a wide spectrum of topics related to Banach algebras, function algebras and infinite dimensional holomorphy.
Professor Rickart has
profoundly affected these areas, and the many participants who have been either associates or graduate students of Professor Rickart would like to join in thanking him for his inspiration. The preparatory work for the conference rested primarily with R. R. Coifman and Phil Curtis, as well as Phyllis Stevens of the Department of Mathematics at Yale University, who lent an expert hand that was invaluable to the conference.
The painstaking job of retyping each manuscript to appear
in these Proceedings fell onto the shoulders of Donna Belli, Caroline Curtis, Mary Ellen Del Vecchio and Bernadette Highsmith ef the Department of Mathematics, under the careful and patient supervision of Regina Hoffman.
To
all those who helped make the conference successful and memorable, as well as to the National Science Foundation ** for its generous support, we wish to express our gratitude. The editors would also like to thank the following conference speakers whose results are appearing elsewhere:
B. Cole, E. Effros, T. Gamelin,
T. Lyons, M. Sibony. With this volume we extend our thanks to Professor Rickart, who has for such a long time served as teacher, researcher, administrator and friend. Fred Greenleaf Denny Gulick
* After
completing his undergraduate study at the University of Kansas, Professor Rickart began his move east, earning his Ph.D. at the University of Michigan in 1941 and serving as Benjamin Peirce Instructor at Harvard University from 1941 to 1943. He joined the Yale faculty in 1943, and became one of the pioneers in the study of Banach algebras. In 1960, Professor Rickart published his classical treatise General Theory .2!. Banach Algebra. More recently his interests have turned to the study of infinite dimensional holomorphy, culminating in his 1979 book Natural Function Algebras, an important contribution to this subject.
** NSF
Grant:
MCS-82l7l28 xi
SIGNATURES OF PARTICIPANTS
'Yl~,.q-~
,/1 /~ .~. .1~
~
t>r.....J-r
~ ~~ ()..-. l-k,~
f}s~ HcL~-.. J ohVl Do/(;;:;,..J ~'j.OJ./ Y
}/tSS I'M
AtJv- .&;~~ [JUt'
9~-J.RJ
k h..()"L/iTl S017
Rtch~, K.lk-~b.~
'6·«1\ __ . .
xiii
P\u
f E.-c~T
Contemporary Mathematics Volume 32, 1984
CAPACITIES IN H. Alexander
1.
INTRODUCTION.
en
*
Various notions of capacity in higher dimensional complex
spaces have been studied during the last few years.
In the classical case of
logarithmic capacity in the complex plane, it turns out that several different possible definitions of a capacity do in fact yield the logarithmic capacity. In the higher dimensional case one obtains different capacities and the problem of relating them to each other. There is naturally a close connection between capacities, pluripolar sets and plurisubharmonic functions.
In particular, Bedford and Taylor have applied
their "local" capacity to obtain results which add considerably to our knowledge about plurisubharmonic functions. The object of this paper is to give a survey of some of these recent developments.
We begin with a list of several equivalent definitions of loga-
rithmic capacity; in higher dimensions these lead to different capacities.
Then
we consider a connection between equilibrium measure and Jensen measure; the local capacity of Bedford and Taylor; Siciak's capacity; the capacities defined from Tchebychef polynomials; and some relationships among the various capacities.
2.
CAPACITY IN
~
I
• We shall recall some of the several equivalent definitions
of logarithmic capacity. 2.1.
Potential theory.
bability measure on
Put
V
measure
= inf ~
IJ.
I(~).
on
defined to be
*Supported
K,
If
The book of Tsuji [12] is a good reference for this. Let
K be a compact subset of
Cl •
If
~ is a pro-
the energy integral is defined by
V<
K such that e-V
~
one shows that there is a unique probability V=
I(~).
Now the capacity of
K,
cap(K),
is
in part by the National Science Foundation
e 1984 American Mathematical Society 0271-4132/84 51.00
1
+ S.2S per page
2
ALEXANDER ~ =
is a monic polynomial 11k of degree kl, where /I • 11K denotes the supremum over K. Then lim ~ k-+""> exists and is equal to the capacity of K. This limit is also equal to 2.2.
Tchebycheff polynomials.
11k
i~f ~
2.3.
;
this follows from the relation
Robin's constant.
singularity at
Let
g(z)
JENSEN MEASURE.
inf {llpIlK:
mj +k
~
mj
p
~.
be the Green's function for
i\K
with
K; 0).
nlen near ~ we have One shows that e- Y is also the capacity of
(assume cap
w
g(z) = loglzl + y + o( Izl). The y is Robin's constant.
3.
Put
K.
Bishop [6] has introduced the notion of Jensen measure
for uniform algebras.
His argument works in the more general context [2] of a
multiplicative semigroup (MSG) of continuous functions; namely, for a compact Hausdorff space ~
X we say that
(a)
ftg E G
(b)
G contains the constants.
f
A closed subset f
E G.
(i)
r of
Suppose that
~(fg)
I
~
~ /If /IX'
C,
boundary for
is an MSG
C C(X)
if
and
X is a boundary for
= ~(f)~(g)
I~(f)
(iii)
• g EG
G
G if
IIf IIr
=
is a complex valued functional on f,g E G,
for all
(ii)
~(1)
IIf "X
for all
G satisfying
= 1 and
Then there exists a probability measure
1.1.
on
r
(a
satisfying logl~(f) I ~ J loglfldl.l.
r
for all f E G. We say that ~ is a Jensen measure for ~. In Bishop's case, G is a uniform algebra on X and ~ is a homomorphism of G. For an arbitrary MSG ~(f)
G,
= exp(J
a functional
~
satisfying
(i) - (iii)
is defined by
X. We shall show that the equilibrium measure of (2.1) can be obtained as a Jensen measure in the following way. For K e t a compact set of positive capacity c, we let P be the MSG consisting of the set of all polynomials n restricted to K. Define a functional ~ on P by ~(p) = an c where n n-l p ( z ) = anz + an_lz + .•. + a O is a polynomial of degree nand c = capacity of K. It is clear that ~ satisfies (i) and (ii). To verify that
10glf/d V),
where
v
is any probability measure on
I~(p) I ~ IIpllK we need only observe that
~
is monic and therefore by n
(2.2) , > m
-
n
> _ cn •
CAPACITIES Now since
We claim that
f(~) = ~ -
for all I(~l) =
z.
z.
(i) - (iii), I-Ll
~(f) =
Then
This implies
I(I-L) ,
and so
3
there is a Jensen measure I-L 1
is the equilibrium measure c, and so (*)
c~ 1
log
I-L = I-Ll
yields ~
I(I-L l )
V
I-L
log c
= 1(1-L) = log
of
J
~
on
(2.1).
K
Let
loglz~ldl-Ll(~)
c. 1
Hence
by uniqueness.
Our interest in this construction is due to the fact that in higher dimensions the notion of energy is not meaningful in this setting and so the equilibrium measure does not arise in this way.
However an analogue can be obtained
as a Jensen measure -- this construction relies on the inequality til
Tchebycheff constant and m is an n n analogously to the construction of Section 2.2.
where
4.
LOCAL CAPACITY OF BEDFORD AND TAYLOR.
c
mn
~
cn '
is a capacity defined below
Bedford and Taylor ([5],[4])
have
developed a powerful theory of local capacity which is based upon the complex Monge-Ampere operator. Let ~ convex domain in Cn ) and let
~.
on
be a ball (or more generally a strictly pseudobe the set of plurisubharmonic functions
p(Q)
Then by a result of Lelong, if
current on
~
u E P(~),
with measure coefficients,
ddcu dC
where
is a positive
= -i(a-a)
In general it is not meaningful to form higher powers of pointwise multiplication of measures. c
(dd u)
k
u E P(~)
c
c
• (dd u) A ••• A (dd u)
n L~l oc (~)
k
C
dd u;
dd c = 2iaa.
this amounts to
However it is shown in [5] that
(k factors)
and that (for
and
1-1
= n)
is well-defined for
(ddcu)n
Moreover the (non-linear) Monge-Ampere cperator
is a positive measure.
u ~ (ddcu)n
is a continuous
(in an appropriate sense) extension from its natural definition on
P(Q)
nt
~
(~).
The MOnge-Ampere operator is a very natural generalization of the Laplacian to several complex variables; in particular, it transforms under a holomorphic change of variables like the Laplacian and, like the Laplacian, it annihilates functions which arise from the "Perron process" [4]. For a compact set capacity of
K of 2,
K with respect to C(KJ~)
= sup{!
Bedford and Taylor define the relative ~
by
(ddcu)n: u E P(~), 0 < u < l}
K
which
To study this capacity they introduce the extremal function is defined as the uppersemlcontinuous regularization of UK{z,Q) = sup{v: v E pm), v < 0 Then
UK* has the following properties:
on 52,
v
< -1
Oll
K}
ALEXANDER
4
The extremal function is used to study plurisubharmonic functions and pluripolar sets; a set E L 5~ is pluripolar i f there exists that E C {z E s;] : u(z) = _GO}.
u E p(SCl), u ,-.... on
The local capacity extends naturally to Borel sets capacity, such that theorem of
=0
C(E,s;])
if and only if
E
E -C SCl ,
SCl
such
giving an inner
is pluripolar.
Extending a
H. Cartan on subharmonic functions, Bedford and Taylor show that
plurisubharmonic functions are "quasicontinuous" in the sense that if then for all
c > 0
uous on
and
Q\W
there exists an open set C(W,SCl) < &.
such that
W C SCl
u
u E P(SCl)
is contin-
They also apply their capacity to settle a well-
known open problem of Lelong on the equality of the class of negligible sets with the class o'f pluripolar sets -- a negligible set is one of the form
* {w < w}
where
w is the sup of an arbitrary family of locally bounded above w*
plurisubharmonic functions and
the usc regularization.
We refer the
reader to the paper of Bedford and Taylor which contains many other interesting results, for instance, a proof of Josefson's theorem [7] on the equality of locally and globally pluripolar sets.
5.
SICIAK'S EXTREMAL FUNCTION.
Siciak in (9] and [10] has studied capacity as
generalized from the Green's function and Robin's constant. done related work in this direction. Let L = {u E P(en ): u(z) :: log Iz I + 0(1) ,n
the Siciak extremal function
U;
as
Zaharjuta [13] has compact in
-+ .... }.
For
K
is defined as follows.
Let
~(z) '"'
z
* the usc regularizatian of ~, is sup {v(z) : vEL, v :: 0 on K}. Then ~, finite when K is not pluripolar. It is easy to see that ~* is the Green's function in the classical case of
n = 1. subharmonic but not pluriharmonic off K.
For n > 1, ~* is in general pluriHowever it does have the following
nice properties:
(b)
* l1<* = 0
(c)
(ddcu;)n
(d)
J (ddcu;)n =
(a)
~(z) =
log Iz on
I+
0(1) ,
i.e ••
l1<*
EL
except for a set of relative capacity
K
is a positive measure supported on C
n
=
K
a constant depending only on
n.
o.
CAPACITIES
5
The analogue of Robin's constant can be defined by
*
(u'. (z) - log 1z I).
'i' = lim z-+""
n > 1.
In general the limi t (vs.
K
One can now define a capacity of
lim)
does not exist for
K as
C(K) - e-Y in direct analogy
*
can be obtained from polynomials
with Section 2.3. Zaharjuta and Siciak have shown that rather than from the more general class
~
L.
This allows one to relate
'"
~
to
the Tchebycheff capacities, which we shall define next.
6.
TCHEBYCHEFF CAPACITIES.
IIPkliK
where
Pk
The idea is to define
is a polynomial of degree
k
~
as the infimum of
which is normalized in some
appropriate way. Then the capacity of K, T(K), is defined as lim ~/k = 11k k~ inf ~ • In the classical case "normalized" means monic. For subsets K of complex projective space a "projective capacity" was obtained in this way in [1], using homogeneous polynomials
Pk • for capacities for compact subsets
Here we want to consider two normalizations K of
¢n.
Pk is normalized if IIPk liB ::: 1 where B is the unit ball, then we obtain a capacity which we shall denote by T(K). 6.1.
If we say that
Pk = ~ + Hk _l + ... + HO as a sum of its homogeneous parts. We can say that Pk is normalized if llI1c. II B:: 1. This yields a capacity that we ,..., shall call T(K). T is translation invariant. 6.2.
7.
Write
SOME COMPARISON RESULTS.
and the local capacity
Having defined the capacities
C(K, ~,
..... C(K) , T(K) , T(K)
one seeks the relations between them.
Siciak
* T(K) - exp(-sup ~(z». Taylor [11] has used this to show 6 Iz I~l A T(K) for all compact subsets of the unit ball, where A and
[10] has shown that that
o
C(K)
~
are constants.
This proof relies on the interesting inequality
f
Iz 1=1 It is easy to check that
*
~ (z) da(z)
T(K) ~ C(K) ~ T(K).
~
ny Some comparisons in which
is dominated are discussed by Levenberg and Taylor [8]. tween local capacity and For
T(K)
K compact in the unit ball
and for all K c {z :
Iz
r < 1
I
~
r}.
T(K)
The relationship be-
is studied in [3] where the following is shown: B in
there is a con£Jant
In,
A - A(r)
such that for
ALEXANDER
6
T(K)
~
exp
~(~~Ba
The work in [3] was motivated by the desire to give a proof using capacity of a
Pl.
It.·nuna of JOHcfson t" ion,
;UiSt!rts that one can find pol.ynomials whicb are small on the set
givL'll
This result, which he proved by an elaborate construc-
~malyth:
polynomials for pll.Lynumia.ls.
funcLion
is I:;n1<.1l1.
It
turn~
out
that
tIlL'
where
Tchebychef [
E alrcady have th<=! dcsired properties of the approximating
rchebycheff polynomials are related to
first one comp~res
log!fl
IInit b~lll.) to ~stimate to
f
E
to the extremal function
C(E,B),
f
L'E*
in the following way: (here
S2 = B,
the
and then the above inequalities relate
C(E,B)
T(E).
REFERENCES
1.
H. Alexander, Projective capacity, Ann. of 1-1ath Studies lOO (1981.), 3-27.
2 ..
H. Alexander, 1.\ note on projective c.apacity, Canau . .J~ }1atll. 34 (198L),
1319-1329.
J.
II. Alexander ,mJ B. A. Taylur, COlllparisun of
tW()
capacities in
\tn ,
to
appear.
4.
Eric Bedford and B. A. Taylor, The Dirichlet problem for a complex MongeAmpere equation, Invent. Nath. 37 (1976), 1-44.
5.
Eric BedfDrd and IL A. Taylor, A new capacity for plurisubharmonic functions, Acta }mth. 149 (1982), 1-40.
O.
E. Bishop, Holomorphic completions, analytic continuations and the interpolation of seminorms, Ann. Hath. 78 (1963), 468-500.
7.
H. Josefson, On the equivalence between locally polar and globally polar sets for plurisubharmonic functions on q:n, Ark. Mat. 16 (1978), 109-115.
8.
Norman Levenberg antI B. A. Taylor, Comparison of a Robin constant with Tchebyshcff constants in a: n , preprint.
9.
J. Siciak., Extremal plurisllbharmonic tunctions in
Ann. Pol. Math.
39 (1981). 175-211. lU.
,J. Slciak, Ext!""~ma1 plurisubharmonic (une tions und capaci ties in Kokyuroku in !,lath. 14, Sophia U., 'l'okyc\, 1982. !;!~limate
(tn, Sophia
11.
1.1. A. Taylor, An 1f'11 " \I., prepnnt.
for an extremal plurisubharmonic function on
12.
N. Tsuji, Potential Theory in :lodcrn Function Theory, Maruzen, Tokyo, 1959.
IJ.
V. Zaharj uta, 1'l"ansfinite di.=imeter. Cebyscv constants and capacity for compacta in [n, }~th USSR Sbornik 25 (1975), 350-364.
'"
itEPAJ.::niE:-.iT 01'
Nl~THE':-lATICS
I:NIVERSITY OF ILLINOIS AT CHICAGO BOX 4348 CHICAGU, IL
60680
"
Contemporary Mathematics Volume 32, 1984
HOLOMORPHIC LEFT-INVERSE FUNCTIONS G. R. Allan
1.
Let let
~n
DOMAINS IN
A be a complex unital Banach algebra (in general non-commutative),
D be a domain in
D such that
fez)
~n
and let
f
be a holomorphic A-valued function on
is left-invertible in
exist a holomorphic A-valued function
g
A for all on
in
z
D with
D.
g(z)f(z)
Does there
=1
throughout
D1 In 1967,
I
D for an
proved a theorem giving a sufficient condition on
affirmative answer to the question.
I shall describe some recent progress
on related questions. The earlier theorem was as follows; write ~(D,A) holomorphic A-valued functions on THEOREM 1 ([1, 2]).
Let
Stein manifold) and le~ in
D, there are elements
there are functions
for the algebra of all
D.
D be a domain of holomorphy in f l , •.. , fk
in e5(D,A)
(or eVen any
be such that. for every
z
k
b l , ..• , b k
gl' •••• gk
Cn
in
Z b i fi(z) i=l such that
A with
in (3(D,A)
= 1.
Then
k
z
gi(z)fi(z)
=1
(z € D).
i=l This theorem is a non-commutative extension of a result of Cart an (31. which gave the result for the case
A
=~.
The original proof of Theorem 1
used the Cartan result together with some Banach-algebraic methods.
Other
proofs have since been given; the.re is a nice account of a proof due to
M. A. Shubin in the survey article [7]; this elegant proof used a theorem on the existence of sections of Banach-analytic vector bundles on Stein manifolds. We now have a number of remarks, complementary to Theorem 1. 1.
Local holomorphic solutions may easily be seen to exist; for suppose
that we are given some
w in
invertible in
fl' ••• , fk Then for
D.
A;
set
z
in ~(DtA) close to
k
and that
Z
bifi(w)
i=1 w, the element h(z)
for
k
= Z bifi(z)
i-I on a neighborhood of w.
= h(z) -1 b i
=1
is
Then
© 1984 American Mathematical Society 0271.4132/84 $1.00 + S.2S per page
7
ALLAN
8 k i~
~
hnlomnrphie and
g.(z)fi(z)
1
=
~
i=l
on that neighborhood.
However, despite Theorem 1, there seems no reason for these particular local solutions D.
gl' .•.• gk
to be analytically continuable.to the whole of
What we can actually prove in this negative direction is the following
Lhl~nrem.
3,
We shull just give the case
IH~low,
wi.l1 make it clear that the case
Theorem 2.
~
could be deduced from
1
Let the holomorphic function
for all
in
O.
f:D
~
A be such that
w E 0; then. provided that
Let
A, for every sufficiently small open polydisc
is a holomorphic function
~
g: 6
A
and
Since
few)
m. say.
0,
011
(z E
g
= h(1.)£(z) = I ~
0
while
g
A. it has distinct left inverses
I
and
h
on
D.
on
0
such that
Then the function
e(z)f(z)
=0
on
O.
= 1.
g(w)
=g
e ~
Let
hew)
- h
= m, while
is holomorphic
be any open polydisc, ~ C
D and
~
e(z)
0
is a domain of holomorphy, we may choose a complex-valued
6
we may so choose
E, we have E
= ah
on
6
~
-
z
as
d
tends to
but we do not need that.)
is holomorphic on
tends to any point of 2.
E
ah
that is not continuable across
E of
that, for some dense subset
~
It(z) I
t
u
from within
from within
h.
In fact
and for all ~.
~(z)e(z)
d(z)f(z) = 1
h.
oh
u
in
(We may even
Now define the function
d(z) = g(z) +
1.
there
~).
holomorphic function
Then
D,
but with
.
~
w, that is sufficiently small so that both
Since
take
E~)
C
It is a simple algebraic corollary to Theorem 1 that there are
e(w)
centre
w E6
with
6
g(z)f(z) = 1 (z
with
is not invertible in
holomorphic functions g(z)f(z)
fez) E Gl is not invertible
few)
not analytically continuable across the boundary of
PROOF.
is the set of left-
Gl
A.
THEOREM 2. z
k
We use the notation of Theorem 1; also.
invertible elements of
in
= 1; the method of proof of Theorem
k
d:6
~
A by
(z E h).
(z E h), but
IId(z)/1 ~
00
as
This concludes the proof.
Using a partition of unity argument, we may glue together local
solutions, obtained as in (1), to obtain global continuous (or even smooth) solutions, without any restriction on the domain solutions, with
f l , .••• fk
given continuous,
D.
In fact, for continuous
D could obviously be any
paracompact Hausdorff space. 3.
Consider the case
k
= 1;
of course, for
A
=
~,
or even for
A any
commutative or any finite-dimensional Banach algebra, the problem has a trivial positive solution, with no restriction on with unique two-sided inverses.
D, since we are then dealing only
However, for general
A that is no longer
HOLOMORPHIC FUNCTIONS
9
the case; in fact we have the following rather strong converse to Theorem 1. We write
H for separable Hilbert space.
THEOREM 3.
Let
D be an open subset of
~D, L(H»
that is pointwise left-invertible on
algebra (9(D, L(H».
Then
(Of course, to say that
f
~
f
in
D is left-invertible in the
~'means
is "pointwise left-invertible on L(H)
for every
z
in
that
D.)
We make use of a well-known converse to the theorem of Cartan
mentioned earlier (see e.g. [4]). k
Suppose that every
D is a domain of holomorphy.
fez) is a left-invertible operator in PROOF.
tn.
1
(f l , ... , f k )
and every k-tuple
functions on
This result states that if, for every of complex-valued holomorhhic
D without a common zero. there exist
gl' ••.• gk
(9 (D)
in
k
L 8i(z)f i (z) = 1 on D, then D is a domain of holomorphy. We i=l shall show that the hypotheses of the theorem imply those of this converse to such that
Cartan's theorem, which will then conclude the proof. Thus, let D.
k ~ 1
Partition the set
Nl • " ' , Nk
and let
has
Xj
and write
Ni
=
{n(i, j): j
on
51' .••• Sk
I}
infinite subsets
k
for
i
= 1, ... ,
H as follows:
k.
Define
represent
H as
l2
(xl' x 2 , ••• ) in H define Si(x) to be that element of H that in the n(i, j)-coordinate and zero elsewhere for i = 1, ... , k, 51
is left-invertible in
H onto a closed subspace, and
spaces
~
have no common zero in
=
Then each
j ~ 1.
of
x
in (?(D)
N of natural numbers into
bounded linear operators and for
f l , •.. , fk
im Si (i
= 1,
•.• , k).
L(H); in fact it is an isometry
H is the orthogonal direct sum of the
Now define the function
f:D
~
L(H)
by
k
fez) Then
f
~
(z ED).
fi(z) Si
i=l
is a holomorphic L(H)-valued function on
no common zero on in fact, given Ti
=
D, it follows that
im Si; then
fi(z)
-1
5., with 1
Ti
Ti
=1
Now let
in
Dt
D;
annihilating the orthogonal
is a left inverse of g:D
~
fez).
L(H)
such that
(z ED). (e )
mm~l
be the standard orthonormal basis of g(z)em
where each
have
fi(z) f 0, and let
so that
By hypotheSis, there is a holomorphic function g(z)f(z)
f l , ... , fk
is pointwise-left-invertible on
z in D, choose i E {I, ..• , k}
be the obvious left inverse of
complement of
f
D; since
=
l2'
Then
(z E D, m ~ 1),
is a complex-valued holomorphic function on
D.
For all
z
10
ALLAN k
=
f(z)e l
i:l fi(z) en(i,l)
and so g(z)f(z)e l
=
k Z
f i (z)
Z
g (. 1) (z)e. P I n 1., ,p
i=l
p
el
and writing
Thus, equating the coefficients of
~
gi
for
gives
8 n (i,1),l
k
Z
=1
fi(z)g.(z)
'11.
(z E D).
~=
The hypotheses of the converse to Cartan's theorem are thus verified, which concludes the proof. We remark that, having used Cartan's result on k-tuples of holomorphic functions to prove Theorem I, we may now use that theorem to generalize Cartan's result to a countable set of functions as follows (and we could also make a similar extension of Theorem 1). PROPOSITION.
Let
D
be a domain of holomo_rphy in
a sequence in {9(D), without a common zero.
in tl(D)
and let
(f i ) i ~ 1
Then there is a sequence
such that, for every compac~ sllbs~
K of
Z
Ilg.II K '1f.:I K
Z
g.(z)f.(z)
i ~ 1
~n
1..
be
(gi)i~ 1
D,
< "" ,
1
while i ~ 1
1
=1
(z E D).
1.
The proof is a straightforward extens:ion of the method of Theorem 3, partitioning
N into infinitely many infinite subsets; of course, the
hypotheses of Theorem 3 are now supplied by Theorem 1.
The details are left to
the interested reader.
2.
TilE LEFT- 11,VERTIBLE ELEMENTS OF A BANACH ALGEBRA
Let
A
be a non-commutative, complex, unital Banach algebra.
the group of invertible elements of
A and let
G.f.
Let
be
G
be the multiplicative
semigroup of all left invertible elements. By elementary Banach-algebra theory ([6]. Chap. 1), both open subsets of
A and
G Is the union of certain whole components of
particular, the identity component of
G..e
Of course, similar remarks apply to the set
A.
G and
are
Gt Gt
;
is just the identity component of G r
in G.
of right-invertible elements of
HOLOMORPHIC
FU~CTIONS
11
We now claim that, for any reasonable definition of infinite-dimensional holomorphy, it is not in general possible to choose, for each left inverse, say
Gt
L(x), in such a way that the function
L
x
in
Gl
. a
is holomorphic on
.
This follows immediately from Theorem 3 because, otherwise, for every open subset D of a: n and every holomorphic mapping f:D -+ G.e' the function Lof
would be a global holomorphic left inverse of
f
on
such a global left inverse need not exist unless
But, by Theorem 3.
D.
D is a domain of holomorphy.
We now ask the somewhat vague question as to whether anything positive can be said about holomorphicity of the left-inverse function in the general case.
We shall indicate one way in which the question can be made precise, and
some sort of positive answer be given.
First we must specify our notion of
holomorphicity.
for complex Banach spaces
We take the simplest:
X, a function
X, Y, and
U
Y will be called holomorphic if it is complex-differentiable in the Freehet sense at every pOint of U. an open subset of
f:U
-+
I
Firstly, relative to our remarks (1) and (2) in §l, we have the following simple lemma.
LEMMA.
Let
(i) K:U
be a left inverse of
= band
K(x)x = 1
I
+
a
-+
(x - a)b E G
Then
and a holomorphic mapping
for all L:G1
a.
x
U;
in
A with
x = a + (x - a) = [1 + (x - a)b]a.
Write
(i)
we have
K(a)
of
U
there is a continuous mapping
(ii)
PROOF.
b
there is an open neighborhood
A with
-+
and let
a E Gt
L(x)x
=
Then for
1
on
Gr.'
IIx-ali < IIbl!
-1
,
and may define
K(x) = h[l + (x - a)b]-l (U)
Since
Gt
is metrizable it is paracompact.
Hence, using (i) , we
{U i }i E I of G.e. and, for each 1, a continuous (in fact holomorphic) function Ki :U i -+ A such that Ki(x)x = 1 on Ui . Now let {si}i E I be a real-valued, continuous partition of unity
may find a locally finite open covering
subordinate to the covering and define for
of course we set
continuous left-inverse function on For each element
Then we have that
x
Gl
of
x £ u. . l.
Then
= {y
E A: yx
= I},
Annt(x)
= {z
E A: zx
= O}.
Annt(x);
therefore connected, subset of
is a well-defined
we set
Invt(x)
=b +
L
G.e.)' where
Gl ·
is a closed left ideal of Inv.e.(x)
= 1 Si(x)Ki(x) (x E
L(x)
G.e.'
A and, for any given
in particular
Inv.e(X)
b
in
Invt(x) ,
is a convex, and
12
ALLAN
If
E
C
Gt
,
define Inv.e.(E)
u
=
Inv.e. (x) •
x E E
COROLLARY. of of
For each component
Ge,
U of
the set
Invt(U)
is a component
Gr , The mapping U ~Invt(U) is a bijection from the set of components G.e. onto the set of components of G . r
PROOF.
By the lemma. there is a continuous mapping
L(x) E Inv.e.(x)
for all
x
is a connected subset of But each
Invt(x)
includes
G
Gr'
If
-+
A such that
U is a component of
then
Gt
that contains a left inverse of each
r
(x E U)
connected subset of
Gt ,
in
L: Gt
is convex and meets
L(U); thus
There is thus a component, say
x
in
Inv.e.(U)
V, of
Gr
L(U) U.
is a that
Inv.e.(U),
A precisely similar argument, with left and right interchanged, shows that there is some single component of
that includes
Gt
obvious notation); that component can only be a left inverse of some element of
V ~ Invr(V)
mapping
U.
U; i.e., V
Invr(V)
(with an
V is
Thus every element of
= Inv.e.(U), Moreover, the
is inverse to the mapping
U
~
Inv.e.(U), which is then a
bijection. A standard way of constructing the Riemann surface of. for example. is to consider the set and then to show that
R of all points
(z, w)
in
,2
that satisfy
z,
R is a connected closed analytic submanifold of
We shall now imitate that process to make a single (infinite-dimensional) complex manifold out of the left inverses of all elements in a single
Gt .
component of Thus, let
Gt and let V = Inv.e.(U) be the G ; then, of course, U x V is an open submanifold
U be a component
corresponding component of
r A
of the complex Banach space
I;t)
of
A.
Now let
M = {(x,y) E U THEOREM 4.
The set
submanifold of PROOF. Then
U
x
V. F:U x V ~ A by
(x, y)
bounded linear mapping
in
(h. k)
in
A
F'(x, y): A
~
~
A ~ A
A.
But then, for any given
= yx,
M, the Frechet derivative of
F'(x, y)(h, k) for all
F(x,y)
so that
M = F-I(l).
U x V of a bounded bilinear mapping
is the restriction to
For each point
V: yx = I} .
M is a connected, closed, imbedded Banach-analytic
Define a function F
x
a
in
A,
=
given by
yh + kx,
F
at
(x,y)
Ax A ~ A. is the
HOLOMORPHIC FUNCTIONS 1
21
F'(x, Y)(2 xa, since
yx
=
I
A
~
A
mapping
for
(x, y)
A;
4
in
M.
13
ay) =
2I
Thus
F'(x, y)
i.e. the identity
I
21
yxa +
of
A
ayx
a,
=
is a surjective linear
is a regular value of
standard applic<.ltion of the implicit function theorem then shows that natural structure as a closed, imbedded, analytic submanifold of To see that
M is connected,
let
(a, b), (c, d)
a, c E U and so may be Joined by a continuous arc in (0
~
t
1).
<
If
L
the lemma, then (c, L(c». joining
t
Also,
t
(a, b)
(c, L(c»
4
to
is the continuous left-inverse function on (y(t), Loy(t» -+
is an arc in
(a, tL(a) + (1 - t)b)
by an arc in
Suppose we write U x V onto
~
~
t
1)
(a, L{a», and similarly we may join M.
We may thus join
which establishes the connectedness of
of
(0
M, joining
(a, b)
CertaInly
holomorphic mappings of the manifold
M has a
-to
M.
Then
yet)
Gl given by (a, L(a» to
is an arc in (c. d)
to
(c, d)
in
M
M,
M.
p. Q for the restrictions to
U, V respectively.
to
t
a
U x V.
be points of U, say
F;
M onto
may think, in Riemann-surface terms, of
P:M
P
M of the projections
U and -+
Q are surjective
and V
respectively.
We
Q
U as the "projection" and
as the single-valued holomorphic left-inverse function on
M.
However, the
analogy with the classi.cal si.tuation breaks down in one important respect, in P
that fibre
is not a covering map (i. e. is not a local homeomorphism), for the
p
-1
(x) = Inv.e(x)
is not discrete (except in the trivial case when A, when
is actually invertible i.n trivial convex subset of
p-l(x)
x
is a singleton) but is a non-
A.
The question naturally arises as to whether we can improve Theorem 4 by exhibiting the various left inverses of elements of a component a covering surface of
Certainly we may take the sheaf
li.
holomorphic A-valued functions points of
Ut and give
surface of
U.
,..,M
f
of
continuations?
f(x)x
U
f(x)x
=
1
the sheaf topology, which makes
M.
N
M of germs of on neighborhoods of
-
M a covering
In other words, given a local holomorphic
= 1, near a point of
G)
li, what are its analytic
of Theorem 2 makes that seem unlikely.
~
being connected)?
The result
Indeed, the method of proof of
Theorem 2 could be used, provided that there is a neighborhood base of A such that, for each Ig(x)l ......
U
that we may obtain all germs of solutions at all points
(which would correspond to the sheaf
N such that
as
~
For example, is it ever the case (except, trivially, when
is a component of in
that satisfy
Gt
However, the interesting question here would be about the
connected components of solution
f
U of
N in the base, there is a holomorphic function aD
as
of points in the boundary of
x N.
tends, from within
0
in g
on
N, to any of a dense set
That presumably involves some restrictions
14
ALLAN
on the Banach-space structure of
A (see [5J), but it seems likely to be
true for many of the standard Banach spaces. Nevertheless, the problem of finding more information about the analytic continuations seems natural and interesting.
For example, is it ever the case
that different germs of left·-inverse functions at the
~
point of
U may be
obtained from each other by analytic t:ontJnuation?
REFEREXCES 1.
G. R. Allan,
Holomorphic vector-valued [unctions on a domain of
holomorphy, 2.
J. London Math. Soc. 42 (1967), 509-513. ideals of vector-valued functions,
•
Proc. London Math. Soc.
(3), 18 (1968), 193-216.
,
3.
H. Carlan, complexes,
4.
R. C. Gunning & H. Rossi, Analytic functions of several complex variables, Prentice Hall, 1965.
5.
P. Noverraz, Pseudo-convexite, convexite polynomiale et domaines d'holomorphie en dimension infinie, Nurth Holland, 1973.
6.
C. E. Rickart, The general theory of Banach algebras, Van Nostrand, 1960.
7.
M. G. Zaidenberg, S. G. Krein, P. A. Kuchment & A. A. Pankov, Banach bundles and linear operators, Russian Math. Surveys 30:5 (1975), 115-175, from Uspekhi Mat. Nauk 30:5 (1975), 101-157.
Ideaux et modules de foncLions analytiques de variables Bull. Soc. Math. France, 78 (1950), 28-64.
UNIVERSITY OF CAMBRIDGE 16 MILL LANE
CAMBRIDGE CB2 ISB GREAT BRITAIN
Contempora.ry Mathematics Volllmf! 32, 1984
GEOMETRY OF PSEUDOCONVEX OPEN SETS AND DISTRIBUTION OF VALUES OF ANALYTIC MULTIVALUED FUNCTIONS
'Ie
Bernard Aupetit **
O.
INTRODUCTION In the last five years the use of subharmonic functions in spectral theory
and in the theory of uniform algebras has given a lot of interesting new results. Perhaps [3] contains the nicest of such striking applications. In the eighties a new and important analytic tool came back to life.
The
origin of this concept goes back to some work of K. Oka related to singularity sets in
E2,
and in some sense, takes roots in the work of Charles Puiseux, a
student of Augustin Cauchy, who obtained, in relation with algebraic geometry, several interesting results about algebroid functions.
Of course K. Oka never
gave applications to functional .malysis. This concept resuscitated in 1980 for the following reasons. settled the following conjecture:
a:
D of
if
f
is an analytic function from a domain
into a Banach algebra .:Ind i f the set of
countable hal:> a positive outer capacity then in
D.
In 1977 I
).,
for which
Sp f ().,)
Sp f (A.)
is
is countable for all
I also explained that this conjecture would solve another one (which is
a generalization of a conjecture due to
A.
Petczy~ski concerning C*-algebras),
which can be expressed in the following way: involution, and suppose that is countable for all
x
in
Sp h A,
duced the
z.
h
= h*. TIl en
Sp x
A has a "particular" algebraic structure. stodkowski had the good idea.
He intro-
followin~:
DEFINITION.
Let
K be a mapping from an open subset
of non-void compact subsets of valued on
A be a Banach algebra with
is countable for all
and
I gave some partial solutions but
let
Em.
We shall say that
U of
En
into the set
K is analytic multi-
U if it satisfies the following properties:
*Ded~cated to Professor Charles Rickart. **This work has been supported by Natural Council of Canada Grant A 7668.
15
Sciences and Engineering Research
AUPETIT
16
s~~i~ontinuous
1)
K is upper
i1)
for every relatively
on
U,
V of
subset
compact_~~~?
U and every function
plurisubharmonic in a neighborhood of the graPE.. of
cP
V,
to
K,
restricted
the function
to. ) = Max{.p (z) : is plurisubharmonic on
zEK(A)}
V.
Then he proved the following remarkable result: THEOREM (Stodkowski).
Let
from
Then
~2
into
U c t
= {(X.,Z):A
t.
be an upper semicontinuous multivalued function
K
is analytic multivalued on
K
E U, z ~ K(\)},
U
if and only if
the compleme.nL of its graph, is a pseudoconvex
~p.en _subset of [2. From this it is not difficult to obtain the following results: i)
is an analytic function from
f
if
x.-i'
then
Sp f (A)
c [
U
into a Banach algebra
is analytic multivalued on
U
,
let A denote a uniform algebra on a compact set
ii)
let
W be a component of
multivalued function
\
-~
analytic multivalued on
a:\f(K). -1 . g(f OJ)
Then for another =
{X(g):X E
f E
and for
K,
g EA
M(A), X(f)
A
the is
= \}
W.
Even recently, T. J. Ransford [13] gave a new example of analytic multivalued functions related with spectral interpolation theory. What is the interest in Stodkowski's result?
At first glance it looks very
strange to start with some spectral problems or somt:! questions related to uniform algebras and then to consider the rather difficult theory of pseudoconvex open sets.
What do we get in doing this?
The main reason is that many pro-
blems in spectral theory and in the theory of uniform algebras are reduced to purely
geometrica~ problems for pseudoconvex open subsets of ~2.
The reader
will understand this very well looking at Theorems 1.8, 1.12, 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 2.1, 2.4, 2.6.
But before that 1 intend to give a very simple
and instructive example (I give only a particular case in order to simplify the situation) •
Suppose that
on an open set
U c t,
.x. . . .
and let
f (\)
Zo
is an analytic family of compact operators E Sp f(\O)
isolated, so suppose that there exist Sp f(AO) B(zO'c)
n B(zO,e) =0
and
=
{zO}
#(Sp f(A)
and
/A -
r > 0
Aul
n B(zO,e» =
<
1.
and
r
/ A - AO I
<
r,
e > 0
implies
Then
Zo
is
such that
Sp f(A)
n
boundary
Then a classical result - due to
aeA)
the Russian school - says that this unique point for
Zo + O.
with
depends holomorphically on
A.
in
Sp f(A) n B(zO,e),
The proof depends strongly
17
ANALYTIC MULTIVALUED FUNCTIONS on the fact that the projection associated to i.e., that
a(~)
is finite dimensional,
is a compact operator on a Banach space.
f(~)
But really we only
need two things depending uniquely on the geometry of the spectra: K(~)
= Sp
f(~)
is analytic multivalued, and that
limit point for all
~.
multivalued and that
Zo
The reason is simple. E
given a few lines above. [4], [7], [13]),
~ ~ K(~)
is single-valued.
K(~O)'
Zo + 0
K(~)
that
has at most zero as a
Suppose that
K is analytic
satisfies similar properties to that
Then by the localization theorem (see for instance
n B(zO,e)
is analytic multivalued on
B(~O,r)
and
So by a famous theorem due to Hartogs it is ho1omorphic.
So
it works very well, even in a very general situationl I must add that during the past three years the new theory of analytic mu1tivalued functions has been increasing very quickly (see [4], [5), [6], [7], lI3], [14], [16], [21]).
1.
DISTRIBUTION OF VALUES OF ANALYTIC MULTIVALUED FUNCTIONS The famous theorem of Picard asserts that a noo-constant entire function
takes all the values of the complex plane except perhaps one point. happens for the union of all the spectral values of function from
~
into M (t)?
f(A)
if
f
But what
is an analytic
This problem was partly studied by E. Borel,
n G. Valiron and G. R~moundos [15], but their arguments are not always very con-
vincing (even H. Cartan in [8] gave some insights on the general situation, but with a false conclusion on the number of exceptional points).
In the first
part of this paragraph I shall describe the work of my student A. Zra1bi [21] about the solution of the previous problem with the help of Nevanlinna theory. In the second part I intend to show the intimate connection between such 2 analytic multiva1ued functions and pseudoconvex open subsets of E. This connection reduces many problems on analytic multivalued functions - hence many spectral problems - to purely geometrical problems on pseudoconvex sets.
This
geometrical idea (inspired by the proof of Tsuji's theorem given on pp. 41-42 of [4]) gives a very simple proof of the generalization of Picard theory to arbitrary analytic multivalued functions. These results are used in the last part to improve the results obtained in [6].
These theorems on analytic multivalued functions are in fact very
strong geometrical extensions of results obtained by F. V. Atkinson, Ju. L. v
Smul'jan and B. Sz.-Nagy, in the fifties, about the distribution of spectral values of
analy~ic
families of compact operators.
18
AUPETIT Let
/zl
be meromorphic for
F
Jo
2;r
N(r ,F) = where disc
net)
f:
< R ~ +~,
Log
+
n(t) -n(O)
IF(re
i9
and let
o<
r < R.
We define
) Ide
dt+n(O) Log r
t
denotes the number of poles, with their multiplicity, in the
B(O,t),
and T(r,F)
= m(r,F) +
N(r,F)
R. Nevanlinna proved the following inequality (see [18] or [21]). LE}U1A 1.1. then for
F
If p
<.
Izl..::
is meromorphic for
R::
+
and if
CIO,
= F(O)
Co
'" 0,00,
r < R we have:
+ + 1 + +1. + + 1 4Log T(r,F) + 3Log r-p + 4Log r + 2Log p + 4Log Log ~ F'
Using the relation
= -
F
F"
+ 16.
and the previous
.-
F'
lemma, it is possible to prove (see [10]) that there exist constant numbers ak, a
k, bk , b k, bk, c k F(k)
1Il(~ '-F- ) < a k
depending only on
+1 F(O): 1 I +b
+
I
k
+ a k Log Log
such that
k Log
+
1
p+
bk, Log
r
.p +
b
k Log
r
r-P
+
c k Log + T(r,F) . The following lemma is a weak form of a theorem due to E. Borel. LEMMA 1.2.
.that
Let
¢l + 4>2 +
Sketch of proof. The
n
functions
be
+ cP n = 1. For all cP i
k
n
linearly independent entire functions such
Then at least one of the
= 1,2, •.• ,n-1
we have
cP i
has a zero.
cP~k) + ... + cP~k) = O.
being linearly independent, the Wronskian
identically zero, where cP l
clJ
¢'
.:[> ,
tjJ (n-l)
r/J(n-l) 2
1
D
cP n
2 2
...
cP'n
=
1
cP(n-l) n
D is not
ANALYTIC MULTIVALUEO FUNCTIONS
19
We introduce also: 1
1
cp (n-l)
¢
1
q, (n-l)
(n-l) 2
1
n
:1>1 Let
D1
be the minor determinant corresponding to the
first line of
~i
and let
0,
consequently
6
m(r,~) < N(r,~)
1
N(r,~)
-
1
vanishing, because 1
N(r ,0) - N(r '0)
,
6
+m(r,6 l ) +m(r,6) +0(1).
= 0/CP2"'¢ 11 ,
we get
N(r,~)
~.
m(r'¢l).
If all the
1
- N(r,X) •
so:
denotes the greatest of the
T(r)
element in the
be the corresponding minor determinant in
We have
If
i-th
T(r,cp ), p
for
p. l, ••• ,n,
applying
the previous inequalities we get T(r) <
O[Log T(r) + Log r]
except perhaps on some segments having a finite total length. So lim T(r) ~ Log r < +., and this gives a contradiction, because this condition implies that the
CPi
are polynomials.
Now we intend to generalize the Picard theorem to finite analytic multi-
,
valued functions by exploiting an idea of G. Remoundos [15].
First we introduce
the notion of multiplicity. LEMMA 1.3. set
D
of
Let
a:
K be an analytic multivalued function defined on an open subsuch that
K(~)
.. {al, ••• ,ap } and i .; j. Then there exist
II(K( A.)
n B(o.i,e» = n i
K()J
e > 0 " > 0 for o <
--
is finite for all
~
in
D.
n B(aj
Let --
,&) -
be such that
B(ai,e)
and intesers
n l , ••• ,np such that and i · l, •••• p.
P"-~ol<"
--
~
for
AUPETIT
20
PROOF.
By the scarcity theorem for analytic multivalued functions (see for
instance Theorem 3.8 in [4]) there exist an integer discrete subset
F of
UK(X)
D such that
=n
n
on
and a closed
D\F
UK(X)
and
<
n
on
If Xo ~ F the lemma is obvious. So we suppose that Au E F and let ~ > 0 be such that B(AO'~) n F = {XC} and B(XO'~) c D. Then K(A O) = {a l ,a 2 , ••• ,a p } with p < n, and let e > 0 be such that B(ai,e) n B(aO,e) = 0 for i ~ j. By the localization theorem (see for instance Theorem 3.14 in [4], or [7], [13]), A ~ K(A) n B(ai,e) are analytic multivalued on a disc B(AO'~l) with ~l <~. So there exist integers n i F.
and closed discrete subsets on
B(AO'~l)\Fi'
#K(A)
Fi
n B(ai,e)
B(AO'~I)
of <
ni
on
such that
#K(A)
n B(ai,e) = n1
Fi' n l + ... + np ... n.
By upper
p
semicontinuity we may suppose that so
This integer
where the 2n-l
0i
=n
for
for
be defined on
E2 ,
i=l
A E B( AO ,'Y)1) \{AO}'
is called the multiplicity of
ai •
are non-constant entire functions.
Then
F has at most
A E E such that
F(X,u) - 0,
except perhaps for at most
u E E there 2n-l
values
u.
PROOF. zeros.
Uo be an exceptional value. Then F(A,UO) is entire and has no heX) Consequently either F(A,uO) is constant or F(X,u O) = e , for Let
some non-constant entire functions distinct exceptional values A
U B(ai ,e>
is included in
exceptional values in the sense of Picard - i.e., for every
exists of
Ai(A)
IIK(A)
K(X)
E~.
h.
First we suppose that
ul ,u2 , •.• ,un
such that
F(X,u i )
F(X,u)
= ki'
has for all
The identities
u
n n
+ Al p . ) u
n-1 n
+. .. + An (X)
define a Cramer system for the unknowns Vandermonde determinant:
Ai(X)
=
k
n
because its determinant is a
n
21
ANALYTIC MULTI VALUED FUNCTIONS n-l
...
0-2
u1
ul
u1
1
u2
1
n-2 u2
n-l
u2
~
u
n-l
un-2
Consequently diction. which
u n
0
0
is constant for all
Ai
Hence there exist at most
F(~,ui)
is constant.
n-1
1
= 1, •.. ,n,
and this is a contra-
exceptional values vO.vl' •••• v
F(~.vi)
where
- e
hiOJ
We have the following identities: hO (\) + A (~) n
=
e
+ A (\) n
=
e
n
Vo h (},,) n
n vn
Let 1
... q
1
=
1 v
n
n
v
n-l
v
o
n
o.
1
where
We have
1
qi
=
v n-1 i _l
v.1.-1
1 .; 0
n-1 v i +1
...
v i +1
1
n-1
...
v
1
v
n
ul , •.• ,un _1
Now we suppose that
tinct exceptional values such that stant entire function.
i
0 .
is also a Vandermonde determinant.
n
So we have
n
hi
are
n+1
for dis-
is a non-con-
AUPETIT
22 n q =
Ghi
(_l)i qi
~
i=O
(l,)
n
n hi 0.,,) i = l: (-1) qi e i-a n =
(_l)i qi e
~
i-a
-Ul ~(J.) v~-k) ]
- i=l Z
(-1)
i
n ~(~) vin-~. ) qi ( k~l
h. (A) 1
because the last term is ho (A) e
hI (A) e
h
e
n
n - va n - v1
(~)
- v
...
n-l a n-l vI
v
n
v
n
n-l n
va
1
vI
1.
v
1
n
a •
=
n
Because
the determinant
can be written
q
and the
~
i=l functions tion.
are linearly independent.
F has at most
Hence
n
So by Lemma 1.2 we get a contradic-
exceptional values of this type, so
2n-l
exceptional values. THEOREM 1.5.
Let
K be a non-constant analytic multivalued function on
Suppose that
K().)
is finite on a set
Then there exists a smallest integer and
a; \
U
K(~)
has at most
2n-l
having a non-zero outer capacity.
F.
n
E.
such that
IIK()..) == n
for all
).. E a;
So outside of
F we
points.
~EI:
PROOF.
The first part comes from the scarcity theorem. where the
F(~,u)
=
n
n (a.(~) - u) 1=1 1
a.
1
are locally holomorphic.
= un + AI( "') un-I + •.. + II.
This function can be extended analytically to all each
Ili (~)
alII:,
with its multiplicity if
A E F.
The
A ("') "n
).. l F.
by Lemma 1.3, counting
a;2
A. (~)
are well defined in
1
and they are entire because they can be expressed as symmetric func-
tion of the
ai
(in fact we use Rad~'s extension theorem).
not all constant because
K is not.
So
u
is not in
is exceptional for
F.
Then we apply Lemma 1.4.
Moreover they are
U K().) ).Ea:
u
for
Let
if and only if
23
ANALYTIC NULTIVALUED FUNCTIONS This result is the best one because, given arbitrary
2n-1
distinct
points, it is possible to construct a finite analytic multivalued function on avoiding these points.
t
Let
be given distinct points, and consider the following
a 1 ,···,B. 2n _1
a:
analytic function from A. Cle +a 1
f{A)
into -C e 2
1
a
0
1
o
o
M (t)
n
A.
C e 3
defined by
A
(_1)n+1 C
n
0
0
2
= a
0
3
o
a
... 1
n
We have
Let
p(z)
=
n-l L Ci(ai+l-z) ... (an-z)+Cn i=l
By induction it is possible to choose ..• (a
C1, •.. ,C
n
p(z) = (an+1-z)
such that
_ -z), and consequently 2n l
Then the analytic multivalued function
avoids exactly the
2n-l
points
~ ~
Sp £().,)
a1,a2, .•. ,a2n_l
=
{z :
det(f(~)
- z) = O}
4
Now we shall be interested in improving the Picard theorem when the analytic multivalued functions take an infinite number of values. First we extend the Liouville theorem and a weak form of Picard's theorem to analytic multivalued functions.
In the following i.e., the union of
THE OREN 1.6. E
=
U K()..) • AElt
for all
\.
Let
K(D A
K(A) K
will denote the polynomially convex hull of
with its holes.
be an analytic. multivalued function on
Then the
K(A),
boundar~
In Earticular if
K
of
E
1s included in the
is bounded then
K ( ".) "
a:
and let
boundar~
of
is constant.
K().,) ,
AUPETIT
24
PROOF.
It is obvious i f
X ~ - Log dist(zO,K(X» 8(zo,e)
n E = _.
E - C.
is subharmonic on
Then
Zo
Suppose that E. ~
- Log dist(zO,K(X»
~
Let
E.
Then
&>
0
- Log e,
be such that
so by the analog of the
Liouville theorem for subharmonic functions we conclude that is constant. Xl
such that
Let
zl
zl
be in the boundary of
is not in the boundary of
and suppose that there exists
E,
K(X I ).
It is obvious that
Zl i K(X l ) because K(A l ) c E and Zl E boundary E. such that B(zl,r) n K(~l) = ~. Let ~2 be such that z2 E 8(zl,~)\E ~~.
There exists r
B(zl'S)
r > 0
n K(X2) = ~
and
We have
dist(z2,K(X 2»
~ ;r and dist(z2.K(A 1 » ~ ~r
and this contradicts
dist(z2,K(X2»
part of the tbeorem.
If
= dist(Z2,K(~*».
sOAwe get the first
E c K(X)
K is bounded we have
K()JA ~ EA
part, and moreover
- Log dist(zO,K(l»
by definition, so
by the first is constant.
K(A)A
This result obviously extends tbe classical theorem of Liouville for entire functions case.
h
K(~)
because
= {heAl}
is analytic multivalued in that
In the second part of theorem we cannot conclude, in general, that
is constant.
To see that let us consider K(~)
.. {z E I:
K(O)
=
K(X)
K defined by
Izi = I},
~
for
1 0
{z E I:
which is analytic multivalued.
E.
THEOREM 1.7. Let K be an analytic mu1tivalued function on K(~)A is constant or U K(~)A is dense in t.
Then either
AEI:
PROOF.
Zo E E and
Suppose there exist
for all
K(i~)A
r > 0
such that
B(zO,r)
n K(~) A = ~
E E. Then u(z) = I/(z-zO) is holomorphic in a neighborhood of for all ~ E t. So by Theorem 3.9 in [4] (see also [1], [13] and [21] ~
for easier proofs), L(~)
is analytic multivalued on constant, so
L(A)A
=
L(O)A.
nomially compact subset of L(~)
A
~
A
u(K(A) )
A
= u(K(O»,
= I:
{u(z) : z E K(A)A}
and
L(A)
C
B(O,l). r
By Theorem 1.6,
It is easy to verify that 1:\8(zO,r) hence
u
L(A)A is
transforms a poly-
into a polynomially compact set. K(A)
A
=
K(O)
A
because
u
So
is one-to-one.
25
ANALYTIC MULTI VALUED FUNCTIONS
From this theorem we can use a very elegant argument due to T. J. Ransford
[13] in order to obtain a nice result previously obtained by the author with the help of more complicated arguments explained in §2. THEOREM 1.8.
K be an analytic multivalued function on
Let
there exists a constant number
C
E.
Suppose that
such that
IRe u-Re vi ~ C
Max
u,vEK(\) for all set
\ E t.
Then there exist an entire function
h
and a fixed compact
E such that
for all
\ E t.
In particular this happens if the diameter of
E.
formly bounded on PROOF.
Let
u(A)
= Max{Re u : u E K(\)} and v(\) = Min{Re v ; v E K(\)}.
The two functions A E t.
So
u-v
is uni-
K(\)
u
and
-v
are subharmonic and
is a constant
u.
But
u
= v~
u(\) - v(\) implies that
~
C for all
u
is simul-
taneously subharmonic and superharmonic, so it is harmonic on all the complex plane.
Hence there exists an entire function \ E t.
for all
Let
h
L(A) • {z-h(\) : z E K(\)}.
such that Then
L
u(\)
=
Re b(\)
is analytic multi-
t. But L(A) is always included in the half-plane {z : Re z ~ By Theorem 1.7, L(\)A is a fixed compact set E, so K(\)A = {h(\)} + E.
valued on
OJ.
A. Zraibi and I obtained in [7] the following generalization of the Picard theorem for analytic multivalued functions: E,
then either
K(A)A
is a
K(\)A
theorem and is rather complicated. that if
K is analytic multivalued on
is constant or the complement of the union of the sets
Go-set having zero capacity.
metric proof.
if
The original proof uses the Frostman
I now intend to give an easy and more geo-
As explained in the introduction, Z. Slodkowski [16] noticed
K is analytic multi valued then the complement of its graph is a
pseudoconvex open subset of
[2.
Moreover, in [16] he discovered the striking
fact that the theory of analytic multivalued functions with values in
E, the
theory of fibres for uniform algebras and spectral theory are locally equivalent. More precisely, if
K is analytic multivalued on an open subset
with values in
then for every relatively compact subdomain
E,
exists an analytic function from
V into
L(l2)
U of V of
E, U there
(the algebra of bounded
AUPETIT
e2 )
operators on the standard Hilbert space all
~
E V,
A on a compact set
and there exist a uniform algebra
V c ~\f(K)
f, g E A such that
elements
K(~) = Sp f(~),
such that
K(~) = g(fl(~»
and
for
K and two
~ E V.
for all
The following lemma will show that it is always possible to associate a lot of analytic multivalued functions to a pseudoconvex open subset of in fact, the four theories of pseudoconvex open subsets of
E,
valued functions with values in
[2,
£2.
So,
analytic multi-
spectra of analytic families of operators
and fibres associated to uniform algebras are "metaphysically" equivalent in the sense that any result in one of these theories will give new information for the others.
This is one of the reasons why I believe that a deeper knowledge
of the geometry of pseudoconvex open subsets of
will have a great impact
[2
on spectral theory. LEMMA 1.9. (AO,a)
Let
E~.
~ be a non-void pseudoconvex open subset of E2 and let D is the open set of
Suppose that
Then the multivalued function
D
K defined on
+ a
K(A)
AE£
such that
(~,a)
E ~.
K(~)
is
~
U {a}
: (~,z) ~ ~}
is analytic multivalued.
PROOF.
It is obvious that
closed because if or
u;Ea
and
u
K(A)
E K(X),
n
lim l/(u -a) n
1
is non-void for
lim u
n
= u,
= l/(u-a).
~
E D.
then either (~,a
But
The set u=a
1
+-) t u -a n
Q
(so
u E K(A»,
implies
(A,a + --) ~ 5"l, so u E K(X). Moreover K(~) is a compact subset of E u-a because if u E K(}..) with limlunl = + 00, it follows from un = zl_a' with n (~,z
n
) ~ 52,
that
us now show that Because
a E
K(~)
lim z =a n
'
so
(X,a)
t
and this is a contradigtion.
S-~.
Let
21 = {(A,Z) ~ E D, z E E, z t K(A)} is pseudoconvex. for all A E D we have (A,a) i Ql for ~ E D, so A E D,
z E a:\{a},
(X 1
'M
+
a) E Q}
= u -l(sa) n D x (1:\ {a})
where
u(z)
because
= z-a 1 +
u-l (52)
a
and
is analytic on D x (E\{a})
D
x
(a:\{a}).
Thus
Q1
is pseudoconvex
are pseudoconvex.
We now need a technical but very important result in the theory of analytic multivalued functions.
ANALYTIC MULTIVALUED FUNCTIONS LEMMA 1.10.
27
K be analytic multivalued on a domain
Let
U of
E.
Then we
have the following properties: i)
either the set of capacity zero or
ii)
X E U such that K(X)
either the set of of outer capacity
= to}
A E U such that zero~E
that case there exists the
Ai
K(X) .. {OJ for all X E U,
holomorphic on
U,
GS-set of outer
is finite is a
Go-set
is finite for all X E U. and in n n-1 u + A1 (A)u + •.. + An(A). with
K{X)
F(A.u)
K(\)
is a
=
such that
K(X)
=
{z : F(A,Z)
= O}
(see Theorem 1.5), iii)
either the set of
A E U such that
Go-set of outer capacity zero or
K(A)
K(A)
is countable is a
is countable for all
A E U.
PROOF. Part i) comes easily from subharmonicity of A ~ Log Max{lzl : z ~ K(X)} and H. Cartan's theorem on polar sets. Part ii) was given in general for the first time by H. Yamaguchi f20] but his argument is not completely convincing. With the help of holomorphic sections of smooth analytic multivalued functions, it was partly corrected in [4]. Now this result is perfectly proved, and even simplified in [7]. [21] or [14].
Part iii)
in 112], but the proof contains fallacies.
was given by T. Nishino
With ideas contained in [6], this
result is now proved correctly and more easily in [13], [14] and (5]. THEOREM 1.11. domain of i)
[
Let
Q
such that
U
either the set of
x
{OJ c Q.
either the set of
{A} x [
C Q
is a
G5 -set of
U x [ c Q,
A E U such that
number of points, is a
U be a
Then we have the following properties:
A E U such that
outer capacity zero or ii)
E2, and let
be a pseudoconvex open subset of
{X} x E c
Q,
except for a finite
Go-set of outer capacity zero or
Ux t
nQ
is the complement of an analytic variety. This 1s obvious by applying Lemma 1.9 with
PROOF.
and ii) if we remark that {A} x
[
c
Q,
{A} x [ c
Q
a
=0
is equivalent to
and Lemma 1.10 i) KOJ .. {OJ
except for a finite number of points, is equivalent to
and that K(X)
being finite.
We are now able to give a generalization of the Picard theorem to analytic multiva1ued functions.
28
AUPETIT We shall say that the analytic multivalued function
there exists an integer
K(jJ.)
n
~
and
1
G{A,Z)
implies
n
K is algebroid if
AI, ... ,An
entire functions
= 0,
such that
G{~.z) = ~n + Al(z)~n-l + •.. +
where
An (z). THEOREM 1.12.
K be an analytic multivalued function on
Let
E.
U is a
If
E\K{AO)' for__~ ~O E E, then either U is a component of E\K{~), for all A E t, or U\ U K{A) is ~ Go-set of outer capacity zero. Ma: . II ~1l particular i f we consider the analyt1c multivalued function K. then either
component of
is constant or C\ U K{~)" is a Go-set of outer capacity zero. MoreII ~E~ over. if K is not cons taut and is not algebroid, then the set F of z for
K(A)II
{~: z E K(A)"}
which
PROOF.
is finite, is a
We may suppose that
~O
= O.
convex, so the same is true for
Go-set of outer capacity zero.
Then Q
=
Q' = {(z,A) :
{(~,z)
(~,z)
E Q}
So we get the first part by applying Theorem 1.11 i). We now prove the last part.
we get the result. the intersection of Moreover
E\K(A)"
FeU.
t\K(jJ.)A
and F
= U.
proof of Theorem 1.5, we conclude that
Is this result the best one?
U x {O} c Q'.
Considering
Let
U
= a:\
n
IJ.EI: is always non-empty,
K(A)II,
the
K{jJ.)A.
Because
U is connecF
is a
But using the argument given in the
"
K
is algebroid.
Given a compact set
C of capacity zero, is
it possible to construct an analytic multivRlued function
f
and
is pseudo-
So by Theorem 1.11 ii) we conclude that either
Go-set having zero capacity or
11:\
K{~)}
U is the unbounded one, so by Theorem 1.6 and Theorem 1.11 i}
only component
ted.
: z f
K on
a:
such that
U K(A)A = C? Is it even possible to do that for K(A) = Sp f(A), where AU: is an analytic function from E into the algebra of compact operators on A. Zralbi [21] obtained the following particular cases:
some Banach space? If
C is a subset of
Ie
not containing
0
and having at most
limit point, there exists an analytic multivalued function
t\
U K{A)"
0
as a
K such that
= C.
ME
If
C is a compact subset of
analytic multi valued function is that
K(A)
t
of capacity zero then there exists an
has holes and the
K(~)A
t\ U
K(~) = C (but the problem ME cover all the plane!).
K such that
ANALYTIC MULTIVALUED FUNCTIONS
29
Almost simultaneously, F. V. Atkinson, B. SZ-Nagy and J. Smultjan (see the references in [6]) obtained the fallowing result: analytic function from a domain
D of
z·E Sp f
(~)
be an
into the algebra of compact oper-
t
ators on a Banach space, and suppose that A E D for which
~ ~ f(~)
let
t
z
Sp f(O).
Then the set of all
is closed and discrete in
with K compact, it is the classical result of F. Riesz.
f (~)
For
D.
0::
For a general
B. Sz.-Nagy believed that this result is deeper than the previous one.
I+>..K,
f, In fact
the three given proofs used extensively the important fact that the projections associated to isolated spectral values of a compact operator are finite-dimensional.
In [6J, using complex function theory, we proved that this result does
not depend really on functional analysis - i.e., that the operator is compact but only an the geometry of the spectra.
K be analytic multivalued on an open subset
Let I-L
E K(~O)'
exist K(A)
a disc
n6
~O
for some
K n K(~O)
is finite for
IX - Aol
< r.
{~}
0::
and an
E.
We say that
K(~O)
is a good isolated point of
E D,
such that
6
D of r > 0
i f there
such that the set
By the scarcity theorem for analytic
exactly
n
points for all
subset.
This. condition of "good isolated point" means geometrically that around
I~
-
~ol <
r,
such that
K(~)
n6
functions there exists an integer
n
~
1
~ultivalued
has
except perhaps on a closed discrete
(AO'~)
the graph of K is an analytic variety. By definition DK(~) is the set of points of
isolated points. A~ DK(~)
K(>")
It is a noteworthy fact that either
which are not good
K(~)
is empty or
is analytic multivalued.
interesting class of analytic multivalued functions containing the
An
spectra of analytic families of compact (or Riesz) operators consists of those K(~)
0 as limit point. K(>") is always finite) or DK(~) 5 {a}. Finally in [6] we got the following:
for which (if
THEOREM 1.13. D in
Let
¢.
has at most
Let z
D
in 1:.
D.
Let
be fixed complex number.
z E K(>")
Let
+0
G
Then every point of
is either isolated or interior.
K be an analytic multivalued function defined on a domain
Suppose that
z
DK(~) 5
K be an analytic multivalued function defined on a domain
{z ED: z E K(>"),DK(>")}
COROLLARY 1.14.
In that case either
K(~)
has at most
be a fixed complex number.
is either closed and discrete in
o
as a limit point for everx
Then the set of
~
in
D or it is all of
D.
~
in
D for which
AUPETIT
30
COROLLARY 1.15.
~ ~ f(~)
Let
be an analytic function from a domain
the set of compact (or Riesz) operators on a Banach space. z ~ Sp f(AO)' for some AO E D. is closed and discrete in D. THEOREM 1.16.
Let
Then the set of all
D into
Suppose that z E Sp f(A)
A for which
K be an analytic multivalued function defined on a domain
D in £. Suppose that R(A) is countable for every ~ in D. Let fixed complex number. Then the set of A in D for which z E R(A) either countable or it is all of
z be a is
D.
Considering the situation of Corollary 1.14, we know that
E(z)is either closed and discrete or it is all of D. But it
{A ED; z E K(A)} may happen that
E(z)
is empty for some
z.
Applying Theorem 1.12, we can
improve Corollary 1.14 in the following way: THEOREM 1.17.
With the same hypotheses as in Corollary 1.14, suppose that
is not constant. that for PROOF.
~
z
Then there exists a closed set
E(z)
F we have
K(A)
F having zero capacity such
discrete and non-void.
See [21] for more details.
To finish this section, I now intend to show that Theorem 1.11 gives a very simple proof of the Tsuji theorem concerning the distribution of values of entire functions of two complex variables ([17. pp. 329-331]).
The original
proof is complicated and uses conformal mapping. THEOREM 1.18 (Tsuji). Let G(~,~) be an entire function on £2 which is not .. \ - eH(~, IJ.), with H entLre . Th en t h ere eX1sts . of the form G( \.,~~ on ..2 ~. a Go-set
IJ. t E there exists
E having zero capacity such that for
satisfying
C(A,IJ.)
O.
C
Moreover if
A in
£
G is not algebroid - i.e., there are no
G(A,~) = an(~)h.n + ... + a1(~)A + aO(~) - there exists a GO-set F having zero capacity such that for IJ.' F there exist an infinite number of A such that G(A,~) a O. entire functions
PROOF. so
for which where domain. £2
U = {IJ. :
Let
G(A,IJ.)
aO, •.• ,an
= A~(A,IJ.), H(O,~)
G(O,IJ.) Let
+O.
+ O.
such that
G(O,~) ~
O}.
U=~
If
for some integer
then
k ~ 1
G(O,IJ.)
for all
IJ.,
and some entire function
H
So we can reduce the general situation to the case ~\U
In that case,
Q = {(A,IJ.) : G(A,~)
1
a}.
is closed and discrete, so
~
by Theorem 1.11, either the set of ~
x {IJ.} c
~
U is a
It is a pseudoconvex open subset of
because it is the complement of an analytic variety.
outer capacity zero or
= 0,
such that
for all
~.
E x
{~}
Then {O} c
~
is a
x
U c~.
So
GS-set of
Suppose now we are in this
ANALYTIC MULTIVALUED FUNCTIONS last situation. G(A,~) - 0
G(~,~) ~ 0
Then
~ E ~\u.
implies
G(At~)
either discrete or that the zeros of
for
~
But for
~
_ 0
E E and
~
31
E U.
fixed the set
as a function of
A.
In other words
= O}
G(~t~)
{A:
is
The first case implies
G are isolated, but this is impossible for an entire func-
tlon of two variables by Hartogs' result.
If now
~ U
~
implies
G(A,~)
= 0,
we can suppose for instance that o f. u, so G(~,O) - O. If we write COl n we conclude that a (0) ... 0, so 1.1. divides a (~) for G(A,~) • 2: a (I-t)A n n n-O n all n. Hence there exists a greatest integer k such that G(A,~) ... ~~(A,~) with
t x {O},
,0.
K(~,O)
K entire and
K(AI~)
Then
and this is a contradiction.
proved.
has isolated zeros on the line
So the first part of the theorem is
The proof of the last part is very similar to the proof of Theorem
1.12.
2.
CONVEX ANALYTIC MULTIVALUED FUNCTIONS Given
K an analytic multivalued function on ~ ~
easy to verify that ~ ~
implies that
co
aK(A) +
K(~)
=
(l-a)K(~)
E and
~
a
1,
~
is analytic multivalued.
(aK(A) + (l-A)K(A)]
U
0
it is This
is also analytic multi-
~~l
valued on
For convex analytic multivalued functions it is possible to
~.
improve Theorem 1.12. THEOREM 2.1.
t.
Let
K be a non-constant convex analytic multivalued function on
n
Suppose that
K(~)
+ 0.
Then
..
ing
Let
a E
n
K(~)
= E.
~EC
~EI:
PROOF.
U
K(A)
and
z E t.
The half-line with origin
a
contain-
AE~
z
that
has not zero capacity, so by Theorem 1.12 there exists z
is" Inthe segment
[a,b].
But
[a,b]cKO..O)'
so
b E K(A O)
such
ZEK(~O)'
As a corollary we get immediately the following result of J. P. Williams [19] •
COROLLARY 2.2. We define x. Then
Let
a,b
be two non-commuting elements of a Banach algebra
W(x) = {f (x) : f E A', U W(e~bae-~b) = t.
II f/l
= f(l) = I}
A.
to be the numerical range of
~Ea:
PROOF.
The multivalued function
~ ~ W(eAbae-~b)
is trivially analytic because
it has entire selections.
Moreover, it is not constant because
W(e~bae-Ab).
So we get the result by Theorem 2.1.
Sp (a) c
n
ab
~
ba,
and
M~
In the case of finite analytic multivalued functions we have the following:
32
AUPETIT
THEOREM 2.3. ~.
Let
K be a finite
analytic multivalued function on
non-cons~ant
Then the union of all the convex hulls of the
K(X)
covers all the plane
except perhaps one point.
discrete set g(A)
K(\) =
By Theorem 1.5 we have
PROOF.
=!n
F,
where the
{al(~)
, ••• ,a n (>..)} outside of the closed are locally holomorphic on t\F. If we define
at
(al(A) + .•• + a n (A»
i\r this function can be extended holo-
on
morphically to all the plane, counting each
F
point of A € 1:.
If
(Lemma 1.3).
So
g
ai
is entire and
with its multiplicity at each g(A) E co K(A),
is not constant, by the Picard theorem
g
point and the proof is finished.
If
g
t\g(£)
is constant, then
for all
has at most one
n
co K(A)
~
0.
AEI:
and we apply Theorem 2.1 to conclude that
U co K(A) Ha:
= 1:.
In fact these last results are particular cases of the following very recent result obtained by T. J. Ransford, with the help of covering spaces and lifts of multivalued functions:
K be analytic multival~~d~ a:, and suppose that K(A) is connected for all A € E. Then either K(A)A is constant or the union of all
THEOREM 2.4. K(X)A
Let
covers all the plane except perhaps one point.
K is a continuous multivalued function such that K(A) is convex for A in an open subset D of a:, then by a classical result of E. Michael (see for instance [9, p. 183]) it has continuous selections. Even for the If
situation we are concerned with, this can be proved elementarily.
LEMMA 2.5. subset I:
Let
D of
for all
K be a continuous multivalued function defined on an open I:
A in
such that
D.
K(>")
is a non-void compact and convex subset of
Then the function
o
defined by
a if
0 E
K(X)
the unique point
Izi is a continuous selection of PROOF.
=
z
of
K(>")
such that
dist(O,K(A»
K.
It is a purely geometrical proof; see [21, pp. 41-42).
All of this suggests the following problems: i)
If
K is a continuous and convex analytic multivalued function on
does it have entire selections?
E,
ANALYTIC MULTIVALUED FUNCTIONS ii)
33
Let K be analytic multivalued on D. Given Zo E aK(AO)' when does there exist (at least locally) a finite analytic multivalued function small?
Zo E L(AO)
L such that
and
L(A) c K(A)
for
IA - Aol
These very difficult problems are, of course, intimately related to the problem of holomorphic support functions for pseudoconvex domains (see [11, pp. 113-114]) and the two problems of H. Alexander and J. Wermer given in [1] and [2].
Anyhow, any solution of one of these questions would have important applications in spectral theory, giving precise information on the distribution and the growth of spectral values.
The only result obtained until now is Theorem 2.6. To state it, we let K be a continuous and convex analytic multivalued function on ~, and let B(K(A» denote the diameter of K(A). We define
OO(A)
6(K(~»
Max
=
I~I
= IAI
which is a continuous and non-decreasing function of nmoREM 2.6. on t.
Let
IAI.
K be a continuous and convex analytic multivalued function
Given an increasing sequence
sequence of polynomials
fR
(Rn)
going to infinity, there e;dsts a
such that
n
MaxlfR (A)-zi ~ 3 oo(Rn ) zEK(A.) n for all
IA.I
~
Rn
and all
n.
r be a circle of radius R, and let r ~ oo(R). Applying the arguments of [1] to r, r and the continuous function a given by Lemma 2.5, PROOF.
Let
we conclude that there exists
fR
continuous for
IAI ~ Rand holomorphic for
IAI < R, such that IAI· R implies IfR(A.) - a(A) I < 2r. By Taylor's theorem applied to {A. IAI ~ R}, we may suppose that fR is a polynomial. Moreover IfR(A.) - zl ~ IfR(z) - a(A) But so
(A,z) ~ IfR(A) - zl IfR(A) - zl < 3r
for
I
+ la(A) - zl
<
is plurisubharmonic and z E K(A.) ,
1).1
3r,
for
z E K(A), IAI • R.
K is analytic multivalued,
~ R.
Of course it is certainly' possible to get more.
But how to do thatl
AUPETIT
34
REFERENCES 1.
H. Alexander and J. Wermer, On the approximation of singularity sets by analytic varieties, Pacific J. Math. 104 (1983), 263-268.
2.
H. Alexander and J. Wermer, On the approximation of singularity sets by analytic varieties II, to appear.
3.
B. Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras, J. Funct. Anal. 47 (1982),1-6.
4.
B. Aupetit, Analytic mu1tivalued functions in Banach algebras and uniform algebras, Adv. Math. 44 (1982), 18-60.
5.
B. Aupetit, Analytic Multivalued Functions: Applications to Spectral Theory and Uniform Algebras, to appear as a book.
6.
B. Aupetit and J. Zem~nek, On zeros of analytic mu1tiva1ued functions, Acta Math. Szeged 46 (1983), to appear. B. Aupetit and A. Zra!bi, Distribution des valeurs des fonctions ana1ytiques multiformes, Studia Math., to appear. , H. Cartan, Sur les zeros des combinaisons lineaires de p fonctions holomorphes donn~es, Mathematica (Cluj) 7 (1933), 5-29. Also in "Oeuvres", vol. 1. R. B. Holmes, Geometric Functional Analysis and its Applications, SpringerVerlag, 1975.
7.
a. 9.
~
10.
King-Lai Hiong, Extension d'un th~or;me de M. R. Nevan1inna, GauthierVillars, 1957.
11.
S. Krantz, Function Theory of Several Complex Variables, Wiley, 1982.
12.
T. Nishino, Sur 1es ensembles pseudoconvexes, J. Math. Kyoto Univ. 1 (1962), 225-245.
13.
T. J. Ransford, Analytic Mu1tivalued Functions, Doctoral Thesis, University of Cambridge, 1983.
14.
T. J. Ransford, Open mapping, inversion and implicit function theorems for analytic multiva1ued functions, J. London Math. Soc., to appear.
15.
G. Remoundos, Extension aux fonctions a1gebro~des du theor~e de M. Picard et ses generalisations, Gauthier-Villars, 1938.
16.
S. Slodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), 363-386.
17 .
M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, 1959. Second edition, corrected, Chelsea, 1975. C. Va1iron, Directions de Borel des fonctions meromorphes, GauthierVillars, 1938.
18. 19.
J. P. Williams, On commutativity and numerical range in Banach algebras, J. Funct. Anal. 10 (1972), 326-329.
20.
H. Yamaguchi, Sur une uniformite des surfaces constantes d'une fonction entiere de deux variables complexes, J. Math. Kyoto Univ. IJ (1973), 417-433.
21.
A. Zraibi, Sur les fonctions analytiques multiformes, Doctoral Thesis, Universite Laval, Quebec, 1983. DE MATHEMATIQUES UNlVERSITE LAVAL QuEBEC, G1K 7P4 CANADA DEPARTEME~
Contemporary MathemaLics
Volume 32, 1984
SOME RESULTS ON RANGE TRANSFORMATIONS
FVNCTION SPACES
BETw~EN
John M. Bachar, Jr.
1.
INTRODUCTION X, Y, Z be sets,
Let B(X,Z) to
A(X,Y)
a set of functions from
B(X,Z)
is a function
For every
(1.1)
The class of all
X to
F:Y - Z
Z.
Y.
and
A range transformation from
A(X,Y)
satisfying
f E A(X, Y), Fol E R(X,Z).
satisfying (1.1) is denoted
F
X to
a set of functions from
Op(A(X,Y)
~
B(X,Z».
f
X
Y
) Y
~lFZ
Figure 1
Y (instead of is usually Z) and a set 1 In such cases, one wishes to choose A(X,Y) as the subset of those
Often one has A(X,Y».
L
A(X,Y l )
functtons in
having range in
Y.
When
X, Y, Y l , Z, A(X,Y 1 )
8(X,Z)
are all understood in a given context, we abbreviate
A(X,Y)
to
Op(A(X,Y)
Ay, -~
and
to
B(X,Z)
B.
Thus
\oIe
will write
A(X,Y l ) ~ R)
Op(Ay
and to
A,
for
B(X,Z».
In a natural way, each range transformation T(F):A(X,Y)-~
B(X,Z),
defined by
(T(F})(f) = PDf
F
induces a map,
for all
f
i=-
A(X,Y).
The basic problem about ranp;c tranRformations is the determination of properties of elements in
Ay,
B,
Op(Ay -, B)
certain elements in
Op(Ay
determination of properties of structure of
Op(Ay
-+
B).
-.
A or
in terms of properties of B),
and the class of all
X, Y, Z, A, T(F),
and the
Ay. in terms of properties of the
See Section 2 [or examples o[ variOUB reRlIl ts on
such problems. Range transformations often have been studied under the following assumptions: (1.2) is either
X is an infinite compact or locally compact Hausdorff space,
R or
closed subset of A,
over
Z,
~
Z,
(real or complex fields, respectively), A(X,Y)
Y
Z
is an open or
is the subset of functions in a Banach algebra,
of Z-valued continuous functions on
X having range in
Y
(i.c.,
© 1984 Amcn..:an Mathcmatkal So.:icty 0271·4U2/84 $1.00~· $.25 pt~r I'ilge
3S
BACHAR
36
Yl = Z and A(X,Y l ) = A), and B(X,Z) of Z-valued continuous functions on X.
is another Banach algebra, over
Z,
Under the assumptions in (1.2), one seeks conditions under which each function in
Op(Ay
B)
-+
is continuous, or holomorphic, or real-analytic, or of Baire
class one, or of Baire class
2,
or Borel measurable, or Lebesgue measurable,
just to cite a small sample of problems that can be studied.
It is the case
that some of these problems make sense only when
Z
Z
=
R,
or
= 1:,
or
Y
is open. In the past, the following terminology has been used. the class
Op(Ay
B(X,Z)
When
the class
-+>
-+>
B(X,Z)
has been called "the functions that operate in
A)
= C(X,Z),
Op(Ay
When
= A,
A. "
the algebra of all Z-valued continuous functions on
X.
has been called "the functions that operate weakly in
C)
A. "
On p. 167 of Rickart's treatise [26], brief mention is made about the problem of which functions operate in a Banach algebra. The main emphasis in this paper will be on conditions ensuring that Op
-+
B) c C(Y ,Z),
OptAy
-+>
B) c Op(Ay
-+
or that C)
Op(Ay>- C) c C(Y ,Z).
is trivially true whenever
will be on conditions ensuring assumptions in (1.2) with
Op(Ay
-+
Since the inclusion B c C,
C) c C(Y,Z).
We will work under the
X compact; sometimes we will give results when
is only assumed to be a set of continuous functions on algebra of continuous functions; mostly, we will assume in
~,
most of the focus
though some results hold for
Z
=R
as well as
A
X rather than a Banach Z
= I:
and
Y open
~.
In Section 2, some examples of earlier results on range transformations will be given.
In Section 3, the main results will be stated.
Section 4 con-
tains proofs of these as well as other theorems of relevance and interest. Section 5 contains a brief survey of other results regarding discontinuity, non-measurability, measurability, and analyticity of functions that operate. full proofs of these and other related topics will appear in a forthcoming research monograph, "Range Transformations Between Function Spaces," by the author. It should be noted in passing that one can define "domain transformations," and even "domain-range transformation pairs," between function spaces. In the most general setting, the definition is as follows. be sets, X'
to
A(X,Y) yl,
functions, (1.3)
and
A'(X',Y')
respectively.
f
X to
X, Y, X', yl Y and from
A domain-range transformation pair is a pair of
to, p) , where 6 :X' For every
sets of functions from
Let
-+
X, p:Y -. Y',
E A(X,Y),
satisfying
pofo(S E A'(X',Y').
RANGE TRANSFORMATIONS
37
f Y
~
X
Xl
The class of all pairs A'(X',Y'».
t6, p)
In the case
X
Figure 2
lp
01
• yl
~fo6
satisfying (1.3) is denoted
= X'
and
= the
6
Op(A(X. Y)
-+
identity map, one sees that we
are dealing with a "pure" range transformation,
p.
In the case
Y
III
and
y'
p - the identity map, we are dealing with a "pure" domain transformation, 6. The subject of domain transformations will not be taken up here.
For an
extensive bibliography and survey of results, see E.A. Nordgren's "Composition Operators in Hilbert Spaces" in {3]. 2.
EXAMPLES OF EARLIER RESULTS ON RANGE TRANSFORMATIONS First. we introduce some notation and state some basic facts. When X is a compact Hausdorff space and Z is ~ or ~. a function
algebra
A on
X means an algebra
A(X,Z)
over
Z of Z-valued continuous
functions on
X whose algebra operations are the pointwise operations of func-
tions on
and which separates the points and contains all constant functions.
X.
A is inverse-closed on on X,
(l/f)
is in
X means that for every
A,
where
a Hanach function algebra
~
II· II
lex)
=1
for all
any non-empty open set Y.
x
For any function algebra
Y as explained above. on X (where
~,
Y in When
A,
X (i.e., H(Y)
Ay
A that never vanishes
for all
x
in
X.
A is
A is a function algebra on
such that
in
in
= l/f(x)
X means that
having a Banach algebra norm fined by
(l/f) (x)
f
11111 ]
= 1,
where
1
EA
is the identity of
X
is deA).
For
denotes the holomorphic functions on
is the set of
f
in
A having range in
A is an inverse-closed Banach function algebra
X is infinite), the A-valued integral.
(2.1)
can be defined (see [26], [13], or [8]).
Here
ber of closed rectifiable curves in Y enclosing its "interior." over Y, that
f (X)
f(~A)'
is identical with
A,
where
and is independent of ~A
inverse-closed on Ff(X)
X;
= Fof(x)
for all
uous complex homomorphism of algebra
B,
A,
because
this fact makes the construction of x E X,
is because "point evaluation at ~anach
in
f)
A.
x"
hence
Ff
y.
Note
is the set of all (neces-
sarily continuous) non-zero complex homomorphisms of over,
(the range of
The integral is defined as the limit of finite Riemann sums
the limit exists in the norm of
f(X)
is a union of a finite num-
y
= Fo£
is a member of
cP A'
Ff
for all
A is
possible. f E
Ay.
MoreThis
and so is a contin-
For any semi-simple commutative unital
Gelfand proved that
B is isometrically isomorphic to a
BACHAR
38
Banach function algebra isomorphism defined on
~
9:B X
= ~A
X·,
A on a compact Hausdorff space,
A is ,. defined by by f(~) = ~(f)
a(f) = f
(for all
(for all,.
~
with the weakest topology under which every
f
..f
the isometric
fEB),
where
is
E ~A)' where ~A is endowed is continuous, where A is
the algebra,. (necessarily a ,. function algebra that ,. is contained in all such f, and where ilfll;;; IIfli (for all f) is the norm on
C(X,t»
of
A.
Two early results on range transformations are as follows. THEOREM 2.1.
Let
A be an inverse-closed complex Banach function algebra on
the infinite compact Hausdorff space
X,
set in
that is, every holomorphic function
y
11:.
Then
algebra
B,
-I>
A),
Let
01;1.
B be any semi-simple commutative unital complex Banach
and let the Banach function algebra
ation.
Then A is inverse-closed on
in
H (Y) c
Q; ,
Y be any non-empty open
A.
operates in
THEOREM 2.2.
H(Y) c. Op(Ay
and let
A be lts Gelfand represent-
and for any non-empty open set
~A'
Y
Op(Ay .... A) •
'rhe latter theorem, often called the Gelfand-Silov Theorem, is thus a corollary of Theorem 2.1; the proofs of both theorems ar.e essentially those found in [26], [13] or IS]. The question of when
OrAAy
-+
= H(Y)
B)
naturally follows from these
two theorems, that is, when it is true that the set of functions operating from one Banach function algebra of holomorphic functions on
Y.
A to another one,
B,
is precisely the set
The following two theorems provide information
in the case where
Y 1s the open unit disc.
THEOREM 2.3 [24].
Let
A be a non-self-adjoint Banach function algebra on the
infinite compact Hausdorff space
X,
whose complete algebra norm is the uniform
X. Let Y = {zl Izl < I}, the open unit disc. Then every complex continuous function, F, on Y that is also in O~Ay .... A) is holomorphic
norm on
.2a. Y. THEOREM 2.4 [1].
Let
A be a Banach funct ion algebra under the uniform norm
9n the infinite compact Hausdorff space
X,
closed function algebra on of complex conjugates unit disc.
Then
(~
X)
ORAy .... B)
=
X
"'
B:> A be another uniformlx B does not contain the se! CA let
and suppose of the functions in
A.
Let
..
Y be the open
H(Y).
It is thus seen that Theorem 2.3 is a corollary of Theorem 2.4, and that the restriction that
F be continuous on Y can be removed.
A result on real-analyticity of functions that operate is the THEOREM 2.5 [27].
Let
fol~owing.
A be the Banach function algebra of all absolutely
39
RANGE TRANSFORMATIONS
T-
{z
E
[llzl = I}
open set in
n=--
1)
on the compact space 00
(the unit circle), with norm
containing
~
Z anz n
convergent Fourier series
~he
of real-analytic functions on
lanl.
n---A)
Then
origin.
Z
Op~ ~
Let
Y be any
is precisely the set
Y.
A result on the structure of A in terms of the existence of a certain element in Op(Ay
~ A)
THEOREM 2.6.[22].
Let
is the following. A be any Banach function algebra on
be assumed 1{lequal" (i.e., homeomorphic) to Define
F(x)
= Vi,
for
x E (0,1]
= Y.
CPA.
Let
and let
X
be self-adjoint on
A
F E Op~J.y ~ A)
If
X, then
A
X.
= C(X,4:).
Another result along these lines is the following. THEOREM 2.7 [9]. ~
X,
and let
of functions in
~
B be a Banach function algebra under the uniform nosm
A" Re B (the space of functions on X that are real parts B).
Let
affine continuous function B "#
C(X)
!h!m.
Op
(Ay
~ A)
Y be any interval in
h E Ox(Ay
~ A)
then
R c [. B
If there is a non-
= C(X).
o
Equiva lently, if
is precisely the set of affine functions on
Y.
No attempt will be given here to list an exhaustive survey on functions that operate.
The reader is referred to the list of references at the end of
this paper, together with the bibliographies contained therein, for more information. 3.
MAIN RESULTS Some notation and definitions will be given first because they are cen-
tral to the exposition that follows. DEFINITION 3. 1.
For any function
f
on a set
X, Xf denotes the range of f. is a set of continuous functions
When X is a compact Hausdorff space and A on X, Zx is the set of f E A such that Xf is infinite and "fII X :5 1, where IIlIx is the uniform norm on X. If A is a Banach function algebra on X whose complete norm II' 'I is not necessarily the uniform norm on X, then GIl
is the set of
f E A such that
Xf
is infinite and
set A of Z-valued continuous functions on f E A, {af-rbla,b E Z} c A. L(Xf,Z) and
For every
Call a
X affine-like if, for every
f E A,
where
denotes the set of all affine functions over
C(X£,Z) For any set
IIfl\::: 1.
A is affine-like, Z restricted to
Xf ,
aenotes the algebra of all Z-valued continuous functions on Xc of Z-valued (Z .. ~ or q:) continuous functions on X conA
taining all polynomials (over
Z)
in
the set of all polynomial functions on to Xf , and P(Xf,Z) function algebra over
f
for every Z,
~-valued
P(Xf,Z)
with coefficients in
denotes its uniform closure on Z of
fEAt
Z,
denotes restricted
Xf • For any Banach functions on X, AO(Xf,Z) denotes the
40
BACHAR GO
set of all absolutely convergent power series,
As stated before, functions on Op(Ay
B)
~
C
X,
OPCAy
c OPCAy
~ B)
~
~
Z a x
n
a
n=O n C),
where
for all
E Z
n C
=
n,
all continuous
C(Y), we necessarily have In oroer to study conditions under which the former inclu-
and so whenever
C(Y).
OPCAy
C)
C
sion holds, a reduction theorem is often helpful, under which 1t suffices to study the more concrete problem of when
Op(AY
~
C')
C(Y),
L
where
L(Xf,Z), P(Xf,Z), P(Xf,Z), or AO(Xf,Z), and C' is C(Xf,Z). is as follows. It will be proved after Corollary 4.19. THEOREM 3.2 (Reduction Theorem).
Z
Y an open set in
Hausdorrf space, and (i)
Let
For any set
= IR
or
A'
is
The theorem
X an infinite compact
C:,
Z.
A of Z-valued continuous functions on
X
that is
.
affine-like, ~ C)
Op(Ay
(ii)
For any set (~
ing all polynomials
OpCAy
=
A
of Z-valued continuous functions on
Z)
in
~ C)
f
= n
for all
Op(Py(Xf,Z)
f
E A,
~
C(Xf,Z».
X conta~
fEl.i
(iii)
For any complex Banach function algebra
plete norm is the uniform norm on
Op(Ay(X,¢) (iv)
~
A
on
X whose com-
X,
= n
C(X,¢»
fE~
op(rY(Xf.~) ~
c(xf,a».
For any complex .l:5anach function algebra A 2!l X,
Op(Ay(X,Q:)
-+
C(X,CC»=
n Op(AO (Xf'~) fEE~1I y
-+
C(X f '¢»·
More definitions are needed before stating the main theorem. DEFINITION 3.3. Let A be any algebra of functions on a set X that separates points and contains ]. For any x E X, M denotes {f E Alf(x) = O}. x
For any complex [unction algebra
Mx
A on a compact Hausdorff space
X we say
is (3.3.1)
nei.ghborhood
LZ
ox
(3.3.2) sequence
(3.3.3)
of
x
such that
f
vanishes on
f
ox'.
EM, x
there is an open
WMOI (has weak mutually orthogonal idempotents) i f there is a
{xn }
that for all
(locally zero) in case for every
C
X\{x}
m,n E ~,
and there is a sequence of functions f
n
(x ) m
=
6
nm
{f }
n
C
M
x
such
(Kronecker delta);
SI¥ (has strong infinite factorization) if for every
f
EM, x
RANGE TRANSFORMATIONS {f(n)} c
there is a sequence
41 {f (n)} C M such that for 1 x f(n+1)f(n+l).
and a sequence
M
x
fen) = 1 1 ' WIF (has weak infinite factorization) if there is an
f = f(l)f(2)"'f Cn )fin )
all n E N, (3.3.4)
and
f E
~
such that the condition in (3.3.3) holds. For any complex commutative normed algebra,
I! II,
with norm
A,
we say
has
A
a B.A.I. (bounded approximate identity) in case either one of the following two
equivalent conditions holds (the equivalence is proven in [7]): (3.3.5)1 There is a x E A,
and a net
K > 0
in
{e\}\EA
such that for all
A
= x.
lim e\x ">..EA
(3.3.5)2 There is a K> 0 such that for every finite set {xl" .• ,xn } CA and every e > 0 there is an element e E A (with liell :: K) such that IIxi - exill <
i = 1, ... ,n.
for
g
We say the complex Banach function algebra
A on X satisfies condition
P
in case (3.3.6) norm over
There is an
X),
sequence ~
=
such that
IIfliX
=1
=1
Inl
(IIIIX
and
{ck }
and
0
n U (ck n..l k=n
+
converging to
Z
C
in
b
00
{c k }
f E A such that
'# 4>.
and every
0,
C
is
A
WI
II fll X< e
e:
and
e > 1~
0,
sup(ck + [aX f + b]) _
and every
e
> 0,
there is
and
1!!
X be an infinite compact Hausdorff space,
fO E A such that
OP(A.}
-+ C (Xf
A.{ !!. Ly(X f ,Z), Py(X f ,Z), Py(X f ,t), 2! AO (X f
o
A is
there are an
0,
Z = R 2! C (except as otherwise noted), ~ Y open in following implies that OpCAy -+ C) C C(Y) : There is an
X.
(weak interpolating) if for every infi-
converging to
Z
THEOREM 3.4 (Main Theorem).
(i)
is an accum-
a
f E A such that for every infinite
z such that \laf+bll x <
raXf +b ])
nite sequence
1s the sup
Xf •
(strongly interpolating) if there is
...
IIfll
A be any set of affine-like Z-va1ued continuous functions on
Let
a
such that
a E~
and there is an
ulation point of
5I
f EA
0
0
Y
,~),
Z. ,Z»
o
Each of the c C (Y) ,
and where
where A
!!
0
defined in (i), (ii) , (iii), or (iv), resp., of Theorem 3.2. (ii)
A is any set of Z-valued continuous functions on
affine-like, and m(Xf ) > 0,
o
(iii)
where
A
~
5I
~
WI,
m is Lebesgue measure in
fO E A
such that
Z.
A is any complex Banach function algebra on
and there is a non-isolated point the
or else there is an
X that is
x E X such that
M
x
X
(~,
Z
= (1:),
satisfies some one of
five conditions (3.3.1) through (3.3.5)1 (or its equivalent, (3.3.5)2)
~
42
BACHAR
Definition 3.3; moreover, in the case of (3.3.4) (i.e., WIF), the element
f
in
(3.3.4) additionally is assumed to satisfy the condition that for every open neighborhood (iv)
Ox
3i
x,
f
is not identically zero on
Ox.
is any complex Banach function algebra on
A
X
that satisfies
condition (3.3.6) of Definition 3.3, (v)
is any complex function algebra on
A
that is complete under
X.
the uniform norm on
4.
X
PROOFS OF THEOREMS Complete proofs of the Reduction Theorem 3.2 and the Main Theorem 3.4
require a number of other theorems ,.,hich now will be discussed.
Some of these
are of interest in their own right. The first proposition disposes of the case where PROPOSITlON 4.1.
Let
X be a compact
Hau~dorff
anl;: function algebra on X. Then: (i) dim A is finite a and dim A
=
card X =
n
(U) (iii) (iv)
is finite. and
X
every point of for everl;:
X
n
is isolated
f E A, X f
space,
X is finite. Z = R .2!.
Q:,
-and
A
= = =>
is finite.
Equivalently, (iv) , (iii)' (ii)' (i)' PROOF.
there exists
f E A
such that
is infinite
Xf
there is a non-isolated point in
X
X is infinite dim A
=
011
The proof is elementary and will only be outlined.
then find, for all
i,j E {l, ••• ,n}
with
i ~ j,
functions
X
If
=
{xl, ••• ,xn },
f .. E A with ~J
fij(x i ) = 1 and fij(x j ) = O. For i E {l, ••. ,n}, put Fi - filfi2···fii···fin then Fi(X j ) = 0ij (Kronecker delta), and {Fl, ••• ,Fn } is a basis for the linear space A. The details of the proof can be finished using only eler;entary topology and other elementary arguments. REMARK 4.2.
X
There is a function algebra
is infinite or
finite range. (iii)')
A of infinite dimension (equivalently,
X has a non-isolated point) such that every
f E A has
Thus, the reverse implication (i)' (or its equivalents, (ii)' or
(iv)' in Proposition 4.1 does not hold. The example is trivial. Take X - {O} U {1/2 n ln • 1,2,3, ••• } and A as the algebra of all Z-valued functions f of O.
=>
on
X
that are constant on some open neighborhood (depending
Though it is known that this function algebra has no
on
complete algebra
norm, this will also follow from the result (proven below) that
any Banach
f)
RANGE TRANSFORMATIONS function algebra
43
on the infinite compact Hausdorff space
is a non-isolated point that there is an
f
x E X with
locally zero, satisfies the condition
M
x
such that
E M
is infinite.
Xf
Because of Proposition 4.1. we shall hereafter assume (equivalently,
dim A
=~
such that there
X
X is infinite
X has a non-isolated point) whenever
or
A is a
X. This is because of the obvious fact that when X is finite, every f E A has finite range, and hence every function F on Y (continuous or not) operates weakly 1n A, and so this case 1s completely function algebra
~
OP(Ay
settled, 1. e.,
C)
-to
We next present
Y.
consists of every function on
an elementary result concerning continuity of
functions
that operate weakly.
PROPOSITION 4.3. (1)
{cn }
C
ni
=
)
there is a sUbseguence
y,
II
X.
!!!.
A
(i)
tion 3.3). and ak
-to
{ck }
-+
O.
C).
WI,
l~m
sup(ak+x f )
¢~.
{aki },
= a ki
with
Choose
Applying
> 0
-to
y
C)
-+
C
WI.
Moreover, for any
C (Y)
Y
a
sequence
lim a ki
there are
X is compact, the closure
is continuous, we must have
ox ,
of
let
15 > 0
for all
x'
in
X,
x
We have
is
then
WI
be arbitrary.
{Xi}
= 0,
Op(Ay
-+
Put
ak
C)
Xf
C
Thus
above).
xi {XI} f (x')
there is an
y
in
X
IIfllX
there is an
such that for all
N
<
Thus
and a subse-
i E fi,
Foh
X.
and is continuous on f(xi) Let
= Xi
X.
(i E fi).
x' E {xi}'
But then there is an
is so BeSince
Because
for every open neighborhood
xi E Ox, for all i ~ N. Now so that o > I(Foh)(x") - (Foh)(x')1
such that
ox I
so
and since
such that Also,
e
x,
g E A and
is compact in
- x.
Thus, let
to the boundary of
f € A such that
lim sup,
C(Y).
C
- c k -y (k E N),
this implies
f.
be arbitrary and pick
x" EO,.
A
By definition of
Since
Xf = f(X),
As for (ii), it
from the definitions of each (Defini-
to be half the distance of
x E X f • Put g = x h = g+y E Ay (see choice of e
cause
WI
we find that there is
and
+ xi'
g
ck
compact,
N.
Op(Ay
-
A is
We shall verify the second condition in (i).
Z
C
Y.
quence
implies
51
Thus we will prove that i f
Let F E Op(Ay
~
such that
}
only if the second condition in (i) holds.
is straightforward that
i
ni
is just the basic fact that a function between twu metric spaces is
continuous if and
x
{c
Z.
PROOF.
that
and every sequence
Y E Y
then
51
either of these two properties fmp1ies that
y € Y
Z any function.
-to
be any affine-like set of Z-va1ued continuous functions on
A
the compact Hausdorff space open in
F:Y
Z,
F(y).
.1!!
(ii)
Y open in
~,
is continuous if and only if for every
F
Y converging to
lim F(c 1
Z = R 2!
Let
N such that
for all
f
44
5
BACHAR
>
l(Foh)(xi) - (Foh) (x') I ~ IF(h(xi»
=
i
and so
=
IF(x-f(xi)+Y) - F(x-f(x')+y) I IF(a k +y) - F(y)
1=
- F(h(x'»
I = IF(C k
IF(g(xi)+y) - F(g(x')+y)I
I
IF(X-Xi+y) - F(y)
) - F(y)
I
for all
i
~
N,
i
ts continuous.
F
We next show that the measure condition in (ii) of the Main Theorem 3.4 implies that
is
A
SI.
~
PROPOSITION 4.4.
A be any affine-like set of Z-valued continuous func-
X.
tions on the compact Hausdorff space m(X f )
0,
where
Let
{c k }
:->
o
PROOF.
If there is an
Z,
is Lebesgue measure in
m
m(X f )
iance of Lebesgue measure,
m(c k + Xf )
=
o
O.
for all
m(aX f ) = m(c k + ax f )
o
k,
for all
()
small as you please by proper choice of
and
IlafOllx
O.
>
o
o
also have
Moreover, it
k.
m(aX f ) = lalm(Xf )
Z,
tn
S1.
By translation invar-
o
a "I: 0
is well-known that for every
A is
then
Z be any sequence converging to
C
.=,su;:;:,;c::;.::h~..:t;.:.:h:::a~t;
fOE A
Thus we
can be made
as
Further, the sets
a "I: O.
00
0< m(aX f ) = m(c +aX f ) U (c k + aX f ) (n E~) satisfy 5n+1 C Sn n 0 k=n 0 o :: m(Sn)' and m(Sn+l) <: m(Sn) for all n. By a well-known fact from measure 00 theory, these cnnditions imply m( n s ) = lim m(S ) ~ m(aX f ) > O. In partiw n=l n n n 0 cular, this implies n 5 = lim sup(ck+aX f ) ~ ¢, as was to be proved.
Sn -
n=l n
k
0
We now turn to the properties mentioned in (iii) of Main Theorem 3.4. PROPOSITION 4.5.
(i)
Hausdorff space,
X.
is
Moreover,
WHOI.
~
A be any complex function algebra on a compact
For any non-isolated
M is x X is metric, then the sequence {x}
if
if
If, in addition,
~
p(x , x n )
0'
where
P
LZ
in
is the metric on
X.
3.4(iii)
=$
is
M x
(iii)
II
A
or is
B.A.I.
PROOF. that
5IF, wrF plus the condition
1
E M x
neighborhood
1/2).
Since
such that Nl(x)
M
x
is
in Theorem
on
in
f
-+ C) C C(Y).
For (i), first select P (x,x 1 ) <
f
SlY
=>
is a complex Banach function algebra and
op(Ay
-
~
M has a B. A. I. => M is SlY x x or WIY plus the condition on f
WMOI.
LZ,
.2!. l4MOT, then
an
Mx
and either
M x
A is a complex Banach function algebra, then
the following implications hold: WIF,
then
C X\{x}
Ll
(3.3.2) can be chosen so that (ii)
x E X,
Xl
t
A separates points and contains
fl (Xl) = 1.
such that
We suppose, inductively, that
(in the metric case, choose
x
f1
Since
M
vanishes on
x
is
Nl(x).
m open neighborhoods,
=X
LZ,
~"'J
Clearly,
Xl ~ Nl(x).
1
points
have been selected such that
{x1, .•. ,Xm } c X\{x}
open
N. (x) (i=l, ••• ,m),
NO(X)
Nl(x)
there is
there is an
have been selected such that
J
lI,
so
Nm(x)'
that
m distinct
RANGE TRANSFORMATIONS Xi E Ni _ I (x)\N i (x) for i=I, .•• ,m,
(4.5.1) that
45
{fI, ••• ,f } eM
m functions
m
have been selected such
x
that
(4.5.2) and fi is zero on Ni (x) for i E U, ... ,m}.
(4.5.3) In case
X is metric, we
Now since
x
is non-isolated, there is an
metric, choose
LZ,
xm+l
* fm+1(xm+1)
such that
such that
= 1.
Since
xm+1 ~ Nm+l (x)
we have Nm+I(x)
for
0
=
N~l(x)
for
p=l, .•• ,m
Nm(X) c ••• c N1 (x».
for
i=l, ••. ,m.
xm+l E Nm(x)\{x};
N~l (x) n Nm(x) ,
Nm(x) , fm+I(X j ) - 0
C
fp(xm+I)
==
i
fm+I
such that
(use the definition of
1
=
1/2
in case
* C-f 1) ••• (D-fm) • fm+1 ;: fm+I
Define
fm+1(xm+l)
<
P(x,xm+I) < l/Zmi-I. Next, there is an
there is an open neighborhood
N~l(X).
p(x,x i )
further assume
fm+l
j=l, ••• ,m
X
is
f~l
E Mx
M x
is
Since
is zero on
fm+l
is zero on
and (4.5.3»,
Nm+l (x),
(see (4.5.1) and (4.5.2», and
(see (4.5.3) and use the fact that
Nm+l(x)
C
This completes the induction, and so (i) is proved.
M have a B.A.I. By Cohen's Factorization x Theorem, for all f EM, there exist g,h E M such that f = gh. Suppose x x (1) (n) we have found f 0.) , ••• , f (n) and f1 , .•• ,f l in Mx such that We now prove (ii).
f =
Let
fi j - 1 )
f(l) ••• f(j)fiJ-l) and
Factorization Theorem to such that that M
x
fin),
f(j)fi j )
is
in
Mx
SIF.
is
SIF
is
WIF.
Finally, we prove that if
M
is
wrF
M
is
WMOI.
Theorem 3.4 (iii),or First, if
lof x
then
SIF
M
x
Applying Cohen's
and this completes the induction and proves
l'
It is trivial from the definitions of that if M
j=2, •.• ,n.
we find that there are f(n+1) , f(n+l) 1
f(n) = f(n+1)£(n+1) 1
for
x
SIF, then is
SIF
x
and
x
LZ,
then
and
WIF
(see Definition 3.3)
plus the condition on M
x
is
WMO!
f
in
by part (i).
Secondly, if
M is SIF and not LZ, then the condition on f in Theorem x 3.4 (iii) clearly holds for every g in M. Therefore it suffices to prove x that M plus the condition on the single element f E M given in Theorem x x 3.4 (iii) implies M is WMO!. x Let be fixed and let {~i} be a positive sequence such
...
Z ~i < 5. The condition on f referred to above is that f is not i=l identically zero on every open neighborhood o of x. Moreover, there are that
x
46
BACHAR
sequences
{f
(1)
, ••• ,f
(n)
, ••• }
(1) (n) {f 1 , ..• ,f 1 , .•. }
and
contained in M
x
such that
= f(l) ••• f(n)fi n )
(4.5.4)
f
Now put
01
(1)
f(n+l)fi n+1 )
=
for all
n. 0 E t.
about
So (0)
1
By the above properties of (1)-.1 hood, f (So (0», of
f,
1 £ (1) (xl) ~ O.
there is a point at wh1ch
x
f
is
Suppose
k
xl
~
x
in the open neighbor-
not zero, i.e.,
f(x l )
~
O.
1
0< If(l)(xl)1 < 0 1 =11/lI f l )lI.
Hence, we have
We now use induction. have been
fin)
and consider the open disk
= 'rI1/11 f1 II
By (4. 5. 4) ,
and
distinct points,
{x1 , ••. ,xk } c X\{x},
such that
s~lected
(4.5.5) and (4.5.6)
If (p) (xp ) I < min {'rI p I 211 f (i) II ·211 f (1+1)" ••• 211 f (p-1) II· 1
= l,2, ••• ,p;
'rIJllf(P)(xj)l. j=l •••• ,p}, p=1,2 ••. .,k.
0k+l = min{TJk+1/211 f (i)" ••• 211 f (k) II
Now let
II fi p ) II ,
'11 f~k+l) II,
i=l, ••• ,k+l;
'rI k+ l I f (k+1) (xj )I, j=1,2, ... ,k}. Now of such
f x.
is not identically zero on Since that
f (k+l)
X is Hausdorff, there is an open neighborhood
Xi
neighborhood.
* Ox
N
x
for
i=l, •••• k.
= 0 x n f(k+1)
f(k+l)(~+l) ~
By (4.5.4),
the open neighborhood
O.
Now
0
-1
(S
(0»
°k+l of x
x
f
is
not identically zero on the
(0», 0k+l Thus we have
of
x.
-1
(S
Select
~+1 ~
x
in
N
x
it
If(k+l) (xk+1)/f(k+l) (X j >I < TJ k+1 , 1 ~ j ~ (k+l) - 1 and If (k+l) (~+l) I < min{'rIk+1/211 f (i) II' 211 f (i+l) II' • •211 f (k) II
.
·11 fi k+1 ) II ,
I
1=1, •.• ,k+l; "k+l f (k+l) (xj ) I , j=l, ••• ,k} = 0k+1' This completes the induction and shows (4.5.5) and (4.5.6) hold Now let sequence in Gk,p Note that
p be a fixed positive integer. M
x
for
= (f(P+l)
for all
We define the following
k = 1,2,3, .•• : - f(P+l)(xp+1)D)'.'(f(P+k) - f(p+k) (Xp+k)])fi p+k ) .
k.
RANGE TRANSFORMATIONS
by use of (4.5.4), (4.5.5) and (4.5.6). Thus, for all k~
and so
00,
ment
j,
{Gk,p}kEN
47
j
00
- Gk+, II < ~ T) +k+i < 2: T) +k+i ~ 0 as ,p J,P i=l P i=l P is a Cauchy sequence, and thus converges to an ele-
IIG k
GEM •
x
p
We check the values of
G
p
at x p ,x p+ l ,xp+2"" • For any y E X. since is a complete algebra norm and
"II
- Gk,p I' since point evaluation at y is a bound complex homomorphism of A. Thus for j=1,2,3, ••. , we have G (y) = lim Gk (y). Replacing y by x p+j p k,P lim Gk (x +,) = 0, and so G (x +,) = 0 for j=1,2,3, •.•• Clearly, p p J k ,p P J G (x) = lim Gk (x) = O. Finally, p k,P G (x ) slim Gk (x) = lim[(f(P+l)(x )_f(p+l)(x »"'(f(p+k)(x) p p k 'P P k P p+l P IG (y) -
Gk
P
,p
(y)1 511G
P
- f(p+k) (Xp+k»]fi p+k ) (x p ) = 1im[f(P+l)(x ) •• 'f(P+k)(x )f(P+k)(x ) (l_f(P+l) (x k
p
p
1
p
p+l
)/f(p+l)(x p
»
"'(l_f(P+k)(x )/f(p+k)(x »] p+k p = lim[f(x )/(f(l)(x )"'f(P)(x })](l_f(P+l)(x )/f(p+l)(x» k p p P p +1 P ••• (l_f(P+k)(x )/f(p+k)(x» p+k p _ (p+k) . (p+k) _ using (4.5.4). Define Ek = f (xp+k)/f (xp ), k-l,2,3, •••• By (4.5.5) and (4.5.6), lekl < T)p+k for k=1,2,3, •.•• We will show {(l-e l )··· (l-£k) }kEN
is a Cauchy sequence which converges to a
~-~
com-
plex number. For
= =
m=1,2,3, ••• ,
1(1-f.1)···(1-ek+m) - (l-el)"'(l-ek>1
1(1-e1)···(1-ek)[(1-ek+l)···(1-ek+m)-1]! l(l-e 1 )···(1-e k )-1+ll·I(1-£k+l)···(1-e k+m)-11
(1+ [exp (! ell+' • '+1 ek 1)-1]) ( exp( Iek+ll+' •• + 1ek+m \>-1) 00 00 ~ exp( ~ T) +')'[exp( Z T)i)-l] ~ 0 as k ~ 00; here, use is made of i=l p ~ i=p+k+l (4.5.5), (4.5.6), and the results in Section 15.3 of [28]. Thus the sequence
!:
00
is Cauchy. for all
k,
I(l-e ).··(l-e )-11 ~ exp( Z T) )-1 < exp 1 k i=p+l i provided B > 0 is chosen sufficiently small, say
Finally,
Thus the limit of this sequence
is non-zero. and so
G (x ) p p
~
B -1 < 1
0 < 1/10.
0
(see the
G (x ) above and the definition of Ek'S). p p We can mUltiply G by a function g which is 0 at xl ••••• x p _ l and p p g (x ) ¢ 0, since A separates points and contains the constant functions, p p and thus we have a function, call it fp, in Mx such that f p (x j ) - 0 for j ¢ p, and such that f (x ) ~ O. We can multiply by a scalar so that we can P P assume f (x ) = 1. This argument works for every p, and so we have produced p p a sequence { f } C M and a sequence {xj } C X\{x} such that p x for all p,j E IN, and so Mx is WHOl. expression for
48
BACHAR
It is easy to see in the
p(x,x j ). ~ 0,
can be chosen so that
{x. } J
induction above that if where
X is metric, then
To prove (iii), it is clear that each of the conditions
WMOr.
Thus, i t suffices to show Let
Put
e = half the distance from
function in
A with norm
nite, where
{xi}
j=l, ••• ,n
to the boundary of
in
k,
Y.
x
WMOI.
~ ~
O.
Let
We will produce a
n {ak } is infi-
f({x i })
condition. clearly,
e/2;
a k , ••• , a k
{ak }
from n
such
a k f.(x.) = a k . i 1 1
clearly,
i=1,2, ••• ,n;
is
is
M
having infinitely
so that
1
for
y
imply
Mx
such that
i
e/2
-+
such that
e
Suppose we have selected
for
1
Also,
1=1,2, ••• ,n.
y
less than
'I 0
a k f l (x1 ) = a k . that
= ck-y
for all
when
ck
~
is the sequence of the WMOI
There is an 1 lIa k fill < i
C) c C(Y)
-+
{c k } be arbitrary, with
y E Y and
many different terms.
1.
Op(Ay
X.
is the metric on
0
and
H _ a f + ••• + n k1 1
IIH n
II
kn
f
lIakn+lfn+lll a
nj Cauchy sequence
for
+1 e-/2n
<:
EM x
n
< e (1/2 + ••• + l/2 n ).
and
and
H (x.) = n J
n.
for
J
kn+l > k n so large Thus, Hn+1 ~- Hn + a k +1 f n+l n. n+l and IIHn+lH·:: dl/2 + ••• + 1/2 ),
has a limit function
.
a
Next select
j=l, ••• ,n+l
and so the IIRII <: e, R(Xj ) = a k ' j=1,2,3, ••• j If we put g = H + y1l, then is continuollS on
~
and so for
g t= Ay
with
HEM
x
F E OP(Ay
-+
C),
Fog
X.
In order to complete the proof, the follOWing lemma is needed. LEMMA 4.6.
b!.E. X be a c.ompact infinite Hausdorff space,
complex function on converging to n
E~.
u.
X,
~
Suppose
Then there exists
open neighborhood
ox ,
~
{an}nEN
{xn}nEN
= an
g(x ) n
for all
=a
and such that every
contains infinitely many
different points of
x' E X such that x
a continuous
a complex sequence of distinct numbers X is such that
C
g
g(x')
{x }. n
PROOF.
The proof is straightforward and involves elementary
topological con-
Siderations, hence we omit it. Using this lemma, we apply it to the sequence {c k }, to g, and to {x.}. We see that g(x j ) = H(X j ) + Y a + y = c jfor j=1,2,3, ••• , and kj
J
so there is hood let that
0
x
I
e > 0
x' E X of
x'
such that
be arbitrary and choose
=
=y
and such that every open neighbor-
contains infinitely many different
e >1(Fog)(x) - (Fog)(x')1
(Fog)(x')1
g(x')
kj
IF(g(x»
- F(g(x')1
0x I ,t':
for all
=
IF(g(x»
points of
(open neighborhood of x E Ox, eO - F(y)f.
Thus, Since
{x.}. J
Next,
x') so
e > I(Fog)(x) 0, x ,F,
contains
RANGE TRANSFORMATIONS
infinitely many pOints of
say
{x j },
{x j }pGN
for some subsequence of
p
we have g
> IF(g(x. » Jp
- F(y) I = IF(C k
) - F(y) j
the following is true:
(depend ing on
for
p=1,2,3, ••••
{ck } of distinct points converging to
for every
there is a subsequence
0,
£. >
such that for
of
I
p
We have shown that for every sequence y,
49
p=1,2, .•• , e > IF(C k
{c k
} j
) - F(y)l. jp
This implies that Thus
Op(Ay
F
is continuous at
C) c C(Y) ,
-+
y.
Since
y
is arbitrary,
F E C(Y).
as was to be proved.
We turn attention now to part (iv) of Main
Theorem 3.4.
We
will employ
part (iv) of the Reduction Theorem 3.2 which will be proven below.
But first
we prove a result on the existence of a bounded approximate identity in a certain maximal ideal in the Banach algebra of all absolutely convergent power series on the closed unit disk. PROPOSITION 4.7. ~ A be the Banach function algebra of absolutely convergent power series on the closed unit disk D, i.e., A = {f E C(D)I there exists
{an} c
a
OD
Z lanl <
such that
-
OD
~
n~
GO
fez)
=
OD
Z anz
n
~
n~
IIfll = r. Ian ,. 1!! ~ be the maximal ideal of functions n=O zED. Then .!a A vanishing at 1. Define ~ E A ~(z) : Z for all a {(l-~) }O
2
with norm
-
£z
~
-
nl
•
011
PROOF.
Let
f
... E. a • 0-0 n
Define
=
n
all zED, and n=O n n Pn(z) = a O + Rn + alz + ••• + anz (z E D) where
E MI'
Thus,
f (z)
n
00
Then
IIf-P II = II-R + Z a z 1\ n n i=n+l n ~[lan+ll + lan+ 2 1 + ••• ] -+ 0 as such that for all n::: n, IIf-p
""
1: a z,
o=
f(l)
=
IRnl + lan+ll + lan+21 + •••
n -+ OD. Thus, given e > 0, there is n e II < el2. f - Pn II < e/ 2 • In particular, II £. n e = O. Thus, there is a polyPn(l) = a O + Rn + a l + ••• + an = i~Oai 00
Note that oomial
Q l'
n-
crucial
of degree
n-l,
such that
P (z) = (l-z)Qn_l(z). n
to the proof later on.
We now show
II (ll_~ctll = 2
for
0 < a < 1.
n
First notice
that
a(a:-1) (o!!. - 1)", (..1L - .1)
a (1-a) (1 _ a) ••• (1 _ ~)
n
n
(4.1.1) (a) "" _ _ _2~_ _ _n;,;;.-....;1~_ _ (-1) n-l (4.7.2)
This step is
1(:>1 =
(_1}n+1(~), n=l,2,3, ••••
2
n-.L
so
BACHAR ""
aa
1 + La I ~)' = 1 + L. (_1)n+1(~), 0 < a < 1. To show the n=l n=l "" latter converges, we apply Raabe's test: given Z a n with all a n > 0, if a n +1 n-1 there 1s a p > 1 such that n(---- - 1) ~ -p for all sufficiently large n, an aa then converges. Thus, La a n n=l a(a- 1 )··.(a-n ) n! 1- 1) n (l (n+1)! a (a- 1 ) ..• (a-[n-1]) - n , - 1) - ~(la-nl-n-1) - n(l an+1 .. ~(n-a-n-1) = ~(-1-a) n+1 n+l n+1
II (I_Z)Cl Il
Thus,
-(1 + ~)
<
verify).
=
for all sufficiently large
I! (lI_met!1 <
Thus
= exp(a
a
shown that
g
n
te1y on
[-1,11
(3ee [11]).
=1
o<
=2
+ (1 - h (1» a
(]_~)a
Thus,
then it can be
-
g
is the ho1omorphic extension of
(D_~)a(:) ... 0,
I' (l_lr)a ll
Next we note that
and
by
[-1,1]
on
h (x) = 1 + a and that the series converges uniformly and absolu-
to the closed unit disk, and
4.7.2)
!l
-1 ~ x < 1,
for
Z (-1) (a)x , -1 ~ x ~ 1,
n~l
ha
g , 0 < a < 1,
g (1) - 0, a can be represented by the infinite series
na
n
aa
In(l-x))
(the last inequality is easy to
0 < a < 1.
aa t
If we define the real functions, g (x)
n
a
so
(1;7)a E
MI'
0 < a < 1.
/. I (~) I - 1 - Z (_l)n(~) (using n=l n=1 (1) = 2 - 0 = 2. Thus lI(l_~)all = 2, - 1 +
a < 1.
{(I-~)ct}Oca
We now will prove f - f - P
Let
+ P
n
Define
n
""f:
e
a
M1 a < 1. Note
is an approximate identity in
E A by
e
- (l_~)ct, 0
=1
a -
<
g
Z I (~) I = Ii (J_~)a!l - 1 = 2 - 1 = 1. Now f - CD-if)af = 0=1 f(D - (l_lr)a) - fe ... e (f-P ) + e P • Hence Ilf - (lJ_?l)afll ~ lie IIl1 f - P a a n an a n
that
lie all'"
&
g
+ lie a (]J-~Q 0 -111 < d2 + liea (l-7)I!'IIQn -111· e: e We now show that for any 0 > 0, there is an with
a
0 < a ~ ao'
l!ea (l-7J)1I < 5.
By choosing
a o> 0
a
=
II e
such that for all
2I1Qn&_111'
the last
&
estimate above reduces to
a<
a ~ a,
which shows
{(l-?l)Cl}n_
~ + ollQn -1!1 ... ~ + ~ - & for
<:
e
1
is an approximate identity.
o vc:;;a< To show this, we compute the following. aa
e (z) .. _ ~ (_l)n(a)zn a
IIf - (1I-3')afll
n- 1
n
= aZ
+ a(1-a)z2+ .•• +
a (1-a) (1 - ct) ••• (1 - -.£..)
2 n
2
and
(e (J-Z»(z) a
= e (z)·(l-z) a
=
az + [a(l-a ) - alz2+ ••• 2
a (l-a)···(l - n:2) n n-l ]z + •.•
a(l-a)···(l -~) + [ n-l n
Thus
lie (1-71) II a
= a[l
+ (1 _ I-C!.) + (I-a _ 2
2
(I-a) (1 _ a) 2 ) + ••• 3
n-l zn+ ••.
RANGE TRANSFORMATIONS (I-a) •.. (1 - ~) _ _ _ _ _ _n_-_l_) +.•.
(I-a)· •. (1 - n~2)
+ ( - - -n-l ----Now the
n
th
n
term of the series in the brackets is
a l l - n~l] (I-a) ••• (1 - n-2) [~ n
liea (l-~) I' ~ '
Thus
2~1
(because
II :: lIea(l-7I) II < rea(.1I-lf)
51
a[l + (l+a)
{--L + 2.1
(l-a)"'(1-n~2)(I+a)
=
n(n-l)
_1_ + ••• +
3.2
+ 3:2 + .•• + n(!-l) + ••. =1).
3a.
Clearly, given for
(5
PROPOSITION 4.8.
a > 0,
0 < a < 0/3, ~
° < a < 1, aa'" a/3
choosing
we have yields
and we are done.
K be any infinite compact set in
!E&
(the closed unit disk), 1 E K,
•
+ ••• }] = a[l + (l+a)]
1
n(n-l)
Since
~ n~:~l)
1
such that
~
is an accumulation point of
KeD ~
K.
A be as in Proposition 4.7, let ~ = {g E C(K) Ig is the restriction to K of some f E A}, and let MI = {g E C(K) Ig is the restriction to K of K
~ f E MI }.
tion to
K),
Then under the map
9:A'" C(K)
defined by
is algebraically homomorphic onto
A
9:f'" f IK
AK' Ml
(restric-
is algebraically
homomorphic onto
Ml ; and under the quotient norm, !I!f+IKili E inf 'If+hli, K hEIK is the closed ideal of functions in A vanishing on K, A/IK ~
~
IK
MI/IK
are Banach algebras.
MlK'
resp.,
under
and
A/TK
and
MlK = {f E AKlf(l)
= o}
is a maximal ideal in
approximate identity in a < 1,
Ml/IK'
{(J-~)
K,
~
are isomorphic with
the natural transference of the quotient norm,
both are Banach function algebras on
o<
Ml/IK
a
111"111,
+ IK}O
and the restrictions to
is a bounded approximate identity in
MI'
to
~
~. ~
Further, ~
HI ' K
is a bounded
K ~
(l_~)a,
Finally,
oti (AK)y'" C)
K
c C(Y)
(Y
PROOF.
All of the assertions are standard facts in Banach algebra theory and
open in
IE).
the verifications are straight forward. is a bounded approximate identity in
Also, verifying that
Ml
{(]-~aIK}o
involves only routine calculations. K
Thus, since implies that
I
is non-isolated in 0
K,
an application of Proposition 4.5 (iii)
p( (AK)y ... C) c C (Y) •
PROPOSITION 4.9.
~
Hausdorff space,
X,
A be a complex Banach function algebra on the compact that satisfies (3.3.6).
Then
OtiAy'" C) c C(Y)
(Y open
!Il. «:). PROOF.
By (3.3.6), we have
Xf C D
(the closed unit disk) with
a
an accu-
X, where lal = 1. Then g = af satisfies (3.3.6) with 1 an accumulation point of XeD. With K EX, K satisfies the
mulation point of
g
g
S2
BACHAR
~
conditions of Proposition 4.8, and clearly,
By Theorem 3.2 (iv) (to be proven below), we get
3.1). C
= AO(Xg,t)
(see Definition
OIi:Ay
C)
4
Op( (~)y .... C) c C(Y).
We next prove some results that will yield the proof of Theorem 3.4 (v) on uniformly closed complex function algebras.
REMARK 4.10. For any compact infinite set K in t, let Coo = {z t KI there is a "finite" polygonal line, P(Z , ... ), which connects z to co, and such that P(z, ... ) c t:\K}. A "finite" polygonal line means a union of a finite number of line segments joined end to end, beginning at It is easy to prove that
Cco
subset
cp),
U c
t
(with
Fr U ;:; {z E: t at
z
U -;:
I
C.
00'
For any
we define the open ball S5(z)
cS > 0,
contains at least one point of
0
centered and also
¢\U
U}.
at least one point of
For any infinite compact set
LEMMA 4.11.
and ending at
is an open and connected subset of
for every
of radius
z
Fr C
K c ~,
GO
is contained in
K
and contains infinitely many points. PROOF.
x E Fr Cco
Let
By compactness.of With
1
open disk
f
.
x
Since
z E: Caa'
C...,
and joining z
x
*
there is an
Caa
since
is open.
Suppose
x
+ K.
z
of
and
C",,'
z :I: x
Clearly, the closed segment
[x,z] c Sf)(x) i..: S5(x) c t\K. there is a finite polygonal line P(z,..,) contained in t'\K (which is not in K) to Attach [x,z] to p (z,ao) at
z,
.10 •
and obtain a finite polygonal line,
...
Q(x,oc),
contained in
and joining
~\K
*
Since by assumption x K, we conclude by definition of Cw that x E: C... , which contradicts the fact that x Ceo' Thus the supposition x K can not hold, so x E K and Fr Cco t; K. x
to
C00
must contain at least one point
S5(x)
since
Clearly,
Xo E: K such that dist(x,K) = Ix-xol :;:. O. the closed disk S''i (x) lies in the complement of K. The
K,
'21 x-x 0 I,
5 =
.
*
*
Next, let Zo E K, and let R(ZO;0) = {zO} U {w E ~Iw -;: zO' arg(w-wO) - O}, 0 ~ e < 2n (Figure 3). TIIUS, R(ZO;8) is the ray beginning
K
R(zO;S) Figure 3
at
zo'
zo and making angle
real axis.
Since
i.e., there is R(zO;9)}.
we
e
relative to a line through
zo
K is compact, there is a "last" point EK
n R(zO;9)
such that
It is easy to verify that
Iwe-zol
Le;:; {w E
clw
the disjOint union of the two "open ll line segments From this it clearly follows that
we
E Fr
e....
we
parallel to the in
K
= sup{lw-zol:w
n
R(zo;9),
EK
n
E R(ZO;S), Iw-wel > O} (zO,we)
and
(We ,00) •
is
S3
RANGE TRANSFORMATIONS There must exist
90 , 0 S 00 < 211,
K = {zO}' contradicting the fact that S :; {a E [0,211) lwei: z} is non-empty. {wa}OES
such that waO~ zO' otherwise K is infinite. Thus the set If S is infinite, then the set
e
is a set of distinct points (Le.,
'I- a'
in
S
wa i: wa'~
implies
K
and {we}eES C Fr Coo. On the other hand, if S is finite, then, since is infinite, there must exist at least one 80 , 0 ~ eO < 211, such that K
n (zO'w e ]
K
n (zO,w e )
is infinite.
It is clear (since
is finite) that
5
o
Fr Coo'
C
and so
Fr Coo
is infinite.
o
DEFINITION 4.12.
Let
as in Remark 4.10. if there is
z
x
to
x EK and there is
E C00
with
IX> ,
P(z
,~)
P (z ,co)
C
x [\K,
-Then
K.
5
C00
{x}.
-
Coo'
S5 (zx)\{x}
that
and
Q(x,oo)
z
Q(x,~)\{x}
c
x
c.
x
of
00
Finally, it is easy to
00
in
K is in
Fr Cao •
(K) be the set of all circularly accessible points of ca is dense in Fr C .
S
Let ca
be
00
x is the finite polygonal line connecting
then the union
(K)
OIl
0/2 > 0, x E Fr C • Then there is x' E S5/2(x) x' K. Since K is compact, there is x" ~ K
Let
PROOF.
So (zx) n K
such that
'\ > 0
see that every circularly accessible point LEMMA 4.14.
C
[x,z] with x is a finite polygonal line connecting x to
z, x Q(x ,ao)\ {x} C G:\K
such that
and let
x
x joined at
P(Z ,(0), x
[
is called circularly accessible from
A point
Furthermore, if
C00 •
be an infinite compact set in
It is easy to see, hy the definition of
REMARK 4.13. L
K
DO
nC
00
,
and
*
clearly,
Figure 4 K
Ix'-x"l = dist(x' ,K) ;:: 5' > 0
such that x E Fr Cao
C
+ lx'-x" I
< 5/2
SS,(x')
nK -
radius
5'
K
on
we have
K,
+ 0/2
~
x'
and
X"'
E (x', x") ;
XII,
dist(x' ,K) ~ Ix-x'i < 6/2.
o. We claim that x"
by the definition of
about
Ca' (x').
=
6' •
ES
If
Clearly, since Also,
ca (K).
C6 ,(x ' )
lx-x" I ~ lx-x' I
Observe that is the circle of
x" E C5 ,(x'), but there may be other points of However, if we let (x' ,x") be the open line segment between
then
x',
(Figure 4).
then
(X',XIl) C
SR'(x').
then SIx '" -x " I (x''')
Clearly.
nK =
{x"}.
(x I ,x") c C.
"" Hence x" E S
ca
Let
(K),
as
claimed. PROPOSITION 4.15. ~ K be an infinite compact set in C, and let P(K,G:) • be the uniform closure of all complex polYnomials in z (see Definition 3.1). ~
of
~A
be the compact Hausdorff space of all non-zero complex homomorphisms... P(K,~), !!!h K imbedded homeomorphica11y in ~A via the standard ;
54
BACHAR
_".. po~i:.n~t:;...;e:;;.v~a::.;l:;.;u_a:;.;t:o.:i:;.:o:;.:no:.'_'.::m:::a:.l;p;..:(..:;s:.;;e_e;...:S_e_c_t_i;.;:::0~n:...;;;2.).:.._.;;:;E.;.:x,;:,t,:e.:;.n,:d-.:ea::.;;.ch... ~A
uous function on C'00 co {Z * P(~A) I z
~
~,
in the usual way.
P(z,~) C t - P(~A)}.
and such that
Suppose there is
Fr C:,
each open disk
K.
entirely outside
1\1. -\0 I <
(since K p(
(1.)
Fr C....
Xs
lies on a
and hence
P(¢A)'
and hence
\) E Fr C!., there is \ E S~/2C>"'0) \ f K by definition of C:.,. Thus 1 so '':)/2. Nmv p - )"lD is never zero on Since
\I(E) = sup I~(f) I.
« p- \
11)
-1
From this we ~ee that
::: II (p-\ D) -1 1K =
)
= l/inflz-~ll ~
zEK (PO E CPA
On the other hand, there is
sup {II zEK
I>
Thus (1) and (2) show the contradiction.
Iz-''ll }
216.
such that
,,«p_\l)-l) = sup I(P_\l)-l(;~) !pE¢A
(2)
and
in! 1\ -z I ";> zEK 1 P(K,¢). In any commutative Banach algebra we have
and
and
C
Also,
C
.5 = dist(~O,K).
is invertible in
vCf) ~.: I'fll
Fr C~
-
which connects
P(z.~),
~
z, z E K,
E C"" hv definition of Coo' By definition of \.0 E Fr e.." contrary to assumption. Thus \0 f K.
\.1 E C!.,
e/2
=
p(z)
Xo
Thus,
we conclude
such that
!z
to a contin-
\0 E Fr C~\Fr Coo. SUEl20se \0 E K. By definition of 56 P"O) contains a point Xc E C!,. By definition of
C!., xl) t p(
Next, let
p
there is a finite polygonal line,
PROOF.
Fr Con'
Define
f E P(K, CC)
p(q)O) = ·\0'
1(p-\l)-l(\'I)O)
so that
1= l/Ip(!po)-\I
Fr C' \. . Fr C
Thus
""
as was to be
",,'
proved. COROLLARY 4.16. PROOF.
Cw
n P(~A) = ¢.
Suppose not.
nition of
C~,
out~ide
K.
crosses itself, as is easy to see. P(~)A)'
Zo
to
~
It can be arranged that Consider
there is a "last" point P(zO'~).
along
polygonal line
Zo E p(9A). Ry defito P(zO'~) connecting
such that
there is a finite polygonal line
wand lying entirely of
Zo E C~
Then there is
P(ZO,m)
zl E P(zo ....)
Note that
zl
f
K.
P(zO'~)
n P(¢A).
n P(ef'A)
never
By compactness
as we proceed from
It is clear that the finite
(in the same zl to direction as from Zo to ~) along P (zO'~) lies entirely outside p(<:PA) with the single exception of the point zl· Thus, it is clear that zl E Fr C' p' (Zl , ...)
obtained by proceeding from
~
~.
But Proposition 4.15 shows contradiction that C.., 'I P(~A)
= ¢.
zl E K.
Fr
C~ C
Fr C....
Thus for every
Since
Zo
Fr C ""
E C~,
Zo
C
K,
we obtain the
* P(~A)
and so
RANGE TRANSFORMATIONS
PROPOSITION 4.17.
~
K be any infinite compact set in
Let
uniform closure on
S5
K of all complex polynomials in
(i)
For every non-isolated point
x
of
(ii)
For every non-isolated point
x
of
and
P(K,t)
the
z.
Fr Cm ,
B.A.I.
PROOF.
By Lemma 4.14, we may select a sequence
(i):
~
in
S (K)\{x} such that x ca n there is a 0 > 0 such that
So
n
{5
SI x -x n
by
g
x
For any
,(xll)\{x} c
n
For each (x~)
nK =
n
n
x" E (x',x), n
cco .
n
n
C~
Since
n(z) = (z-xn )/(z-x"), n
51" x -x n
n P(~A) is in
n
=
n
there is
xn '
n
Ixn -x'n I.
n =
x.
{x·} of distinct points
all
{x }, n
an
and
Here,
n E N.
I(x") nK= {x} n n ¢,
x' E C n ""
and
the function
defined
P(K,I).
n
Figure 5
Co
(x~)
n
'-x) n n Note that g II(X ) = 0 and gx" maps I\S5, (x') n x n n n n Figure 5) into the shaded area in Figure 6.
(which is shaded in
Figure 6
g ,,(x )=0 x n n
The sup of
gx"
over
12(xn-x') n 1/ Ix n+x"-2x' n n I > 1. along
(x~)
a:\So
n
n
Thus,
is assumed at
SUPzEct'~
(x • x' ) •
n n Now fix an
on
x' + (x'-x ) n
Since
lim
Ix"-x I~ n n
n
and is equal to as
(x') Igx"(z) I\, 1 n n
for
Since
n
x" n
~
x n
we have
Kc
P(~A) c
(x~),
I\So
we
n
x"E(x x') n n' n
conclude that
lim
Ix"-x I~ n n x" E(x
x')
n n' n Consider the sequence
{x
+l'x +2""}' nO nO
We have
lim x +j = x. Let nO Z e :: k < 1/10. For
coj-j=l j
BACHAR
56
j ...1,2 ••••
,
pick
x" +jE{x +o,x' +0) nO nO J nO J
so near
Ig x" (x n ) - 11 0 no+j
(1)
and
x no+j
< e·
that
j
eo.
(2)
J
IIg" 11K < 1 + eo. Now define fj:: (p-xl1)gx" gx" •• ·gx" , Xno+j J nO+1 nO+2 nO+j We have fj(x ) = {x -x)g" (x )···8" (x). By a fact nO nO xnO+1 nO XnO+j nO
Notice that j=1,2 •••••
on products of complex numbers (see Section 15.3 of [28]), we have that for
I {1+£ 1 )···(l+e j )-11 ~ (1+le 1 1)···{1+l e j l)-1
(*)
~ exp( If-II + •. ·+1 e 01 )-1. J
If
(**)
Ifj(x
nO
)/(x
nO
~
(1+£I)···(I+e. j )
1 , ••• ,e j are positive, then this, we deduce f.
eXP (f. 1+···+e j ).
Applying
-x)-ll = I (1+[g " (x )-I])"'(l+[g" (x )-1])-11 xnO+l nO xno+j nO GO
=:: exp{lg"
(x
xnO+1
nO
)-11 + ••• + I~II (x )-11)-1 ~ exp( E e j ) - 1 nO+j nO j-l
=:: exp(1/10) - 1 < 1.106 - 1 ... 0.106.
We now show that
{fj(x )/(x -x)}oQU is a Cauchy sequence, and this nO nO J.;a'f will then show that the limit, PO' of this sequence must therefore satisfy Ipo-11 ~ 0.106.
which implies
Po ~ O.
By using (*), (**), (1), (2) above,
we see that I(x -x)-l(f j {X )-fj+k(x »1 nO nO nO • Ig" (xn ) •• 'g" (x n )1·ll-g x II (x n )."g" (x)1 x x x n no+l 0 no+j 0 no+j+1 0 no+j+k 0 j+k ~ :: II g" • • • g" Ii K [ exp ( ~ e i ) -1] =:: (1+£ 1 ) • • • (1+e . )[ exp ( ~ e. i ) -1 ] xnO+1 xnO+j i=j+1 J i=j+1 j
GO
GO
:: exp( Z f.i)[exp( E e i )-l] i-I i=j+1 Letting
j
~
that for all
GO,
~
we conclude that for every
k E IN,
We next show
I (f. (x Je
{fj}jQN
Z £i)-11.
exp(1/10) [exp(
i=j+1
e
>
0,
there exists
je
».
) - f. +k(x (x _x)-ll < e.. nO Je nO nO is a Cauchy sequence. Let y E K.
Then
such
RANGE TRANSFORMATIONS
57
I (fj-f·+k)(y)I = Iy-xll gx " (y)···gx" (y)-gx" (y)···gx (y)1 l nO+1 nO+j nO+1 nO+j+k ::: Iy-xlllgx" ···gx" - gx" ···gx" 11K nO+1 nO+j nO+1 nO+j+k ::: Iy-xl [llgx " 11 K'. '1Igx " 11K + IIg x " 11 K" 'lIgx " 11 K] nO+1 nO+j nO+1 nO+j+k <
ly-x l[(l+&l)···(l+e j ) + (1+e 1 )···(1+e j +k)]
"K
IIg" < 1 + e. for all j, and the latter is xn +j l"" ~ 2Iy-xl'exp( Z e.) < 2Iy-xlexp(1/10) < 2.2l2Iy-xl. i=1 1 Thus, given e > 0, for all y E K n SeI2.212(x) (i.e., for y E K such
because
that
Iy-xl
As for
< eI2.212),
we have
y E K lying outside
SE/2.212(x),
=
M> 0
there is
i=l The latter goes to <
F..
for
F.
j,k=I,2,3, •..•
Iy-xl ~ E/2.2I2,
i.e.,
we note
0
as
such that
"" Z si) - 1].
::: M exp( Z ei)[exp (
kEN
1<
suply-xl ~ M, and so yEK Iy-x I Jgx " (y). "gx" (y) J 'Il-gx " (y) .. ·gx" (y) I nO+1 n O+j n O+j +1 nO+j+k
K,
that by compactness of I (fj-fj+k)(y) 1
1(fj-f j +k )(y)
i=j+1
j
~
=.
Thus, there exists
je
such that for all
and all y E K such that Iy-xl ~ e/2.212, we have I(fje-fje+k)(y) This concludes the proof that {fj}jeN is a Cauchy sequence.
I
~im"f-fj I!K = O. Clearly, ln ~ nO' f(x n ) = ~:: fj(x n ) = 0, since
Hence, there exists
f E P(K,«:)
such that
) = lim f.(x ) ~ 0, and for nO j__ J nO fj (xn ) - 0 for all j > n. In addition, f(x) = 0 since f is continuous on K and x ~ x. Since separates points and contains the constants,
f(x
n
there is a function
* O.
xl, •.• ,xn -1
o
and such that
hex
nO
F = [(l/f(x »f][(l/h(x »h] is in Mx , nO nO nO F (x) = 5 (Kronecker delta) showing M is WHOI. nO n nOn x There is an x' E C~ such that S5(x') n K - {x}, where
Therefore the function
and we have (ii) : 5 ..
h which vanishes at
Ix-xl I.
Define
~,,(z)
-
(z-x)/(z-x~),
~
x
z E K,
where
x~
for
E (x',x),
n
n=I,2,3, ••• , (n=1,2, .•. ) an n
and where
x" n
along
(x' ,x).
IIf-gx"fllK n
Ilgx"Il K n
is bounded, and that for any
such that
It is easy to show
f EM
= I'f (1I-g x ,,) 11K < e.
x
and any
e > 0,
there is
The properties of the
n
g" x n
derived in the proof of (i) enable one to demonstrate this, and the straightforward details are omitted. PROPOSITION 4.18. Hausdorff space
Let X.
A be a Banach function algebra on the compact
~:
)
58
BACHAR
dim A
(i) •
(ii)
let
x E X that is non-isolated.
X is infinite.
(iv)
PROOF.
(linear space dimension).
There is an
- (iii)
..
=~
There is an
f E A such that
Xf
is infinite •
-
By Proposition 4.1, it suffices to show only that (ii) - (iv).
x E X be non-isolated.
Thus,
M is LZ, then M is WHO I by Propox x sition 4.5, and by the construction used in the next to last paragraph there, one obtains an element H
If
in
A
with infinite range.
If, on the other hand,
M is not LZ, then (se"e 3.3.1) there is an f E M such that for every x x open neighborhood 0 of x, f does not vanish on O. Using the Hausdorff x
x
property, it is easy to prove that
f
has infinite range.
COROLLARY 4.19.
If the Banach function algebra
Hausdorff space
X has the uniform norm on
there is an element
f
~
A such that
A on the infinite compact
X as its complete norm, then.
Xf
is infinite.
We can use the previous results to prove the Main Theorem now.
However,
we first will prove the Reduction Theorem. PROOF OF REDUCTION THEOREM 3.2.
The proofs of each of the four parts are
essentially the same, so only part (iv) will be done in detail. F E Op(Ay(X,£)
First, let ~
f E i. 1I
II'
F E Op(AOy (Xf,t:)
-I-
~ C(X,~».
We must show that for every
T~
C(Xf'G:».
this end, let n
g E AO (X£, t:)
...
be
Y
Z Ia I < and n-O n n=O We must sh2w Fog E C(Xf'~)' i.e., Fog is continuous on Xf' n ~ n ~ Now the element G E Z a fn E A since " Z anf II ==: z lanillfli S Z laIlI
arbitrary.
Then
g
must have the form
Z anz , z E Xf'
GO
110
Furthermore, for
But
knowing that if Fog
G(x)
GO
n
GO
n
a f ) (x) - Z a [f(x)] (since point n=O n n=O n evaluation at x is a continuous ~-valued homomorphism) and so we see that "" range G • range g c Y. Also, for all x E X, G(x) = Z a [f(x)]n - g(f(x» n=O n - (gof)(x), so G = gof. Moreover, since F E Op(Ay(X,~) -I- C(X,C», we have FoG E C(X,t:).
x E X,
=
( L.
= Fo(gof) =
FoG
(Fog)of
(Fog)of,
is continuous and if
The question boils down to f
is continuous, then is
continuous? f
continuous, onto
x X,Xf are compact Hausdorff (Fog)of continuous
59
RANGE TRANSFORMATIONS Let
(hence compact) in f
X;
hence,
is continuous; finally,
since
~;
be an arbitrary closed set in
V
is onto.
f
Thus
[(Fog)of]
f([(Fog)of]
-1
(V»
is thus closed
(V)
is compact in
Xf
since
= f(f-l«Fog)-l(V») =
f([(Fog)of]-l(V» (Fog)-l(V)
-1
is closed and so
(Fog)
(Fog)-l(V)
is continuous,
as was to be proved. F E
Second, let we must show
n~
Y
is continuous on
Foh
is true, so assume
h
(Xf'~) ~ C(Xf'~».
Op(AO
fEZIIP
X.
If
has infinite range.
and the element
h
has finite range, then this
The element
G(z) :: I'hl!z, z E Xf ,
h E Ay(X,t),
For any
is in
f:: (lPlhl!)h so
AO (Xf'G:),
FoG
is in is
Y
continuous on
Xf
•
Now for all
= I!hl!{{l/l!h!I)-h(x» = Foh. FoG
so
h(x) ,
x E X, (Gof)(x) - G(f{x»
Gof
= h,
so
=
Fo(Gof) == Foh,
G«l/lIhl!)h(x» so
(FoG)of
=
Thus, since compositions of continuous maps are continuous, and since and
are continuous, we have that
f
Foh
=
{FoG)of
is continuous.
PROOF OF MAIN THEOREM 3.4. This is obvious because of Theorem 3.2.
(i)
This is the content of Proposition 4.3 (ii) and 4.4.
(ii)
This is the content of Proposition 4.5.
(iii)
This is the content of Propositions 4.7, 4.8 and 4.9.
(iv) (v)
Since
X
is infinite, Corollary 4.19 implies there is an f E A
Xf is infinite. Proposition 4.17 shows Mx is WHOT for every non-isolated point x of Fr C~ (which is infinite [and compact] by Lemma such that
4.11).
Hence part (iii) (or else Proposition 4.5 (iii»
Op(Py(Xf'~) ~ C{Xf'~»
5.
C{Y),
C
implies
and the result follows from Theorem 3.2 (ili).
SURVEY OF OTHER RESULTS AND SOME UNSOLVED PROBLEMS The following (5.1, 5.2 and 5.3) will appear in [2]:
(5.l)
There is a function algebra
X such that
Op(Ay
~
C)
A on an infinite compact Hausdorff space
contains an element
tinuous and non-Lebesgue measurable on
F
that is everywhere discon-
Y.
This algebra is an algebra of polynomials on a certain "thin" space of course has no complete algebra norm.
When
X,
and
A has a complete algebra norm,
we have: (5.2)
For every Banach function algebra
space
X,
each
(5.2.1)
F E Op(Ay The set has empty
DF
~
C)
with
A on an infinite compact Hausdorff
Y open in
of points of
C satisfies
Y at which
F
is discontinuous
interior.
In order to state the next result, recall that a Banach function algebra
60
BACHAR
A on
X
is defined to be "natural" if
"point evaluation map" (5.3)
Let
space
X. (i)
x EX
t-+
X is homeomorphic with
IPx E 4>A'
where
q>
x
(f)
= f(x),
via the
~A
f E A.
A be a Banach function algebra on the infinite compact Hausdorff i=1,2,
f-1 = ±I,
If
(5.3.1)
there
f l ,f 2 E A such that
are functions
o E int(elX f
pair
and if
1
+ EZX f )
~ ~
for some specific
Z
(el,e Z)'
then (5.3.2)
every
F E Op(Ay
~
C)
is of Baire class
1
(or equivalently, is the pointwise limit of some
{Fn } of continuous functions on V). (ii) If there is an f E A such that Xf has an infinite connected component, then 0 E int(X f + iXf } ~~, which, in turn, implies (5.3.2). (iii) If A is natural and if there is f E A such that Xf contains a countably infinite sequence of pairwise disjoint closed subsets, E (n E ~), sequence
n
with dist (X f \En ,En ) > 0 for n E N, then Op(~~ -I> C) C C(Y}. If there exists --y no such f, then (5.3.2) holds, even when A is not necessarily natural. In [6] the following results are proved:
(5.4) X,
If
A is a Banach function algebra on the infinite compact metric space
and if
is natural, then either Op(Ay
A
F E OpCAy ~ A)
of
~ C) C C(Y),
or else every
is locally Lipschitz on some dense open subset (depending on F)
Y.
(5.5)
There is a Banach function algebra
is an
F E
Op(Ay
-I>
single point of
A)
(F
A on
X
= [0,1]
such that there
operates "stronglyll) which is discontinuous at a
Y.
Since it is possible that a function that is locally Lipschitz on a dense open subset of
Y may actually be non-Lebesgue measurable (such an example is
easy to construct), we see that (5.3) (iii), together with the fact that Op(Ay
~
A)
C
Op(Ay
-I>
C),
actually strengthens the conclusion of (5.4) to the
fact that (5.3.2) holds also. In view of the above results, the following unsolved problems arise. Ql.
For every complex Banach function algebra Hausdorff space F E
O~Ay
-I>
Baire class Q2.
C}
a
Same as Ql, with
X,
and for
Y open in
A on an infinite compact t,
is it true that every
is Lebesgue measurable (or Borel measurable, or of
a)?
for some Op(Ay
-I>
C)
replaced by
Op(Ay
-I>
A).
61
RANGE TRANSFORMATIONS
Q3.
Does there exist points in
F E Op(Ay
~
A)
with infinitely many discontinuity
Y?
In view of the above results and those of Section 4, a negative answer to Q1 could obtain only when (2)
f E Ay, m(X f ) = 0, condition (3.3.6) fails for every
(3)
conditions (5.3) (i) [i.e., (5.3.1)] and (5.3)
(4)
A is not natural,
(5)
every candidate
(1)
for all
f E A,
(ii) fail for all
f E A,
F E Op(Ay
~
able must be such that int DF = ~. In [2J we prove that an F E Op(Ay
C) ~
that might be non-Lebesgue measurC)
exists which has infinitely many
discontinuity points on Y in the case where A is the Banach function algebra of absolutely convergent power series restricted to a very rapidly convergent sequence,
X - {xn } c {zl Izi < I},
converging to
O.
REFERENCES 1.
J.M. Bachar, Jr., Composition mappings between function spaces, Thesis, UCLA, June 1970.
Ph.D.
2.
J.M. Bachar, Jr., Range Transformations Between Function Spaces, research monograph, to appear.
3.
J.M. Bachar, Jr., Hilbert Space Operators, Lecture Notes in Mathematics 693, edited by D.W. Hadwin and J.M. Bachar, Jr., Springer-Verlag, 1978.
4.
P.C. Curtis, Jr., Topics in Banach spaces of continuous functions, Lecture Note Series No. 25, Matematisk Institut, Aarhus Universitet, December 1970.
5.
P.C. Curtis, Jr. and H. Stetkaer, A factorization theorem for analytic functions operating in a Banach algebra, Pac. J. Math. 37 (1971), 337343.
6.
H.G. Dales and A.M. Davie, Quasi-analytic Banach function algebras, J. Functional Analysis 13 (1973), 28-50.
7.
R.S. Doran and J. Wichmann, Approximate identities and factorization in Banach modules, Lecture Notes in Mathematics 768, Springer-Verlag, 1979.
8.
N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theorv. Interscience, New York, 1958.
9.
o.
Hatori, Functions which operate on the real part of a function algebra, Proc. A.M.S., 83 (1981), 565-568.
10.
H. Helson and J.P. Kahane, Sur 1es fonctions operant dans les alg~bres de transformees de Fourier de suites ou de fonctions sommab1es, C.R. Acad. Sci. Paris 247 (1958), 626-628.
11.
E. Hewitt and K. Stromberg, Real and Abstract Analysis. Springer-Verlag, 1965.
12.
H. Helson, J.P. Kahane, Y. Katznelson, W. Rudin, The functions which operate on Fourier transforms, Ac~a Math. 102 (1959), 135-157.
62
13. 14.
BACHAR
E. Hille and R.S. Phillips, Functional Analvsis and Semi-Groups. Am. Math. Soc. Colloquium Publ. 31, Providence, 1957. , , , J.P. Kahane, Sur un theoreme de Wiener-Levy, C.R. Acad. Sci. Paris 246 (1958), 1949-1951.
15.
J.P. Kahane, Sur un theor~me de Paul Ma11iavin, C.R. Acad. Sci. Paris 248 (1959), 2943-2944.
16.
J.P. Kahane and Y. Katzne!son, Sur le reciproque du theoreme de WienerLevy, C.R. Acad. Sci. Paris 248 (1959), 1279-1281.
17.
J.P •. Kahane and W. Rudin, Caracterisation des fonctions qui operent sur les coefficients de Fourier-Stie1tjes, C.R. Acad. Sci. Paris 247 (1958), 773-775. I , , Y. Katznelson, Sur les fonctions operant sur l'a1gebre des series de Fourier absolument convergentes, C.R. Acad. Sci. Paris 247 (1958), 404 ... 406.
18.
,
,
~
19.
Y. Katznelson, A1gebres caracterisees par les fonctions qui operant sur elles, C.R. Acad. Sci. Paris 247 (1958), 903-905.
20.
Y. Katzne1son, Sur Ie calcu! symbolique dans quelques a1gebres de Banach,
,
Ann. Sci. Ecole Norm. Sup. 76 (1959), 83-124. 21.
Y. Katzne1son, A characterization of the algebra of all continuous functions on a compact Hausdorff space, Bull. Am. Math. Soc. 66 (1960), 313315.
22.
, , ' . Y. Katznelson, Sur les algehres dont les elements nonnegatifs admettent des racines carres, Ann. Sci. tcole Norm. Sup. 77 (1960), 167-174.
23.
Y. Katznelson and W. Rudin, The Stone-Weierstrass property in Banach
algebras, Pac. J. Math. 11 (1961), 253-265. 24.
K. de Leeuw and Y. Katznelson, Functions that operate on non-self adjoint algebras, J. Analyse. Math. 11 (1963), 207-219.
25.
p. Malliavin, Calcu1 symbolique et sous-algebres Math. France 87 (1959), 181-190.
26.
C. Rickart, General Theory of Banach
27.
W. Rudin, Fourier Analysis on GrouEs, Interscience, 1962.
28.
W. Rudin, Real and Complex Analysis, McGraw Hill,
29.
S. Sidney, Functions which operate on the real part of a uniform algebra, Pac. J. Math. 80 (1979). ~65-272.
30.
W. Sprag1in, Partial interpolation and the operational calculus in Banach algebras, Ph.D. Thesis, UCLA, 1966.
31.
J. Wermer, The space of real parts of a function algebra, Pac. J. Math. 13 (1963), 1423-1426.
de Ll(G), Bull. Soc.
Algebras, Van Nostrand, 1960.
DEPARTMENT OF MATHEMATICS CALIFORNIA STATE UNIVERSITY AT LONG BEACH LONG BEACH, CA 90840
1966.
Contemporary Mathematics Volume 32, 1984
RECENT RESULTS IN THE IDEAL THEORY OF RADICAL CONVOLUTION ALGEBRAS William G. Bade In this survey I will discuss some problems concerning the structure of the family of closed ideals in certain radical convolution algebras on the
positive integers.
I
shall give background to these problems and describe
exciting results that have been found in the past two years.
In conclusion,
I shall briefly discuss the corresponding problems on the half-line.
A real-valued function w defined on ~+ = {n E ~:n ~ a} + and if function if wen) > a for all n E ~, w(m+n) 5 w(m)w(n)
for
m,n
w is radical i f
We say that the weight function
is a weight
E7/. lim w(n)l/n n-+OO
= 0.
For con-
venience we assume that w is non-increasing and tnat w(O) = 1. An example _n 2 for n E ~+. Denote by el(w) of a radical weight is given by wen) - e the set of all complex-valued functions x on '11+ for which l!x!1 = i Ix(n) Iw(n) < Then .e l (w) is a Banach algebra for the convolution n=O multiplication IXI.
(x*y)(n)
n
~ x(j)y(n-j)
=
for
n E ~+,
j=O withunit
e- [1,0,0, ••• ]
and generator
we can write
=
[0,1,0,0, ••• ].
If
1
x E t (w),
00
x -
and regard
z
tl(w)
volution of
as a Banach algebra of formal power series.
with
x
Z x(n)z n , naO
z
yields the right shift of
(z*x)(n)
= x(n+l)
n
=
{x:x(i)
These ideals, together with
1. QUESTION.
If
= 0, (0)
for and
i < n}, t
1
(w),
for
by one coordinate place: n E ~+.
x E t 1 (w),
for
There are certain obvious closed ideals in
M
x
Note that con-
t 1 (w), n
namely the ideals
= 1,2, ••••
are called the standard ideals.
w is a radical weight, is every clesed ideal in
a
standard ideal? © 1984 American Mathematical Society 0271-4132/84 SI.OO + S.25 per page 63
64
BADE
This question, and its analogue in the continuous case, are our main 1 concerns in this paper. The question for "(w) is attributed to Silov in 1941. If x E ,lew) and x ¢ 0, write «(x) - inf{i:x(i) ~ O}. For a radical weight w the following are equivalent: (a) all closed ideals in "l(w) are standard, 1 1 (b) for each x E "(w) with x ~ 0, x*" (w) - M ( )' _~_ 1 a x 1 (c) for each x E t (w) with a(x) ~ 1, the ideal x*t (w) contains (d)
some power of z, the closed subspaces of
"l(w)
z
invariant under convolution by
are totally ordered by inclusion. In view of (d) we call a radical weight w unicellular if any of the conditions (a)-(d) hold. The ideal question for
I t (w)
is part of a larger problem for weighted
1 + "(Z) by setting S(e) - Xnen+1' where ~n > 0 and en - {6kn :k E ~+}, for n n - 0,1,2,.... Then S is equivalent to the operator of convolution by z 1 1 in the space t (w), where wen) - ~0.~r .. ~n-1. In general, t (w) will not be an algebra unless conditions are placed on the ~n's. However, when w is a radical weight, the closed ideals of "l(w) are the closed subspaces invariant under cODVolution by z, and correspond to the closed subspaces of tl(~+) which are invariant for S. The early results concerning the ideal question for radical weights appeared between 1968 and 1974 in papers by Nikolskii [11]-[13], Grabiner [5]I8J, and Belson [9]. We describe a few of these below.
shift operators.
Let
S be the weighted shift defined on the space
DEFINITION. A radical weight w is a basis weight if for each r 2. there exists a constant Cr such that w(m+n+r)
~
C w(m+r)w(n+r), r
~
1
for m,n E ~+•
The condition says that every left shift of w is essentially submulti2 plicative. An example of a basis weight is wen) - e-n • The following elementary but important result is due to Niko1skil [11]. We give a proof by Belson [9].
3.
THEOREM.
1
Suppose 1 £' (w) is a non-standard closed ideal. define a multiplication by
PROOF.
M1
Every basis weight is unicellular.
(xty)(n) - (x*y)(n+l),
for
Then
On
n E J'+ and x,y.1ft.
Then (M1,.) is s Banach algebra with unit z- [0,1,0 •••• ]. Its unique maximal ideal is M2 , and I is a closed ideal of (M2 ,G). Hence I ~ !f2" An inductive argument shows that
RADICAL CONVOLUTION ALGEBRAS
65
00
len M n
=
(0),
n=l
so that
I
is unicellular.
For
w to be a basis weight it is sufficient that In w be concave. w(n+k) decrease to o This condition is equivalent to the condition that wen) Grabiner [6] has shown that for a basis weight every (non-closed) principal ideal x*t1 (w)
as n
~ ~
for
~
k
1.
contains a power of
Basis weights are quite special.
z.
In the negative direction, Niko1skii [13] constructed a class of weight sequences
t 1 (w)
w for which there were non-standard closed subspaces in
invariant under the right shift.
It was believed that his method yielded
algebra weights of the type we are considering for which there were non-standard closed ideals.
However, M.P. Thomas showed in 1979 that Nikolski! 's
argument did not work for algebras.
Thus the question of whether or not there
existed radical algebra weights yielding non-standard ideals became of great interest. In the past two years there have been two major results concerning this question, both of which are due to M.P. Thomas.
The first of these gives a
new and important class of weights having only standard closed ideals. The second is the construction of a difficult pathological weight which has a nonstandard ideal. I shall try to explain both of these results in an intuitive way. For the first theorem we say that a radical weight the function
w(n)l/n
decreases monotonically to zero as
comes from the fact that the region below the graph of from the origin. if an element
y
w
is star-shaped if
n -+
IXI.
This name
In w is illuminated
Thomas shows that star-shaped weights have the property that =
00
L y(n)z
n=O cation by a power of z,
n
of
t
1
(w)
is shifted to the right by multipli-
one can get a sharp estimate for the tail of the
resulting series: (II)
if
k:!: 1,
then
1\
y(n)zn+k ll 5 w(m)k/mllyll,
Z
for
m E IN.
n=m+l One does not know whether all star-shaped weights are unicellular. However, this is true with a small additional assumption on the rate of decrease of wen) l/n. 4.
THEOREM (Thomas [16]). is star-shaped and nw(n) lIn
All closed ideals in -+
0
as
n
To give an idea of the proof, let and
x(l) = 1.
We wish to show that
~
tl(w)
are standard if
w
00.
1
x E .e (w),
x*tl(w) = MI'
and !=Ittppose that Let
c = {c{n)
the unique complex sequence (called the associated sequence to
x)
t11=0
x(O)
=0
be
such that
66
BADE
GO
Z c(n)z n=O
n
satisfies the equation
...
GO
( ~ c(n)zn)( ~ x(m)zm) n=O m-l
in the algebra
=z The sequence
of formal power series.
G:[[z]]
c
will not in
1
t (w).
The strategy is to show that there exists a sequence of GO n Z c(n) z partial sums of the series such that in t 1 (w) we have n n-O p n lim x*( L c(n)z ) = z. n=O p-n-l k We must estimate the distance from z to x* Z c(k)z. Actually, it is 3 k=O n-l k+2 to x* ~ c(k)z • This is more convenient to estimate the distance from z general be in
k=O
1
sufficient because x*t (w) is standard if it contains z3. Since n-1 k X1c To c(k)z agree on [0,1,2, ••• ,n], we have, using (N), that
z
and
k=O
liz
3
n-1 k+2 - x* l c(k)z II ~o
~
= IIQ
n
n-l k+2 +3(x* Z c(k)z )/1 ~O
n-1
( z Ic(k) Iw(n+l)
k+l n+.r
1
)w(n+l)
n+r
k=O
(where for any series
y
00
n
= Z y(n)z,
Z Ic(k)lw(n +1) k=O
GO
n=m enters.
It suffices
p
= O(n ) P
3
z.
Thomas proves the existence of this
He shows that if there is no such sequence,
then w cannot be a radical weight. esting argument.
~
such that
p
in order to approximate
sequence by an indirect argument.
n
n +1
p
~....
n
Z y(n)z ).
~
n -1
p
as
{n p };c1
to prove that there exists a sequence
as
...
we write
n=O 1/ n Now the condition that nw(n) ~ 0
IIxll
I
shall not try to describe this inter-
Clearly more investigation is needed to get an effective CD
construction of
{n} p p=l· Such a construction would seem to be the key to further positive results of this type. THEOREM (Thomas (181). There exists a radical weight w and an elemen"t x E t 1 (w) such that x*t 1 (w) is a non-standard ideal. 5.
I will try to give the flavor of Thomas' construction, but I cannot ade-
quately convey the difficulty of his argument.
The paper [10] of McClure
gives an illuminating discussion of the problems that must be surmounted • Let such that n(j)
...
{n(j)}j=l n(1)
partition
=1 ~.
be a strictly increasing sequence of positive integers and
n(j+1) > n(j)(n(j) + 1)
for
j
We may assign values of the function
~
1.
The integers
w on
110
{n(j)}j=l
RADICAL CONVOLUTION ALGEBRAS
so that
{w(n(j»};=l
is strictly decreasing and
67
0
wenCk»~ ~ 1.
<
When this
is done, we define w for intermediate values by
= wenCk»~ t w(j),
w(tn(k)+j) for each
t
satisfying
~
n(k)
for
0
~ j
tn(k) + j < n(k+1).
< n(k)
The further assumptions
that w(n(k+l»n(k) < w(n(k»n(k+l)+n(k) and
ensure that Thus
lim w(n(k»l/n(k) = 0 kw(s)w(t) for all sand t.
~
w(s+t)
This is shown in [17].
w is a radical weight which is far from being either a basis weight or
star-shaped.
Let
x
be the lacunary power series
= 1, x(j) = 0 unless
x(n(l»
-(k+l)
x(n(k»w(n(k»
= 2
is one of the sequence
j
Notice that
•
x =
x E
1
e
(w),
j
~
r. x(j)z, where
ncO ~ {n(k)}k_l'
and that
x
and
and
ware
00
completely determined by the choice of the sequence
{w(n(k»}k=l. ~
We now make further assumptions on the rate at which creases to
O.
wenCk»~
ensure also that there is an extremely large drop in n(k).
hence of
x,
w and
~
c
~
=
n=O proof of Theorem 4. and
c(n)z
n
The value k
~
x,
which is
that was introduced in the c(O), •.• ,
wenCk»~
-
11
>
~lx(n(k+1» I •
is standard and
x*t (w)
nomials for which
is a
It is a general fact about the associated sequence that
non-standard ideal? if
so
we have
How does this last condition enter into the proof that 1
and
The key is to tie
Inductively we choose the terms
Ic(n(k+l»
(*)
n(k) - 1
depends only on the numbers
c(n)
x(l), •.• ,x(n+l).
small that for each
c*x = z
fast and
{w(n(k»}k=l'
w to the associated sequence for such that
~
w between
within the constraints made so far.
the inductive definition of
c(n-l)
decreases
There is still great freedom in the choice of
the series
de-
Thomas makes some assumptions of this sort which are too com-"
p1icated to give here, but which ensure that and
{w(n(k»}k=l
"
p (z) =
m
lim p (z)*x = z,
~m
~
La
(m)
k=O then
lim a(m)(j) = c(j)
(k)z
for
k
is any sequence of po1y-
j E~,
~
is the
jth
coefficient of the associated sequence for
x. Using this fact, together with (*), Thomas uses a remarkable recursive argument to
where
c(j)
prove that if in
x*"
1
(w)
x*tl(w)
Ml ,
is the standard ideal
which is sufficiently close to
infinitely many of the coefficients
a(j)
z
then for any element
x*a
in norm, it must be true that
in the expression
a =
r a(j)zj
j=O
68
BADE
are necessarily non-zero.
But since
a
can be replaced by a suitable par-
tial sum of its series, we obtain the required contradiction. Finally I would like to mention briefly the continuous analogue of the problems we have been considering.
Suppose
function on R+ for which w(s+t) w(t)l/t .... 0 as t -+ co. The space
II [1/
f~1 [(t) Iw(t)dt <
=
~
w is a positive continuous
w(s)w(t)
+ s,t E tl,
for
l,l(R+,w)
and for which
of all functions
such that
f
is a radical Banach algebra under the product
co
(f*g)(t) =
t fOf(t-~)g(s)ds.
A standard closed ideal is an ideal of the form M a
=
{f:f(x) = 0
a.e. on
One can ask again whether all closed ideals in
[O,aJl.
+,w)
1
L (R
are standard.
This
is a very difficult problem, and until recently it was not known whether there was any weight for which all closed ideals are
~tandard,
nor was it known
whether there existed any weight yielding non-standard ideals. investigations on these problems are [lJ and [2].
Two recent
In 1981, Y. Domar [4]
proved a remarkable generalization of the 'l'itchmarsh Convolution Theorem from which it follo\01s that for a class of very nice weights such as with
a > 1,
all closed ideals are standard.
wet)
=e
_to:
,
In the opposite direction,
H.G. Dales and J.P. McClure [31 had proved in 1979 that if there exists a weight
w on
7/
for whiche 1 (w)
has a non-standard closed ideal, then
this weight can be used to construct a weight has a non-standard closed ideal.
w on
IR+
for which
Ll(R+~W)
Thus the example of Thomas shows that non-
standard ideals can also appear in the continuous case. I.t is nut known, nowevcr. whether or not cae. I I
+) t-. E [1( , ~,w
wi t 11 11l [ l:iUpp [=:) l
aUti
f*L1(1R+.W) = L1(R+,w)
each
radical weight
[or
w.
REFERENCES 1.
G.R. Allan, Ideals of rapidly growing functions, Proc. Internat'l. Symp. on Functl. Anal. and its Applics., Ibadan, Nigeria (1977), 85-109.
2.
W.G. Bade and H.G. Dales, Norms and ideals in radical convolution algebras, J. Functl. Anal. 41 (1981), 77-109.
3.
H.G. Dales and J.P. McClure, ~onstandard ideals in radical convolution algebras on a half-line, unpublished manuscript.
4.
Y. Domar, Extensions of the Titchmarsh Convolution Theorem with applications in the theory of invariant suhspaces. Proc. London Math. Soc. (3), 46 (1983), 288-300.
5.
S. Grabiner, A formal power series operational calculus for quasi-nilpotent operators, Duke Math. J. 38 (1971), 641-658.
6.
S. Grabiner, A formal power series operational calculus for quasi-nilpotent operators II, J. Math. Anal. App1. 43 (1973), 170-192.
RADICAL CONVOLUTION ALGEBRAS
69
7. S. Grabtner, Derivations and automorphisms of Banach algebras of power series, Memoirs Amer. Math. Soc. 146 (1974), 1-124. S.
S. Grabiner, Weighted shifts and Banach algebras of power series, Amer. J. Math. 97 (1975), 16-42.
9.
H. Re1son, Invariant subspaces of the weighted shift, ColI. Math. Soc.
Janos B01yai 5 (1970), 271-277. 10.
J.P. McClure, Nonstandard ideals and approximations in primary weighted t 1-a1gebras, these proceedings, 177-185.
11.
N.K. Nikolskii, Basicity and unicellularity of weighted shift operators, Izv. Acad. Nauk SSSR Sera Mat. 32 (1968), 1123-1137 (also in Math. USSRlzvestija 2 (1968), 1077-1089).
12.
N.K. Niko1skii, Non-standard ideals, unicellularity, and algebras associated with a shift op~rator, App. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov 19 (1970)~ 156-195 (also in Sem. Math. V.A. Steklov Math. lnst. Leningrad 19 (1970), 91-111).
13.
N.K. Niko1skii, Selected problems of weighted approximation and spectral analysis, Trudy Mat. lnst. Steklov 120 (1974), 1-270 (also in Proc. Steklov rnst. Math. 120 (1974), 1-278 -- as an A.M.S. Translation).
14.
M.P. Thomas, Closed ideals and biorthogona1 systems in radical Banach
15. 16.
17. 18.
algebras of power series, Proc. Edinburgh Math. Soc. 25 (1982), 245-257. M.P. Thomas, Closed ideals of e1 (w) when {wn} is star-shaped, Pacific J. Math. 105 (1983), 237-255. M.P. Thomas, Approximation in the radical algebr~ t 1 (wn ) when {wn } is star-shaped, Radical Banach Algebras and Automatic Continuity, (ed. J. Bachar et a1). Lecture Notes in Mathematics 975, Springer-Verlag, 1983, 258-272.
M.P. Thomas, A non-standard closed sub algebra of a radical Banach algebra of power series, to appear in J. London Math. Soc. M.P. Thomas, A non-standard ideal of a radical Banach Algebra of power series, to appear in Acta Mathematica.
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFOF~IA BERKELEY, CALIFORNIA 94720
Contemporary Mathematics
Volume 32, 1984
MODULE DERIVATIONS FROM COMMUTATIVE BANACH ALGEBRAS
w. 1,
G. Bade and P. C. Curtis, Jr.
PRELIMINARY REMARKS Let
that
U
be a commutative Banach algebra with unit
is semi-simple.
/J
p:U
-+
d(m).
D:U
-+
m
Let
m
be a Ie ft-Banach U-module with module action
Our aim is to study the structure and continuity of derivations
under various additional assumptions on
~~t
We do not assume
e.
~(O)
U
and
m.
be the separating space for the derivation
the associated continuity ideal.
of these spaces see [2].
0, and
J(D)
For definitions and elementary properties
We will follow the notation ':ind terminology of tllat
paper. We shall consider one or more of the following additional hypotheses on
li and tne associated module action (A)
U
(B)
the spectrum
p.
is singly generated by an clement ~(/J(z»)
of
~(z)
z.
as an operator on
m
iH at
most countable. p(z)
m.
has no non-trivial divisible subspaces in
(C)
The operator
(D)
M2
=
M
for each maximal ideal
M
(E)
M2
=
}f
for each maximal ideal
M of
of
U. U.
See [3] for a discussion of divisible subspaces for bounded operators in a Banach space. with
0t.t(z)
Under assumption
ill the usual way.
(A)
Assumptions
(D)
or
(E)
Our principal results are the following. (D), the range
the closure (D)
1-,'
tions.
Under ;'lssumptions
U .
(A), (B),
of a module derb 7ation, continuous or nut, lies in
of the maximal divisible subspace for
is replaced by
Thus, under
D(U)
-U
imply respectively
that there are no point derivations or no continuous point derivations on
and
o
we identify the structure space
(E), then
(A), (B), (C), and
p(z).
nell) £ W for all continuous (E)
If assumption derlvation~
D.
tuere are no non-zero continuous deriva-
Under tlle latter set of assumptions, the structure of all disconi:. i.nuous
module derivations can be given (Theorem 2.8). © 1984 American Mathematical Society 0271-4132/84 $1.00 + $.2S per page
71
72
BADE AND CURTIS In Section 3 we apply these results to separable Silov algebras
saLis[ying
(e).
and
(A)
For these algebras every module derivation is
blllllldetl if for each m:~ximal ideal 1..1.
We
THEORE:-1.
dl.~rivaLion.
as~ume
a(p(z) p(p(z)
W(J(D»
m
-+
be a discontinuous
'jeD»)
)~(D)
C
p
of least degree
such that \.J(,S(D», p(z)
divisible subspace in the
$(D).
This is a direct application of Corollary 1.3 of [3].
PROOF.
5
D:11
is the closure of the largest
separating space
let
Let
(B).
There exists a non-constant monic polynomial
whose ruots lie in
where
and
(A)
ha~ finite codimcnsion.
H. HZ
=
D. Ta = ?a, Rm = p(z)m, where
a f 11
and
m
~
m.
In that result
Then
(RS - ST)(a) = p(z)D(a) - D(za) = p(z)D(a) -
= ilnd
RS - ST
1.2.
\.Je aSSUm(~
THEOREM.
11
of
to
(A),
m.
p(a)D(z)
-p(a)D(z),
is continuous as an operator [rom
derivation (rom p
f')(z)D(a) -
(B) and
(C).
11 Let
to
n be
m. a discontinuous
Then there exists a non-constant monic polynomial
least degree with roots in
= {O}.
p(u(z»,S(D)
This follows immediately from
PRl)()F.
'l'lll~(Jrem
1.1.
Under the assumptions for_ Theorem 1.2, the continuity ideal
1. '3.
CUROLLARY.
J(O)
for the derivation
D is a closed ideal of finite codimension in
11,
and J(D) = p(z)l.l •
Rel:all that
PKUOF.
Sinee J ( [),
p
J (D)
=
{a Ell: p(a);i.i (D)
=
{O}} is a closed ideal in
is the monic polynomial of least Jegree for which
P(z ) U c
J (D) •
10'0 r
q E f: p. J
t
Q;[\J/p·
'[X-I/p • G:[:\j
C:[~J
+ J{D)]
isomorphically onto a dense subalgebra of
is finite dimensional,
lies in
he rna p
[q + P • £l~]J ~ [q(z)
map!'>
p{z)
U •
3(0)
Since
ll/J(D).
has finite codilllension in
11.
MODULE DERIVATIONS Thus
n
J(D)
{q(z): q E ([t..J}
{r(z): rEp •
dense in
is
Therefore
e[~]}.
J(D)
73
J(D)
and consists of
p(z)u.
~
Under the same assumptiOIlti as above let
=
P (A)
\I.
TIll
( A-).. )
factorization of the polynomial are disjoint open and
ei
sets E(e. )
p.
Then
clo~ed
be the
1
i
1=1
= e1
U ••• U en' where the a(p(z» and A. Eel· Let 1
a(p(z»
subsets of
be the corresponding spectral projections,
1.
We next consider reductions for our situ3tion by exploiting these spectral projections commutes with
pea)
E(e. )m.
Sel
for all
Moreover
1
a
a(p(z)
D
i
LEMMA.
o(p(z)
I$(D»
E U, Di
The_.separating
Since
E(e.) 1
is a derivation whose range lies in
spac~
for
1::: 1
~ 11,
whicJl...contalns no root.o.£.
PROOF.
To see that
D(x)
-+
n
in.~a.riant
is
>(D)
{"-l' ••• '\n}' the set of ro~~s of
=
= E(ei)~(D)
dence
I ~ i ~ n.
for
!m.) = e.,l!:i~n. 1.
c(p(z»
and
= E(ei)D,
1
1.4.
S(D t )
B(e i ).
s.
$(D)
Then
zx
n
and i f
e
,..;(z),
and
Mo~~over
p.
subs~t.
is an open anet closed
p, then
E(e)$(D)
is invariant under ~
for
=
{Ole
o(z), let
{x} C U, x 11
-
a and D(zx) n = P(z)D(x n) + p(xn )D(z)
~
=
~
n=O
p(z) n .x.n +1
is countable, we deduce that The integral formula for
E(e i )
$(D)
and these subspaces are closed. Since !$(D» e
p
cr(p(z»
••• $
{Ole
=$(D).
\.
Hence
E(e )$(D), n
It follows that
p.
which contains no root of
such that the vector function
=
E(ei)$(D)
is the minimal polynomial for
s E $(D).
J(p(Z»
is invariant w1der R(}..;,o(z», for all
is just the set of roots of
U for each E(e),s(D)
0
pI. I > dp(z) II.
if
shows that $
U of
~
p(z)s.
By analytic continuation of the resolvent, using the fact that
subset of
n
is invariant for
~(D)
R(:\jp(z»
cr(p(Z)
£.t
~ ~
$(D i ) = p(Z)
Let p.
e
on
E(ei)~(D).
$(D),
be an open and closed
Then there is a neighborhood
R(}..;P(z»s
is analytic
It follows from the integral formula for
E(e)
in that
74
BADE AND CURTIS
1.5.
for
The polynomial
REMARK.
p(z)
PROOF.
J(D i )
on
Since
p
v.
=
Pi ()..)
(A-A.) L
is the minimal polynomial
~
and
is the minimal polynomial for
p(z)
on
SeD)
and
a(p(z) 1$(0» = {~, ••• ,An}' the first conclusion follows from general spectral theory [5, Theorem VII, 3.20]. The argument in the proof of Corollary 1.3
= Pi(z)U.
J(D i )
shows that
These remarks show that p
Nithou~
loss of generality we can suppose that
has only one root, which we can take to be zero.
Then
p(A). AV.
Thus
we are reduced to the case of a discontinuous derivation whose continuity ideal is primary and
0 E a(z)
and
0 E a(p(z».
Of course
0
may be a
limit point of other points in the countable spectrum a(p(z». Under the same assumptions we next make some remarks about module action
If
i).
A E aU(z) -
~ll'
we denote by
M(A)
U and the
the associated
maximal ideal. 1.6.
LEMMA.
If
n ~ 2.
PROOF.
n
Let
sum of products of a sequence
where
rk
PROOF. 1.8. m ~ 1.
=M(A)n.
If
n a E M(A) , a
b 1 ••• b , where b i E M(A). Each b E M(A) n qk of polynomials in z of the form
pCa) ~
=
amu
= am+1U
for
is the limit
some
m ~ 1.
p(a)m+1PUff.
~
pea) P\U,
Let
= pea) mp(ll) = pea01..Ll) = Aarn+r::: U) A E °U(Z) , and suppose that
- pea) M(A)m
Then (p(z) - AI) m... m - (p{z) - AI) m+1 m
PROOF.
is a finite
Both conclusions now follow.
a E U, and suppose
Let
THEOREM.
(z-Ae)~
Clearly
is a polynomial.
1.7. LEMMA. Then
1.
>
By the above lemmas, (p(z) -
AI) ~ P\U)
= (p(z)
- AI)
m+l-
~(ll).
m+l~
P,UJ.
= M(A)m+l
for some
75
MODULE DERIVATIONS Hence there exists a sequence (p(z)-U)
m
qk
of polynomials such that
... lim (p(z)-U) k
m+l
,::>(qk(z».
Hence
the reverse inclusion is clear. 1.9.
subspace of
point of ~1~
of
in the above by any closed
cr(p(z». p(z)
Assu;.;,e
(A). (B)
Suppose for some
of order
~
PROOF. that
1.11.
=
1.12. a(p(z»
Since
E(X-)m.
By
(C),
Assuming II
into
~(E(eO)D)
COROLLARY.
A
is a
A.
= (~(zrAl)k+lE(A)m.
(A). (B)
m.
=
(P(z)-AI)kE(A)
Since
~
p.
I
eO
Then
= E(eO)~(D),
and
(C)
1
E(\)D
p(z)
at
p(z) - divisible
{a}.
let
D be a disc:J ..• -::inuous
be an open and closed subset of E(eO)D
cr(p(z»
is a continuous derivation.
this result follows from Lemma 1.4.
Under the above assumptions let
Then
eigenvectors for
Then
{O}.
A be an isolated point of
which is not a root of the minimal polynomial
K(}I/ = M(A).
PROOF.
M(A)k = M(A)k+l.
lies in the closure of the largest
which contains no root of
PROOF.
A be an isolated
{A} , the Mittag-Leffler Theorem [3, Theorem 1.1] implies
THEOREM.
derivation of
=
(p(z)-U)kE(A)m
(P(z)-AI)kE(A)M
subspace in
1,
is the spectral projection associated with
By Theorem 1.8,
a(p(z)IE(A).)
~
(C). and let
k, that is, k
E(A)
and k
(p(z)->..I) E(A) where
p(z)-invariant
m.
PROPOSITION.
1.10.
m
We can replace
REMARK.
is continuous, and
E().,)m
p.
Suppose
consJ._sJ:'s of simple
X-.
This is an immediate consequence of Proposition 1.10 and Theorem 1.11.
76
2.
BADE AND CURTIS
U
ALGf;BRAS
FOR WHICH
M(A.)2 == M(A)
OR
M(A)2 = M(A.)
Throughout this section we shall assume conditions
(A)
and
(B).
We shall assume in addition one of the conditions 2
=M
(D)
M
for each maximal ideal
M of
11 ,
(E)
M2 = M for each maximal ideal
M of
II •
or
au
with (z), .md that a(p{z» C cr (z).
Recall that we identify For
A E 9u(z), we will wrlte
Actually, conditions ideals
(D)
mId
A € cr(p(z».
for which
M( A)
W denotes the closure of the largest
In what follows
m.
subspace of
will be needed only for those maximal
(E)
p(z)-divi~ihle
The proof of the next reRult follows the argument of
[6, Lenuna 3.1]. THEOREM.
2.1.
Then if
D(U)
is
0
lies in
PROOF.
Since
a~y
R
11
that
=
m
and
satisfy conditions 11
derivation from
to
(A), (B), and (D).
m, continuous or not, the range
W. a(p(z»
in a sequence Let
U
Let:
n
I.
Define
D(U) c Vel).
An
such that each element
{A } n
~(Z)-A
is countable, we may list the elements of
~
If
V(O)
=m
and
Vel)
S aU(z)
E a(p(z»
and
appears
a(p(z»
infinitely often.
= RIm = RIV(O).
We assert
a Ell, then
....
= a(~)e
a
and
D(a) == D{w).
fore
D(a)
=
D(w)
Since
=
M(Il)
+ w, 2
= M(~),
~n
D(U)
D(a) E (p(z)-;LI)iI1'. C
-
V(m)
= RmV(m-I)
Taking
u i vi' ui,v i E M(Il).
for each
= V(2),
m. n
m=l V(w).
There-
However,
qk; P(vi)D(u i ) is a similar limit. ThereIl = AI' we get 0(11) ~ Rl V(O) = V(l). The
D(U) ~ R2 Vel)
D(U) c which we define to be
n
w == ~.
(,o(u.)D(v.) + P(v.)D(u.». 1 1 1 ~
for some sequence of polynomials same argument yields
M(~),
1=1
1=1
fore
wE
Hence Vern) ,
and ind'''!ctively
MODULE DERIVATIONS a
Generally, if Then by induction
D(U)
is a limit ordinal we define ~
V(a + m)
V(a)
and
= RmVCa +
m - 1),
There must exist a smallest ordinal We assert that RnW
=W
VCr)
for all
= W,
y
m ~ 1.
for
such that
V(r)· V(r + 1) = ••••
the largest closed subspace of
S~nce
n.
77
m ~uch
that
V(y) - V(y + 1) - R1V(r) - V(y + 2) -
R2V(y + 1) = R2V(y) - ••• = RnVer), for n E N, it follows that Rn VCr) =(P(Z)-An I)V(y) = V(r) , for n E N. Also if Y is any closed subspace _ of
m for which Rn Y .. Y for all n, then
Continuing by induction, we have 2.2.
COROLLARY.
If
2.3.
C
V(y).
m satisfy
U and
then the only derivation
Y
D from
conditions
U to
= AC[O,l],
(A), (B), (C), and (D),
p(z).
An
example which illustrates
the absolutely continuous functions on
for each maximal ideal
bounded approximate identity).
[0,1].
This
IIfli = IIfll ... + j~ If' I, is singly generated,
Banach algebra. with norm given by
=M
= w.
m is the zero derivation.
countability assumption on the spectrum of
and M2
V(y)
Theorem 2.1 and its corollary depend essentially on the
REMARK.
this is U
Therefore
M (since each maximal ideal contains a
L1 [O,l] is a Banach Umodule under ordinary function multiplication, and it may be verified that p(z)
The Banach space
has no nonzero divisible subspace.
derivation of L1 [0,1] D from
U into
Yet
are bounded AC[O,l]
Ll[O,lj
to
THEOREM.
Let
U and
continuous derivation from
PROOF. Let
is a bounded
Ll • Necessari:y all derivations from AC[O,l] to [1, Corollary 2.10]. Therefore all module derivations have the form
D(f) - (df/dx)D(z), where
is an arbitrary bounded measurable function on 2.4.
= df/dx
D(f)
[0,1].
m satisfy (A), (B), U to m, then D(U) C
The argument follows that of Theorem 2.1. tl E: a(p(z», a E U, and
and D is continuous,
a
..
= a(iJ.)e + w,
D(z)
and
(E). If
D is a
W.
We assert
w E M(IJ.).
S1.nce
Dell)
~
T"":
D(w) 1.S approximabLe by sums ot the form
M (,.L)
Vel).
= M(~l.)
78
BADE AND CURTIS n
+ P(vi)D(u i »,
~ (P(u.)D(v.)
i-1
1.
But
P(ui)D(v i ) -
qko
Hence
1.
for
li~(P(z)-~)qk(z)D(vi)
(p(z)-~I)m.
D(a). D(w) c
ui ' vi E M(~).
for some sequence of polynomials
The rest of the ~roof proceeds exactly
as in Theorem 2.1.
2.5.
COROLLARY.
U and
If
only continuous derivation
2.6. D:ll
THEOREM. ~
m
D from
to
U
U and m satisfy
Let
m is the zero derivatiOll..
(A), (B), (e), and
(E), and assume
is a discontinuous derivation for which the minimal polynomial
has only the single root subsets of Then
(A), (B), (e), and (E), then the
m satisfy
a(p(z»
= M(O)
J(D)
p(z)
vectors for
O.
Let_ {e}
and
to},
nn=le n -
~
D(U)
MO. 2
and
be any sequence of open and closed m
n
for which
and set
Furthermore
= {Ole
D(M(O) )
discontinuous linear map satisfying
p
mO
MO = n n=lE(en)m. consists of simple eigen-
In addition, if
D(e)
D:ll
= D(~(O)2) = {OJ,
4
DO
then
is any D is a
derivation. eA = a(p(z}}
Let
~
e. Then E(e')D is continuous by Theorem 1.11 n n and hence E(e')D - {OJ by Gorollary 2.5. Therefore D(a) = E(e )D(a), for n n n E N and a E U and consequently D(U) c mO = n lE(e )m. Also n= n PROOF.
GD
=
a(p(z)l m )
a
{Ole
If
is the restriction module map from
Po
then (B) and (C) hold in simple eigenvectors for
=
mO.
Hence by Proposition 1.10,
p(z) in mo' Therefore if
=
DOHO) 2) = {O},
D(ab) - p(a)D(b) + p(b)D(a)
=0
p(M(O»mO
....
b
let
a, b E M(O).
'lben
and consequently
p(z)m O • {OJ,
= {O}.
Now let satisfies
consists of
Since ~(D) mO' it follows that v p(z) = A, then v = land
p(z)$(D) p(z)Mo = to}. J(n) = M(O). To show that
since
MO
II
D be any discontinuous linear map from
D(e)· D(M(0)2) = {a}.
= b(O)e +
v, for
u, v E M(O).
If
a,b E ll,
then
;
II
=
mO which a(O)e + u, to
We have by linearity that ..
A.
D(ab) • a(O)D(v) + b(O)D(u) = p(a)D(v) + p(b)D(u) - p(a)D(b) + p(b)D(a), so
0
is a derivation.
last remark follows.
Since
D= 0
is the only continuous derivation, the
MODl~E
2.7.
If
Let
COROLLARY.
p(z)
has
m
and
79
satisfy conditions
(A), (B), (C), and
m,
no one dimensional invariant subspaces in
derivation from
If
U
DERIVATIONS
II
p(z)
to
m
(E).
then the only
is the zero derivation.
has nontrivial eigenspaces, then a discontinuous derivation
must be a finite sum of the "generalized point derivations" discussed in Theorem 2.6. 2.8.
(A), (B), (C) and (E), let
Under assumptions
THEOREM.
U into
derivation of
m.
Then
D is necessarily discontinuous, f'V
ro i
i f we set
A.. E a(p(z»
finitely many
1
D. (U) c 'fl(.
an d
1.
1.
--
D. (e) 1
By the discussion in Section 1,
PROOF.
D be a nonzero
If {A.l, ••• ,An } = a(p(z» I$(D)~ and Di Di(U) C 'ilt'., and Di (e) = D. = to}. 1. 1. 'i
== {m
Em:
= Di(~
)
= to}.
i
a(j)(z» 1.j(D»
=
and for
is necessarily finite.
E(ei)D, then by Theorem 2.6,
(M\ )
3.
MODULE DERIVATIONS OF SILOV ALGEBRAS
In this section we consider the following question:
What conditions
make all module derivations of a Silov algebra necessarily continuous?
By a
SHov algebra we mean a commutative Banach algebra U which, modulo its radical, is a regular algebra of functions
Our principal result,
CPU'
OLl
Theorem 3.3, applies only to sefarable Silov algebras.' We will introduce the separability assumption when needed.
The first result
concerns arbitrary
commutative Banach algebras. 3.1.
LEMMA.
Let' M be a maximal ideal in a commutative Banach algebra 2
for which
M
every
n.
Consequently if
sion,
J2
has finite codimension as well.
P~OF.
has finite codimension.
We assert that
J
Mf
Then
MP
U
has finite codimension for
is a primary ideal in
has finite codimension
M
in
with finite codimen-
U
for each
n ~ 1.
M = ~ ~ V and dim V < ~, it follows that M2 = M3 + (M2 ffi V)V = M3 + v2 • Continuing inductively, we have ~1 = MP+l + and dim Vn < w. Therefore
Since
vn
for each
n,
Since consequently for Some follows.
k
n M J
M/J
has finite codimension in is primary,
J
.n-l M
, and the result follows.
is contained in a unique maximal ideal
is a finite dimensional radical algebra.
and consequently
HK c J.
Therefore
Hence
M, and
(M/J)k
M2k E. J2, and the result
= {OJ.
80
BADE AND CURTIS THEOREM.
3.2.
Let
be a separable commutative Banach algebra for which
11
has fillite codimension for each maximal ideal
H2
derivation from
11
in
is continuous.
then
11,
PROOF.
0
Since
J(D)
more, since algebra. {f.
Let
Di
J(O)
is closed,
= l, ••• ,k}
i
+ g, for
= P(fi)D.
D be a
has finite codimension
U/J(D)
is
a finite dimensional BanacH
Hence there exists a family
.k d = 2.: i =l fi
Then
If
Let
11.
is cofinite, it has finite hUll
J(O)
+ J(O) :
1.
m.
to a Banach ll-module
M of
such that
is a family of orthogonal idempotents in g E
J(D). and
fj E
n il=jMi
l1/J(D).
with
J(D i ) = {a : af i E J(D)}, and hull D is continuous it suffices to show that each D.= p(f.)D is
prove that
Then
1.
continuous.
Since we know that for each
J(D.)
i,
need only consider the case of a derivation
To
1.
is a primary deal, we
1.
D whose continuity ideal is a
primary ideal. If J(D)2
J(D)
has finite codimension.
the separability of [2, Lemma 2.4],
THEOREM.
1£
M2
J(D)2
and
J(D)2.
Since
are closed. J(0)2
By
has finite cc);!ll1lension,
U. Let
be a separable Silov algebra satisfying
11
(A)
and
(C).
}, E :Pll' then every derivation
m iii continuous.
PROOF. J(D)
that both
11
has finite codimension for each
M( \) 2
D:ll ...
It follows from Christensen's theorem [4] and
D is bounded on
D is bounded on 3.3.
M, then by Lemma 3.1,
is contained in the unique maximal ideal
Suppose
D:ll
has finite
-+
ill is discontinuous.
hull [1, Theorem 2.9].
a non-constant polynomial
p
By
Since
11
is a Silov algebra,
[3, Corollary 1.3] there exists cr(p(z) I~(D»
whose roots lie in
= hull(J(D»
such that p(p(z»~(D)
where
W(~(O»
is the closure of the largest p(z)
~(O).
Since
nence
p(z) E J(D).
therefore that yields that
: W($(D»,
J(D)
p(z)-divisible subspace in
has no nontrivial divisible subspaces , The proof of Corollary 1. 3 sho.,'s has finite codimension.
D is continuous.
J(D)
p(p(z»$(D) - {O},
= p(z)l1
and
An application of Theorem 3.2
MODULE DERIVATIONS
81
REFERENCES
1.
W. G. Bade and P. C. Curtis, Jr., The continuity of derivations of Banach algebras, J. Functional Analysis 16 (1974), 372-387.
2.
W. G. Bade and P. C. Curtis, Jr., Prime ideals and automatic continuity problems for Banach algebras, J. Functional Analysis 29 (1978), 88-103.
3.
W. G. Bade, P. C. Curtis, Jr. and K. B. Laursen, Divisible
4.
J. P. R. Christensen, Codimension of some subspaces in a Fr'chet algebra, Proc. Amer. Math. Soc. 57 (1976), 276-278.
5.
N. Dunford and J. Schwartz, Linear Operators, New York, 1958.
6.
B. Johnson and A. Sinclair, Continuity of linear operators commuting with continuous linear operators II, Trans. Amer. Math. Soc. 146 (1969), 533-540.
subs~)ac~s and problems of automatic continuity, Studia Mathematica 68 (1980), 159-186.
Pa~t
I, Interscience,
DEPARTMENT OF MATHEMATICS
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA 94720
UNIVERSITY OF CALIFORNIA LOS ALGELES, CALIFORNIA 90024
Contemporary Mathematics Volume 32, 1984
CRITERIA FOR BOUNDEDNESS
~.ND
COHPACTNESS
OF HANKEL OPERATORS
F. F. Bonaall 1.
INTRODUCTION
Let
[7].
A be a not necessarily bounded Hankel operator on a Hilbert space
We prove two pairs of related criteria for boundedness and compactness of
A. Let
D deuote the open unit disc in the complex plane,
classical Hardy space of functions on vector in
H2
normal basis for zED.
zED
let
the
v
be the unit
z
given by
It is known [1] that if for
aD, and for
H2
A is a Hankel operator relative to the usual ortho-
H2, then
A is bounded i f and only if
nAv II
z
is bounded
Using an idea from Luecking's very simple proof [4] of the
Coifmall-Rochberg theorem on the Bergman space for
D, we show that with a
sequence {z } of points of D appropriately chosen in terms of the pseudon is hyperbolic metric, A is bounded i f and only if the sequence {IIAv II} z n bounded. If also limlznl = 1, then A is compact if and only if n--
limllAvz
n-
iI = o.
The proof involves a characterization of functions of bounded
n
mean oscillation in terms of the sequence
{z }. n
Our second pair of criteria is of an entirely different kind.
It is
limited by needing non-negativeness of the matrix elements, but is more general in being applicable to any symmetric matrix with non-negative not just to Hankel matrices.
elements~
It involves a sequence of sub-matrices
an.d
{B } n
the construction of which was suggested by the proof of Paley's inequality due to Kwapie~ and Pelczydski [3]. I am indebted to S. C. Power and D. H. Luecking for bringing to my attention the crucial idea used in the proof of the first pair of criteria. © 1984 American Mathematical Society 0271.4132/84 SI.oo + S.25 per page
83
84
BONSALL
2.
CRITERIA IN TERMS OF A SEQUENCE OF UNIT VECTORS In this section we use the notation of [1].
dellote the usual Lebesgue and Hardy spaces llf functions on
oc.
~
is a complex sequence with
11
2
.
la n
n=O
and
HP
dO,
an),
(n E 'e, ~ E
{a }
LP
In particular
I
<~, and
A
is the not necessarily
bounded Hankel operator given by
,. '"'
AX. = J
,'.s in [1 ], we identify
.
a,+.X. (j E 2+)
w
1=0
J
~
~
A with its extension 00
H
2
-+ H ,
given by
where
cP = l:
... k=O
a. Xl.' J:(k = ...·-k (k E 2), and K. -
is the orthogonal projection
P
-I(,
onto The pseudo-hyperbolic metric
and, with
a
~
D and
0 <
K(a,~)
Given a function for by
z EV
on
d
K(a,~)
D
is given by
~
< 1,
=
{z ED: d(z,a) ~ ~} •
fELl (= Ll(on»
denotes the ball
we adopt the usual convention that
the harmonic extension Or Poisson integral of
f
at
z
is denoted
fez).
The following theorem was suggested by Luecking's proof [4] of the Colfman-Rochberg theorem. tHEOREM 1.
Let
and let
that the pseudo-hyperbolic balls f
E L2,
{z} n
K(z ,r]) n
be a sequence of points in cover
D.
D such
Then, for all functions
HANKEL OPERATORS
PROOF.
Let
f E L2.
We note first that for
85
= r ~~, we have
Izi
(1)
In fact 2
l-r Ie i9-z 12 and
so 2 2 If I (z) - If(z) I
1
"i~ ~n
fo2rr If(e1 ) "9
2rr
=
2
2
l-r Ie i9-z 12
fez) I
ion 1f(e i6) - f(z)
de
12 de
= i:g {lfI 2 (O) - f(z)f(O) - f(z)f(O) + If(z) 12}
since f(O) - f(O). Let
- i:g
{lfI 2 (O) - If(O) 12 + If(O) - fez) 12},
Thus
(1)
a E D and, for ~(w)
is proved.
w
~
--1
-a
,let
= (w+a)/(l+aw)
•
Then an elementary calculation gives, for g('t"(w»
1
=-
1
f2IT g('t"(e it »
"e 2
0
= f('t"('»
(, E
aD)~
2 l-iwi Iw-e it l 2
- If('t"(w» 12
g
0
de
dt; 't"laD
(2)
at
w.
so that also
IFI2(~) = IF(,) 12 = IfI2('t"('». Then (2) IfI2('t"(w»
wED,
2
1~(w)_e1 I
is the harmonic extension of F(,)
and
l-I~(w)1
fo
2rr
Now let
g(e i9 )
2rr
=-
that is, g('t"(w»
Zrr
g E Ll
gives
= IFI2(w) - IF(w)1 2 (w ED).
(3)
In particular, (4)
86
BONSALL
By (I), applied to
F,
that is, by (4)
Since the pseudo-hyperbolic metric
D onto
D, taking d(z,a) 5
= ~(w),
z
~ -
d(w,O) 5
d
is invariant under the mapping
(3)
we have, by
~
of
(5),
and
~
(6)
Finally. there exists
n E N with
d(z n ,a)
5~,
and then
and the theorem is proved. REMARK. f E L2
It is well known (Petersen [6]) that for which
IfI2(z) - If(z) 12
BMO
is the set of functions
is bounded on
this can be replaced by boundedness on
{z
n
D;
Theorem 1 shows that
: n E fN}.
Our first critorion for boundedness is now an immediate corollary. COROLLARY 2.
Let
i f the sequence
PROOF.
be as in Theorem 1.
{z}
-
n
{IIAv z
II}
2
is bounded.
..
2
that with 2
14> 1 (z) -Icp(z) 1
Let
A .. R
cp' (7)
(z E D).
The corollary now follows from Theorem land COROLLARY 3.
A is bounded if and only
n
It is known ([1], formula (5» II Av z 11
Then
[1, Theorem 1].
{zn} be as in Theorem 1 and suppose also that
limlznl n ........
Then
A is compact if and only if
limllAv z 1/ =
n--
PROOF.
Let
lim/lAv z II n-+'" n
= O.
n
°.
By (7),
liml~12(z n ) - Icp(zn ) 12 = o.
u--
= 1.
HANKEL OPERATORS
¢ E VMO
Given
0
as
~
Izl
(:arason [8, p. 50)] and hence that
e > 0, there exists
1.
This
A is compact.
N such that 2
(n ~ N),
I¢(z n ) I < e and we choose
~
1¢1 2 (z) - I¢(z) 12
We show that this implies that will prove that
87
(8)
with
p
(9)
We prove that there exists z E K(at~)
whenever
with
~ lal <
(j
<
(j
< 1
1.
such that
Iz I < 1
p <
To see thiS, note that the pseudo-
K(a,~)
hyperbolic ball
is the Euclidean disc with centre and radius (1_laI2)Tl(1~2IaI2)-I. Thus if
(1_~2)a (1_~2IaI2)-1 and
°
with
(j
cr
~ lal
< 1
z E K(a, T), we have Izl ~ {(1_~2) lal - (1_laI2)~}(1_~2IaI2)-1
~ {(1-T)2)(j + «(j2_l)~}(1_T)2(j2)-l. Since the last expression tends to
as
1
tends to
(j
1, we can choose
(j
with the properties stated immediately after formula (9). Given
a
with
Iz I> P, and, by
~
(j
lal < 1, there exists
(9), n> N.
n
1¢1 2 (z) - I¢(z) 12 On the other
Thus 2
and
RA. 'f'
14> I (z) -
I¢ (z) I
2
~l30d,
~ i~{1¢12(Zn)
~
°
as
0
let
Izi be
A
Izi
as
zn E K(a,T).
Then
~
- I¢(zn) 12} < i!g e.
1, and so
-+
1, and so
cj:.
limllAv n--
complete.
is compact.
A
-cp
Since
compa~t.
are both compact, and so
R¢
~
with
Therefore, by (6) and (8),
1¢1 2 (a) - I¢(a) 12 Thus
n
is analytic,
E VMO. z
II
=
R¢
= 0.
Therefore 0, and the proof is
n
We end this section with an elementary proposition which tells us when the linear span
X of
{v
z
: n Em}
is dense in
H2.
If
X is dense in
n
H2, then, of course, every bounded (compact) linear mapping of
X into
HZ
extends by continuity to a bounded (compact) linear operator on
H2.
PROPOSITION 4.
D and let H2 if and
Let
{z} n
denote the linear span of only if
Zm
n=l
(l-Iz I) n
= ~.
be a sequence of distinct points of {v
z
: n E IN}. n
Then
X
is dense in
X
88
BONSALL
PROOF.
h E H2
Let
(h,vz ) = O.
with
By Cauchy's integral theorem,
n
(1-lzI 2 )
(h,v) = h(z) z
and so
h(z) n
=0
[2, p. 18]) and so X is dense in On the other hand, i f
,
~~ (1-lz I) = ~, we have n=1 n
If
EN).
(n
\
h
with
(l-Iz I) <
ZOIJ
n
Then
~).
(h,v
zn
)
=0
(n
EN), and
X is
MATRIX CRITERIA Let
{e
n
, there exists a Blaschke
~
a2 •
not dense in
3.
h(z) = 0 (n E
(see Duren
H2.
n=l product
h = 0
n
a be a separable Hilbert space with infinite dimension, let
: n E 4l+}
be an orthonormal basis for
span of this basis. between
Ho
denote the linear
We are concerned with the well-known correspondence
mappings
~inear
H and let
A : H u"~
i=O
~
o
and infinite matrices
H
Iai' 12
<
(J- E 2+),
~
J
which is implemented by (i,j E a+),
a ij = (Ae j ,e i ) Ae,
J
As usual, we say that
= L.
~
a, ,e i (j E 2+). i=O 1J
(a ij )
(11)
(10) H, which is of course equivalent
is the matrix of a bounded operator if
A on
holds for some bounded linear operator
to the boundedness of the linear mapping (a ij )
(10)
A :
ao
~
H
is the matrix of a bounded operator, we use the same symbol
a
denote both the linear operator on Given
f
and its 7.estriction to
= ZW
a,e, E H, we denote by j=O J J f
for which we obviously have
tl
f#
When
given by (11).
the vector in
A
to
ao • H given by
00
=
Z la-Ie" j=O J J
II filII
=
II f iI.
We omit the completely elementary proof of the following lemma. Nordgren, Radjabalipour, Radjavi, and Rosenthal [5, Lemma 1), for a more general result.)
(See
HANKEL OPERATORS LEMMA 5.
Let
(a ij )
be the matrix of a bounded operator
lb .. I ~ (b ij )
A and let
(i,j E ~+).
a..
1J
~J
Then
89
is the matrix of a bounded operator (f
Band
E H).
The next lemma is reminiscent of Lebesgue's theorem of dominated convergence. LEMMA 6. (b ij )
It is likely to be known,
Let
(a ij )
n
4
i,j E Z+,
be matrices such that, for all
J
B
lim b~~)
1.J
1.J
n~
:0:
b
ij •
(n)
(b ij ), (b ij ) are the matrices of bounded operators B in the strong operator topology as n ~ ~ •
PROOF.
(n)
That
(b ij ), (b ij )
clear from Lemma 5.
Let GO
(a ij ), and let
( b(n» ij ,
be the matrix of a bounded operator and let
Ibi(~) I ~ a .. , Then
we are unable to quote a reference.
b~t
n'
B,
are the matrices of bounded operators
and
B ,B n
A denote the bounded operator corresponding to ajej E H.
f=X,.
B-
Then
J=O I!Af i/ l1 2 .. X,
GO...
(X,
i=O Let
e > 0, and choose
n
i=n Then choose
o
I) 2 .
J
E N such that
o GO
(X,
j=O
a i . Ia j
]·-0 -
a 1 . Ia j J
I)
2
(12)
< e •
nl E N with 00
X,
a ij Ia.
.
j=n
J
1
1
I < "4
k
E.
\I
(n)
(1= O,l, ••• ,n -1).
(13)
Ibij-b~nj) II a.l) 2.
(14)
o
0
We have ) fl/ 2
II (B-B n
Since
Ibij-b~;) I ~
2a ij ,
~
X,...
i=O
(12)
(X,GO
J
j-O
and
(14)
show that (15)
is
90
BONSALL
Now choose
such that, for all
N EN e
n 1- 1 (n) Z Ibij-b ij
j-O
Witn
Ila j
n
~
N , &
1 & k2
I < 2:(n)
(1 :: O,l, ••• ,n -1). o
0
(13), this gives ~
< (.L) n o
II(B-B
and (lS) now gives
n
)fil 2
(i :: O,l, ••• ,n -1), o
< 5&
whenever
n
~ N&•
The following construction was suggested by the proof of Paley's theorem in Kwapie6 and Pe1czy~ski [3].
I am grateful to Chandler Davis for a
remark that has made the construction simpler than the one I had first adopted.
Given a matrix
let
E
o
..
to},
n
,e n
denote the finite rank bounded
1
and, for
defined as follows.
1J
n E IN, let
E
n
- {k E tl : 2 n- 1 =:: k < 2 n }.
n" 0,1,2, ••• , let
Then, for
(i+j E E )
(n) a ij
={ {
b (n) ij
=
::
1J
THEOREM 7.
(i)
(n)
a ij
n
(all other (1 :::
i,j)
j)
(i < j) (n) _ ben)
ij
a ij
If ,£110
(liB fIl2+IIC*fIl2) <: 00
no. O
(a ij )
:ij
. 0
c~~)
then
n
(b ~nj) ), (c ~~»
operators with r.atrices Let
A ,B
n
n
(f
E H),
is the matrix of a bounded operator (f
then
IIAII
(16)
A; and if
E H),
(17)
=:: 4M%..
(ii)
Conversely, if
matrix of a bounded operator
a ij ::: 0 (i,j E a+)
A, then
(17)
holds with
and
(a ij )
M = 411A1I2.
is the
HANKEL OPERATORS PROOF.
(i)
We observe that when B f
m
1 Bn g,
m
n+2
~
91
f,g E H, we have
and
1 c*n g.
c*mf
(18)
In fact, IX>
( B e., B e k ) mJ n
b~;) = 0 unless
and
i=O
for all
b(.m.>-b(.n) k ~J
~
2m- I !: i + j < 2m with
m ~ n + 2, we have
With b(n) - 0 ik
=
... ~
Thus
k.
B
i:'! j, and therefore unless
bi~) = 0 unless 2n
f.l Bn g,
C*f m
and similarly
m
From the orthogonality relations
(18)
< _ .1.,
in
which case
1 Cn*g.
we now have for all
f E H,
(19)
and (20)
together with similar identities involving Suppose now that
..
00
B
1.
n=O A
n
(16)
= Bn
Z C* ncO n
and
n
holds.
~onverge
+ C , it follows that n
{B 2n+1 } and
Then, by
(19)
and
(20), the series
in the strong operator topology.
Z
A
Since
couverges in the weak operator topology
ncO n
to some bounded operator
A.
Given
n , and then
i,j E Z+, we have
_{a
o
(A e. ,e.) -
nJ
1.
ij
II
i
+
j
EE n
for some o
= no )
(n
(n :; n )
o
IX>
Thus
a 1J.
= ZncO (Ane.,e.) = J ~
ed operator
(Ae.,e.), and J
~
(a;j)
A.
If the inequality (17) holds, (19) gives P 1
B2n+1 •
similarly for
liz
p
cnll
*
p
= liz
ncO
converges to
is the matrix of the bound-
~
n-O
Therefore ~
liZ
n-O
IIZP
Bn ":: 2M"i.
Cn " ~ 2M , and it follows that
A In the weak operator topology.
B2nf1l2!: M//fIl2, and
ncO
Similarly,
IIAII ~
I
4MJi , since
p
Z An n-O
92
BONSALL (ii)
of a bounded operator 60 + 8 2 +••. + B2n •
and
Ill.
den) -~ a. ij iJ
<
is the matrix
( 1,J . . E z+) ,
~
constant for large
""
H2n
""
82nll:: IIAI!,
l.
n=O
n=O B 20+1 '
...
118 2n fl/ 2 :: IIA1I211fll2.
n=O
this gives
0
is the matrix of the bounded operator
Jl.
c*. and n
argument applies to
With a
/lB fll2 ~ 2liA/l2 11fil 2,
Z
(a .. )
Therefore, by
...
Therefore by (19),
similar inequality for
n.
converges in the strong operator topology
n=O
REMARK,
(a ij )
Then
6 and 5, the series
Since
and that
be the matrix of the operator
1J
i,j, we have
nnd, with gIven
Z+)
(i,j E
(d~~»
Let
A.
Ll
L~mmas
a ij ~ 0
Suppose now that
(17)
holds with
M =
4J!AiI2.
A with matrix
Given a bounded operator
also the matrix of a bounded operator :IA* fjl
* a similar A,
(a ij ), the matrix
(a ji )
is
At and
= IIA t-. f.1
,
... where, for
*
C n
=Z
ake k , we take k=O t can be replaced by C, f
Therefore in Theorem 7(i),
n
COROLLARY 8. for all and let
Suppose that either
= u,
i,j.
""
(i;
If
Z""
n=O
6
=
n
a ..
J1
= 8 1.J ..
max { Ia .. JJ
for all
I:
2j E
liB f/l 2 < ... (f E H), then
E } 0
(aiJo)
i,j
or
for all other
n E l
+,
is the matrix of a
0
bow\ded operator. (ii)
unit ball of
PROOF. Let
I f the series
H, then
We assume that
D
n
(a ij ) a ji
~
aa
liB fil2 converges uniformly for n n=O is the matrix of a compact operator.
= a ij
for all
i,j, the other case being similar.
be the finite rank bounded operator with diagonal matrix
given by
= a JJ ..
in the
f
(2j E E ), n
(d~~» 1J
HANKEL OPERATORS and
d~~) = 0
for all other
i,j.
93
on ,
Plainly
and
cnt
=
B
n
Dn •
Therefore
and so
Thus (i) follows from Theorem 7(i)
and the above remark.
converges uniformly on the unit ball u of "" then the same is true of the series .z IIC!fIl Z , ai.ld hence of the series n=O Also, i f
Given
e > 0, there exists
2: P
liB fll2 < e
Therefore, by
n=q
ZCD
C*
n-O n
(aij )
:= N , fEU). e
~ e~
(p > q ~ N ).
e
B2n+l , so
is the matrix of a compact operator. Let
REMARK.
{a} n
be a complex sequence with
2
00
Then
la 1 < "", and take n=O n for some with n 1, on = ~
1'.
+
CD
(a i +j
(i)
and (ii)
of Corollary
and
for,
k E E , and so n n
n CD
Z 02 ~ L la 12 < n n= O n · O n= Thus
such that
ZCD Bn converges in the operator norm. n=O converges in the operator norm, and since A = B + C , n n n
similar result holds for
Similarly,
e
(19) ,
IIZP BZn ll A
(p > q
n
n=q
N
H,
co.
8 are applicable to every Hankel matrix
>· Corollary 8 can be strengthened when the matrix elements are non-nega-
tive.
BONSALL
94
COROLLARY 9. (i)
(a ij )
is the matrix of a bounded operator if and only if 00
Z
/lBnf/l2 <
(f
00
E H).
(21)
n-O (ii) (21)
(a 1j )
is the matrix of a compact operator if and only if the series
converges uniformly for Since
B
* IiCnfll
H.
in the unit ball of
Cn'* is obtained from the matrix by replacing diagonal elements by zero. Therefore, by Lemma 5,
PROOF. of
f
n
• a ij ,
8 ji
~he
matrix of
IIBnf II 1/, and Theorem 7 completes the proof of (i).
~
used in the proof of Corollary 8(ii) of a compact operator if the ball
of
U
seri~s
(21)
(a ij )
is the matrix
converges uniformly on the unit
H.
Conversely, suppose that A, and let
now shows that
Also the argument
P
H onto the linear span of
be the projection of
o
(eO,e1, ••• ,e n _ 1 ).
limllA - P AP
Then
n
n-
A - (AO+Al + ••• + An)
1s the matrix of a compact operator
(a ij )
n
II = O.
Since the matrix elements of
are do.ainated by those of
A - P n-l AP n-l' Lemma 5 2 2
shows that limllA - (AO+Al +••• + An) Ii
=
O.
n--
Again, by Lemma 5, p
p
liz
n=q
Bn II ~
liZ
n=q
Anll,
00
Z
and so
B n=O n
converges in the operator norm.
By (19), it follows that
00
Z
n=U
/lB
fll2
converges uniformly on
20
U, and similarly for
B2n+l •
Corollary 9 is applicable to any Hankel matrix with non-negative entries, and constitutes our second criterion.
As an example we prove the following
corollary. COROLLARY 10.
Let
integers, let
a
m E
~,
let ~
m
F = m
=0
nk
{n k } be a strictly increasing
= -1k
(k E N)
and
a
n
=0
m {k : 2m-I -c:: Ok < 2 }, and let
otherwise.
.If
of .loo-nesa t 1ve + for all other o E ~ • For ~
s~quence
if m • card Fm!min Fm
F ; '/J m
HANKEL OPERATORS
2
GD
~
then
{a} n
PROOF.
is the coefficient sequence of a compact Hankel operator.
We take
=
a i +j , aLld let
IITnll" 1
m
liBmII
::
IIAmII ::
and, by Corollary 9, the
!L
m
denote the elementary Hankel
n {Ok.} , where
=~
E ..!.. < kEF k. -
0:
=
1
lID
,
and
O~
=0
(k ; n).
1
-T kEF k n k m
~
m
REMARK. 1
Tn
and A
Therefore
(22)
!J: <00) m=1 m
operator with coefficient sequence We have
95
•
m
u.2 <
'1:.""
If
m=l
'm
card F
m
=1
liB 112
r,GD
m=l
Hankel operator with matrix
(22) holds in particular if
then
(a i +j )
for each
<
GO
m
is compact.
m,
for then
= -m • This is the case considered by Kwapieri and PelczyJski [3]. REFERENCES
1.
F. F. Bonsall, Boundedness of Hankel matrices, J. London Math. Soc., to appear.
2.
P. L. Duren, Theory of HP spaces, Academic Press, New York, 1970.
3.
S. Kwapien and A. Pelczydski, Some linear topological properties of the Hardy spaces HP, Compositio Math. 33 (1976), 261-288.
4.
D. H. Luecking, Representation and duality in weighted spaces of analytic functions, preprint, 1984.
5.
E. A. Nordgren, M. Radjabalipour, H. Radjavi, and P. Rosenthal, On invariant operator ranges, Trans. Amer. Math. Soc. 251 (1979), 389-398.
6.
K. E. Petersen, Brownian motion, Hardy spaces and bounded mean oscillation, London Math. Soc. Lecture Note Series, 28. 1977.
7.
s. C. Power, Hankel operators on Hilbert space, Research Notes in Mathematics 64, Pitman, London, 1982.
8.
D. Sarason t Function theory on the unit circle, Lecture Notes, Virginia
Polytechnic Institute and State University, Blacksburg, Virginia, 1978.
DEPARTMENT OF MATHEMATICS UNIVERSITY OF EDINBURGH EDINBURGH, SCOTLAND EH9 3JZ
Contemporary Mathematics Volume 32, 1984
ALGEBRA AND TOPOLOGY IN BANACH ALGEBRAS by H. G. Dales
A Banach algebra has. of course, both algebraic and topological structures, and these structures are related, apparently loosely, by the requirement that the two algebraic operations be continuous.
III many cases,
this relationship turns out to be subtler and deeper than is at first apparent: there are some striking results and there are some seemingly difficult points which remain open.
It was Professor Rickart who made the first substantial
contribution to the programme of investigating this relationship, and it was the work and questions in his famous treatise [31] that aroused my interest in the topic.
So I am very pleased to be able, on this occaSion, to report on
progress that has been made in the programme since [31] was written, and in particular to discuss the questions which remain open. The topic is however already well surveyed, and it seemS to be inappropriate to reproduce details easily available elsewhere.
Some comments on
the literature are made at the end of this article. Let me first mention a quite elementary result in this area. belong to a Banach algebra
(A,
v(a), the spectral radius of the other hand,
11·11
of
A.
11·11
replace
limllan"l/n
1:·11).
Then
o(a), the spectrum of
Let
a
a, and
a, are defined in purely algebraic terms.
On
depends on the topology given by the norm
(And only on the topology:
we obtain the same number i f we
by an equivalent norm.)
of Banach algebras"?)
that
Yet we know ("the fundamental theorem v(a)'" limllanil i / n •
The serious part of the programme can be attacked through the following "automatic continuity" problem. 9: A
4
B be a homomorphism.
ensure that each such
e
Let
A and
B be Banach algebras, and let
Which algebraic conditions on
is automatically continuous?
1/'11)
has a unique complete norm i f each norm
a Banach algebra is equivalent to the given norm
A Banach algebra
with respect to which
11·11.
B
A particularly sharp
form of the question is the "uniqueness of norm problem". (A,
A and/or
A is
The problem of decid-
ing which Banach algebras have a unique complete norm is a special case of the © 1984 American Mathematical Society 0271·4132/84 $1.00 + $.25 per page
97
98
DALES
above automatic continuity problem.
If a Banach algebra
unique complete norm, then the algebraic structure (If the topology of
A does have a
A actually determines
A.
There are many other. automatic continuity questions.
One involves the
continuity and nature of derivations from a Banach algebra
A into a Banach
space which is an A-modlal.e: see [7] for a discllsRion of this question. My claim is that the resolution of the problems which have been solved has given valuable inSight into the structure of Banach Algebras, and has also focussed attention on types of Banach algebras that are of interest and which would not otherwise halle been investigated.
It must surely be expectL':
that this will also be the case when the remaining problems are resolved. Let us fir.st consider the uniqueness of norm problem. earliest result was given by Eidelheit in 1940 ([12]): space, then the Banach algebra has a unique complete norm.
B(E)
if
Probably the E
is a Banach
of all bounded linear operators on
At about the
Same
E
time, Gelfand ([17]) proved
that every homomorphism from a Banach algebra into a commutative, semi-simple Banach algebra is continuous
that every commutative semi-simple Banach
~:md
algebra therefore has a unique complete norm.
About 1948, Rickart raised the
problem of whether or not every (possibly non-commutative) semi-simple Banach algebra has a unique complete norm topology.
Although unable to solve this
problem, he did show, for example, that every primitive Banach algebra with miniloal idempotents lIas a unique complete norm, and that a homomorphism from a Banach algebra onto a semi-simple Banach algebra with a unique complete norm topology is automatically continuous ([300, [11, §II.5]). That all semi-simple aanach
algebra~
do have a unique topology was
eventually proved by Johnson in 1967 ([20]), and this is still the most important result of this type.
Between 1965 and 1974, the uniqueness of the
complete norm was proved for Arens-Hoffman extensions of commutative semi-simple Han3ch algebras ([28]), Banach algebras of formal power series ([29]), and the radical convolution algebra
Ll(D,I) ([19]), and a number of other
example!:! are now known. To discusR the proof, we introduce a basic notion in automatic continuity theory, that of the separating space. between Banach spaces
e; (T)
::0:
E and
{y E F:
x
n
~
F.
Let
Then the
there exists 0
and
r(x) n
T: E
-+
~~parating
-+
y}.
be a linear map
space of
(x) c E with n
F
r
is the set
ALGEBRA AND TOPOLOGY The use of the separat:i.ng space to study
99
of the continuity of linear
qu(~stions
operators is very old (e.g., [18]), but its first explicit use in our situation seems to be due to Rickart in [30]. Note that G(T)
= {O}
G(T)
if and only if
if
G(T)
is
in
B, then G(e) Let
"small".
~A
T
is continuous.
8:A
If
-l-
So
T
be the set of characters on a Banach
quasi-nilpotent elements in
= lim
~(b)
In
= lim
n
~(e(a
c
-~
n
» = o.
= rad
~(B)
But what if
B.
algebra
~
E ~A
Of course
A.
is continuous.
~(A)
for the set of
is a homomorphism and that
0
in
A. Take
E CPs.
Then
bE G(e), say
e
E CPA' and so
Thus it is always true that
this tells us little, but, if
g~neral,
G(e)
11
a
is dense
A.
9: A-+B
e(a ), where
SeA)
B.
we have the automatic continuity result that each Write rad A for the (Jacobson) radical of A and
b
is "close to continuous"
is a homoDlOrpilism and if
B
is a (two-sided) ideal in
Suppose that
F, and that
is always a closed linear subspace of
In particular, i f
B is commutative, it shows that B is semi-simple,
0
is continuous.
B is not commutative?
QUESTION 1.
Let
9: A -+ B be a homomorphism.
It is necessarily true that
G(e) c ~(B)?
If
G(O)
C ~(B)
and i f
automatically continuous. each homomorphism
9(A) - B, then
G(O)
rad B, and so
C
e
is
A positive answer to Question 1 would show that
0: A -+ B for which
It is easy to show that, i f
8(A)
b E G(e), then
is semi-simple is continuous.
a(b)
is a connected set contain-
ing the origin, but nothing further seems to be known. The positive answer to Question 1 in the special case that
S
is an
epimorl'h •. .:>nt is Johnson's uniqueness of norm theorem.
Aupetit has a new proof
of this theorem ([3]); it uses the fact that, if
U -+ A is an analytic
function on an open set
U
c t:, then
v
0
f
;
f
U -+ R
is subharmonic.
The
proof does not use any theory of representations, and so is an "internal proof".
The exact form of Aupetit's result is that, if
homomorphism, then
G(e) n SeA) C
~(B).
9; A
-+
B is a
This solves Question 1 if
but we cannot go to the general case in any obvious way from this: difficulty is that the spectral radius function on
B ([2]).
v
e(A) - B,
the
is not necessarily a continuous
100
DALES
We do seem to know rather a lot about the uniqueness of norm problem. Yet we cannot even characterize the commutative Banach algebras which have a unique complete norm. The most obvious example of a Banach algebra with non-equivalent complete algebra norms is an infinite-dimensional Banach space with all products taken to be zero.
Such an algebra is nilpotent and therefore equal to its
prime radical.
example due to Feldman shows that a Banach algebra with a
An
one-dimensional radical can have two non-equivalent complete norms ([16] ; see [5,§6]). radical.
In this example, the radical is equal to the prime
One way to exclude both of these examples is to consider only semi-
prime Banach algebras. a
=
0
or
b
= 0,
and
(An algebra
A is prime if
A is semi-prime if
aAa
aAb
= to}
implies that
= to} implies that a = O.
A commutative algebra is prime if and only if it is an integral domain.) QUESTION 2. norm?
Does each semi-prime Banach algebra have a
unique complete
In particular. does a commutative Banach algebrn which is an integral
domain have a unique complete norm? This question has been studied by Cusack ([8]), who showed that it leads to consideration of topologically simple algebras. is topologically simple if than
to} and A.
dim A > 1
and if
A Banach algebra
A
A has no closed ideals other
A commutative, topologically simple Banach algebra is
necessarily a radical algebra which is an integral domain.
Cusack shows that,
if the answer to Question 2 is negative, then there are commutative, topologically simple Banach algebras. QUESTION 3.
Is there a
commutative, topologically simple Banach algebra?
My guess is that such algebras do exist. example will not be easy:
But the construction of an
it seems to be at least as difficult as the con-
struction of an operator on a Banach space without a proper, closed, invariant subspace.
For a discussion of this problem, see [15].
In the above automatic continuity problems, we imposed conditions on the range algebra of a homomorphism. conditions
Let us now consider questions where
are imposed on the domain algebra.
It is natural to concentrate
on the case in which the domain algebra is a C*-algebra.
An important tool is the continuity ideal of a homomorphism.
e: A ~ B be a homomorphism with separating space G(e). l(e)
= {a E A :
6(a)6(6)
=
6(9)6(a)
= to}}.
Then
Let
101
ALGEBRA AND TOPOLOGY
The set
I( e)
is the continuity ideal of
e.
Clearly,
A, and it is rather straightforward to check ([22, 1.3]) l(e) .. {a E A : the maps x A~
e(ax),x
~
is an ideal in
that
a(xa) ,
B, are both continuous}.
e
The intuition is that "large n in
~
1( e)
is "close to continuous" if I (e)
is
A.
Let uS first consider homomorphisms from commutative C*-algebras. X be a compact space, and let
on
X.
C(X)
Let
be the set of all continuous functions
Rickart ([31, p. 76]) mentions the question whether or not each
homomorphism from
C(X)
is automatically continuous.
The first results are due to Bade and Curtis in 1960 ([5]).
To recall
their theorem, we first give some standard notation. Let
Thus
X
be a compact space, and let M x
= {f
E C(X)
f(x) = O},
J
= {f
E C(X)
£-1(0)
x
Then
x E X.
is a neighbollrllood of x} ..
is a maximal ideal of C(X), and J c j .. M. x x x x phism is a non-zero homomorphism v from a maximal ideal M
that
v(M) x
is a radical Banach algebra.
radical homomorphism, then Let
a
either
a:
C(X)
~
v IJ
:II
x
A radical homomorM
x
of
C(X)
It is easy to see that, if
0, and so
B be a homomorphism.
v
such. v
is a
is discontinuous.
Then Bade and Curtis proved that
is continuous, or there is a non-empty, finite subset {xl, ••• ,xn }
of X, a continuous homomorphism
v1, ••• ,v n : C(X)
~
B such that
~:
C(X)
~
B, and linear maps
9 ... ~ + v l +••• + vn
and
v .IM l. xi
is a
radical homomorphism. Let
1(9)
e.
be the continuity ideal of the above homomorphism
n ••• n J , and so 1(9) - M n ••• n M • Thus G(e) xl ~ xl ~ is an ideal with finite codimension in C(X): we shall return to this fact
Then
l(a) :> J
below. There is another positive result about homomorphisms that is an
a: A ~ B be a homomorphism with separating space G(e), (a) be a sequence in A. Set Gn - e(an ••• a l )G(9). Then clearly n
important tool. and let
Gn+l (N
C
Gn •
Let
A form of the stability lemma asserts that there exists
depends on the sequence
(a
n
»
such that
Gn = GN for n
~
N.
N ER This
102
DALES
principle was used by Johnson and Sinclair in [23]; see also [24] for a general form of the result. theorem
The result leads to the following prime ideal
([33,11.4J, [6,2.7]).
commutative Banach algebra ideal
K of
Let
9: A -,. B be a
A such that
9(A)
B such that the homomorphism
homomorphism from a
= B.
A4 B
Then there is a closed 8/K
4
has a prime kernel.
Combining this fact with the theorem of Bade and Curtis, we see that, i f
e
there is a discontinuous homomorphism is a maximal ideal Banach algebra
M of
from an algebraC(X), then there
C(X), a prime ideal
R, and an embedding
P with
P; M,
a radical
M/P -.. R.
Eventually, it was shown (see [11],[9,9.6]) that, given such an and such a an
P
em~edding
in M/p
C(X) 4
such that
Hlp
has cardinality
'~l'
R for certain radical Banach algebras
The question concerning homomorphisms from study of ideals and order structure in study of radical Banach algebras. ed the radical Banach algebras
e(X)
e(X)
M
there does exist
R.
thus led to a deep
and other algebras, and to a
Indeed, Esterle (see [14]) has characteriz-
R which can occur in the above result, and
he has also given a comprehensive classification scheme for commutative, radical Hanach algebras. The result shows that, if the eontinuum hypothesis be assumed, then there is a discontinuous homomorphism from each infinite, compact space
C(X)
into a Banach algebra for
X.
It is a remarkable fact that the assumption of the continuum hypothesis cannot be dropped from this result: from each algebra Solovay.
C(X)
models of
ZFC
in which each homomorphism
are continuous have been constructed by Woodin and
See [lO,§4] for a discussion of this aspect of the story from the
point of view of an analyst. Now let us turn to consideration of homomorphisms from possibly :l.oncommutative C*-a1gebras. 4.
~UESTION
For which C*-algebras
A is it true that each homomorphism from
A into a Banach algebra is automatically continuous? Let
A Je a C*-algebra,
homomorphism.
I(e)
nC
B be a Banach algebra, and
A.
A.
Thus, if
4
B a
It follows from Bade and Curtis's theorem, as above, that
has finite codimension in
of
9: A
C for each commutative C*-subalgebra
But this is sufficient to show that A is an infinite-dimensional
ideal of finite codimension, then I(a)
c
A.
I (6)
C
has Unite codimension in
C*-al~ebra
with no proper, closed
If, further,
A has an
ALGEBRA AND TOPOLOGY identity, then
e
I(9) - A, and so
of proving that each homomorphism
103
is continuous.
B(H)
-+
This is the easiest way
B is automatically continuous
([22],[33,12.4]).
A slight modification of the method shows that each
homomorphism
-+
K(H)
B is automatically continuous:
C*-algebra of compact operators on a Hilbert space that the multiplier algebra of
here,
K(H)
denotes the
H, and one uses the fact
K(H), B(H), has no proper closed, cofinite
ideals. A second technique for proving that each homomorphism from certain C*-algebras is continuous was also introduced by Johnson ([21]):
it applies
to C*-algebras with many projections, and. shows, for example, that each homomorphism from a von Neumann algebra which has no direct summand of type I is continuous. Let
A be a C*-algebra.
a *-homomorphism
IT
from
A representation of dimension M (t), the algebra of
A into
n
n x n
n
of
A is
matrices.
The representation is irreducible if {O} and
Mn (~)
are the only linear
subspaces of
n(A).
Two irreducible represent-
M «() n
ations of dimension
which are invariant for n
are equivalent if they have the same kernel.
It is proved in [1] that, if
A is a C*-algebra widch has infinitely
many non-equivalent irreducible representations of dimension n E N, then there is a discontinuous homomorphism from made that otherwise all homomorphisms from
n
for some
A, and the guess is
A are automatically continuous.
By combining the two techniques mentioned above, and by using technical decomposition theorems for C*-algebras, it can be proved that the guess is true for a rather large class of C*-algebras, the so-called the class includes
(i)
each
AW*, and hence each von Neumann, algebra;
(ii)
each closed ideal in an AW*-algebra;
(iv)
the algebra
A ® K(H)
AW*M-algebras:
(iii)
for each C*-algebra
each commutative C*-algebra; A.
However, many C*-algebras are not of this c:ass.
It seems likely that
a new idea ,,,rill be necessary to resolve the question for these algebras. is a particular challenge.
Let
Here
A be a (topologically) Simple, infinite-
dimensional C*-algebra without identity.
Is every homomorphism from
A
automatically continuous? We do have an analogue of the Bade-Curtis theorem and of the prime ideal theorem for non-commutative C*-algebraR: from
[32]~
[271, and unpublished work of Cusack.
tinuous homomorphism from a C*-algebra subspace algebra of
the following result is taken
F of A.
A such that Further,
F
~
e = '"" +
morphism which coincides with
9
A.
on
9: A ~ B be a discon-
Then there is a finite-dimensional
1(9) = A and
'J, where
Let
~:
F + 1(9),
A
F + I(G) -+
is a dense sub-
B is a continuous homo-
and
\/: A -" B is a linear
104
DALES
vII(9)
map such that ideal from
P
in
M such that
~
G(9) P
is a homomorphism.
Finally, there is a prime
is the kernel of a discontinuous homomorphism
M into a Banach algebra. We saw at the beginning of this article that the uniqueness of norm
problem is closely related to the question of the automatic continuity of epimorphisms onto Banach algebras. QUESTION S.
Consider the following question.
Is each epimorphism from a C*-algebra onto a Banach algebra
automatically continuous? It was first proved by &ster1e ([13]) that each epimorphism from a COlmnutative C*-algebra is indeed continuous: proof.
see [14] for a very elegant
In [1], the same result is proved for AW*M-algebras, and it has been
proved by Laursen for epimorphisms with commutative range (see [26]): proofs rely heavily on &sterle's result.
these
However, the question is open for
general C*-algebras. Let me conclude by mentioning some sources where more detailed information can be found.
The theory of the automatic continuity of linear operators
between Banach spaces is given in Sinclair's book [33].
The survey [9] con-
centrates on the theory of homomorphisms between Banach algebras and of derivations into modules, and the article [251 discusses the automatic continuity of "intertwining operators", a class of linear maps which includes both ilOmomorphism and derivations: Banach algebra on [0,1].
c(n)[O,l]
the main applications are to maps from the
of n-times continuously differentiable functions
More details of the theory of homomorphisms from C*-algebras are
given in [10].
Finally, let me mention the volume [4], the proceedings of a
conference held in 1981, where many relevant articles can be found.
A longer
list of open questions can be found at the end of that volume.
REFERENCES 1.
E. Albrecht and H. G. Dales, Continuity of homomorphisms from C*-a1gebras and other Banach algebras, in [4], 375-396.
2.
B. Aupetit, Propriet's spectrales des alg~br~s de Banach, Lecture Notes in Maths., 735, Springer-Verlag, 1979.
3.
B. Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras, J. Functional Analysis 47 (1982), 1-6.
4.
J. M. Bachar et al (ed.), Ra~ica1 Banach algebras and automatic continuity, Proceedings, Long Beach, 1981, Lecture Notes in Maths. Springer-Verlag, 1983.
975.
105
ALGEB RA AND TOPOLOGY ~
W. G. dade and P. C. Curtis, Jr., Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 851-866.
6.
W. G. Bade and P. C. Curtis, Jr., Prime ideals and automatic continuity problems for Banach algebras, J. Functional Analysis 29 (1978), 88-103.
7.
W. G. Bade and P. C. Curtis, Jr., Module derivations from conmutative J:Sanacll algebras, these proceedings, 7l-8l. J. Cusack, Automatic continuity and topologically simple radical Banach algebras, J. London Math. Soc. (2) 16 (1977), 493-500.
8. 9.
H. G. Dales, Automatic continuity: 10 (1978), 129-183.
a survey, Bull. London Math. Soc.
10.
H. G. Dales, Automatic continuity of homomorphisms from C*-algebras, in "Functional Analysis: surveys and related results III" (ed. K.-D. Bierstedt and B. Fuchsteiner), North Holland, 1984.
11.
H. G. Dales and J. Esterle, Discontinuous homomorphisms from Bull. Amer. Math. Soc. (2) 83 (1977), 257-258.
12.
M. Eidelheit, On isomorphisms of rings of linear operators, Studia Math., 9 (1940), 97-105.
13.
J. Esterle, Theorems of Gelfand-Mazur type and continuity of epimorphisms
from
C(K) , J. Functional Analysis
36
(l9~O),
C(X) ,
273-286.
14.
J. Esterle, Elements for a classification of commutative radical Banach algebras, in [4], 4-65.
15.
J. Esterle, Quasimultipliers, representations of H, and the closed ideal problem for commutative Banach algebras, in [4]', 66-162.
16.
C. Feldman, The Wedderburn principal theorem in Banach algebras, Proc. Amer. Math. Soc. 2 (1951), 771-777.
17.
I. M. Gelfand, Normierte Ringe, Mat. Sbornik
18.
F. Hausdorff, Zur Theorie der linearen metrischen Rlume, J. Reine Angew. Math. 167 (1932), 294-311.
19.
N. p. Jewell and A. M. Sinclair, Epimorphisms and derivations on
...
Ll(O,l)
9 (1941), 3-24.
are continuous, Bull. London Math. Soc.
8 (1976), 135-139.
20.
B. E. Johnson, The uniqueness of the (complete) norm topology, Bull. Amer. Math. Soc. 73 (1967), 537-541.
21.
B. E. Johnson, Continuity of homomor~hisms of algebras of operators, J. London Math. Soc. 42 (1967), 537-541.
22.
B. E. Johnson, Continuity of homomorphisms of algebras of operators (II), J. London Math. Soc. (2) 1 (1969), 81-84.
23.
B. E. Johnson, and A. M. Sinclair, Continuity of linear operators conunuting with continuous linear operators II, Trans. Amer. Math. Soc. 146 (1969),533-540.
24.
K. B. Laursen, Some remarks on automatic continuity, Lecture Notes in Mathematics 572 (1976), Springer-Verlag, 96-108.
25.
K. B. Laursen, Automatic continuity 01 generalized intertwining operators, Diss. Mat. (Rozprawy Mat.) 189 (1981).
26.
K. B. Laursen, Epimorphisms of C*-algebras, in "Functional Analysis: surveys and related results III" (ed. K.-D. Bierstedt and B. Fuchsteiner), North Holland, 1984.
2~
K. B. Laursen and A. M. Sinclair, Lifting matrix units in C*-a1gebras II, Math. Scand. 37 (1975), 167-172.
106
DALES
28.
J. A. Lindberg, A class of commutative Banach algebras with unique complete norm topology and continuous derivations, Proc. Amer. Math. Soc. 29 (1911), 516-520.
29.
R. J. Loy, Uniqueness of the complete norm topology and continuity of derivations on Banach algebras, ~hoku Math. J. 22 (1970), 371-378.
30.
C. E. Rickart, The uniqueness of norm problem in Banach algebras, Ann. of Math. 51 (1950), 615-628.
31.
C. E. Rickart, General theory of Banach algebras, van Nostrand, Princeton, 1960.
32.
A. M. Sinclair, Homomorphisms from C*-algebras, Proc. London Math. Soc. (3) 29 (1~74), 435-452; Corrigendum, 32 (1976), 322.
33.
A. M. Sinclair, Automatic continuity of linear operators, London Math. Soc. Lecture Note Series 21, Cambridge University Press, 1976.
SCHOOL OF MATHEMATICS UNIVERSITY OF LEEDS LEEDS, LS2 9JT ENGLAND
Contemporary Mathematics Volume 32, 1984
t-UTl'AG-LEFFLER METHODS IN THE THEORY OF BANACH ALGEBRAS
AND A NEW APPROACH TO MICHAEL'S PROBLEM Jean Esterle 1.
INTRODUCTION The aim of this p;lper is to describe two useful tools in the theory of
Banach algebras (the Mittag-Leffler Theorem on inverse limits and the notion of bounded approximate identity), and to apply these ideas to operator theory and to tne theory of entire functions of several complex vari abIes. This paper does not intend to present a survey of recent progress ill thi:! theory of Banach algebras.
Works as important as Dale's construction of
discontinuous homomorphisms from C(K) [14J, Domar's characterization [17] of closed ideals of Ll(IR+, e- t2 ), and Thomas's construci:ion [47] of a weighted radical Banaell algebra of power series with nonstandard ideals (see Bade's report in this volume [5]) will not he discussed.
We just lvish to
presl~nt
a
more-or-less connected set of ideas appearing in works of Arens [3], Allan (1). Sinclair L45j, Dixon [16], and in various papers [18]-[22J of the autill)r, which lead to one of the known constructions of discontinuous homomorphisms
C(K)
frolll
and yield some progress in the general theory of Banach algebras.
We also show how a slight reformulation of the Mittag-Leffler Theurem on inverse limits, given in Theorem 2.1, gives a tool which can be used to prove Cohen's Factorizatiun Theorem [12J (see Theorem 4.3). Sec::ions 2-5 are essentially expository.
In Section 2 we present the
Mittag-Leffler Theorem on inverse limits in a slightly more general form than usual, and show how this theorem includes the Baire Category Theorem.
We
!illOW
also the connection between the abstract version and the classical MittagLeffler Theorem about meromorphic functions with prescribed poles and Singular parts. Section 3 gives some applications to the theory of Banach algebras. first one is a step of G. R. Allan's construction of an embedding of into Banach algebras [11.
The
t[[X]]
The second one is a part of the author's classifica-
tion of commutative radical Banach algebras [21].
107
(We show that a rauical
© 1984 American Mathematical Society 027]·4132/84 $1.00 + S.25 per page
ESTERLE
108 Banach algebra
R possesses a nonzero element
x
such that
2 x E [x R]
if
R possesses a nonzero rational semigroup.)
and only if
In Section 4 we present a new "Mittag-Leffler approach" to Cohen's Factorization Theorem, and some standard consequences of this theorem which seem like the theorem itself
to have been ignored by many distinguished
analysts and specialists of Lie groups. Sectlon 5 gives some "spectral mapping theorems" which do not use the Mittag-Leffler Tlleorem but instead are based upon a game involving topological divisors of 7.eros related to bounded approximate identities. SeLl ion 6 gives anew, unpublished approach to Michael's problem l34] about continuity of characters on Fr/ehet algebr-as. to P. G. Dixon and the author.) l~rechet
commutative
n
VI
We show in particular that all characters on
algebnlfi would be continuous i[ it were possible
to const ruct a sequence such that
(F n> n::l ?
0
(This approach is due
•••
1'n(q:-)
0
of entire functionfi of
= 0.
(:2
into itself
The existence of such a sequence is
n
unclear. and it even seems to be an open problem to decide whether for each entire function
from
2
n Fn (( 2 )
~ ~
Examples due to Fatou [26] and Bieberbach [6J show that some one-to-one entire functions F: a: 2 -+ «;2 F
t
into i.tself.
1, have nondense ranges.
whose Jacobians are identically equal to
The usual
approach consists of considering solutions of some functional equations (references, which go baCK to Poincar~'s tilesiti, are given in Section 6), but we present here
Cl
new appraoch which consistl:i tlf constructing sequences of
«;2
analytic automorphisms of
with respect to the a-topology.
of Jacobian 1 which have a nonsurjective limit (This idea is intimately related to the nOLion We construct in particular a
of Cohen clements l19] for Banach algebras.) one-to-one entire function
{( x , y) E (
2
IRe
F: ¢2
~ a:2 l:;uch that
x ::- 1, Re y > I}
Jacobian, such that lar, G
(D 2 )
avoidl:i the sets
')
an d
{ (x, y) E
allows us to obtain a strange entire function -1
2 F(a:)
inf
(I u I, Iv I>
== 1
('" IRe Co: 0;2
for each
This
x .;: 1, Re y < l}. -+
(;2
with nonvanishing
(u, v) E G(a;2).
In particu-
meets the ranges of all nonconstant entire functions (where
denotes the closed polydisc of radius for the construction of
l~,
2).
D2
Theorem 2.1 is usad as a basis
and the Mittag-Leffler Theorem (applied to a rather
unusual system of complete metric spaces) is used to establish the connection between
~1ichael' s
problem and entire functions of several variables.
Michael's
problem seems in [act to be the starting poInt of this circle of ideas, since it was used as an essenti:ll tool in Arens' approach [3] to continuity of characters on Frechet algebras. "
MITTAG-LEFFLER METHODS
109
In this paper we have tried to show how the theory of Banach algebras can still bring results or ideas to other branches of mathematics, and how some general structure results can otill appear inside the theory. indebted to C. E. Rickart.
I am deeply
The results, the methods, the conception of
mathematics given in his treatise [41] were and are a constant reference for my research in mathematics.
2.
THE MITTAG-LEFFLER THEOREM ON INVERSE LIMITS Let
(E ) n rel
be a countable family of sets, and assume that for each
elements
x
= for each
projection from if
x E F, F
~
11 :::
n E
Also, if
E • m
d(x,F) .. inf
1T
(E,d)
is the set of all
n
satisfying
>1 n n_
We will denote by
1.
n E onto >1 n n_ E, then we set
n
+-
of the cartesian product
en (xn +1)
xn =
11m (E ,6)
Its projective limit
is a projective system.
m
the mth coordinate
is a metric space and
d(x,z).
zEF We have the following theorem.
,e ) be a projective system, where En is a complete n n metric space with respect to a metric dn for each n ~ 1. Assume that the
THEOREM 2.1.
Let
(E
following conditions hold:
dn (9 n (x),9 n (y»
(1)
for
S dn +1(x,y)
x,y E En +1 , n
1
~
00
(2 )
)..
Z
n
n=l
<
where
GO,
A = sup n
for
n
~
1
xEEn GO
lim (E
Then K :::
+-
,e ) ; 0, nn
d [x, K
1T
K
(lim (E ,6 )] +-
nn
~
Z )..
m
for
x EE , K
1.
Fix of
g
n n:::2
Fix
and
k::: 1.
> E
n
Then for
0 and let
x EEL.
By induction we define an element
satisfying the following conditions:
n::: k+1
we have d (y , e (y +1» n n n n
< ).. n
ESTERLE
110
This shows that the sequence
Ek for each k ~L for each k ::: I, we have
= lim n--
d l (x, 91
o en-l(Yn»n~+l
0
Denote by
in
dl (x,x l )
(9k
its limit.
xk
(E ,9 ). (xk)k~l E lim ofn 0
is a Cauchy sequence
Since
is continuous
9k
Also
9n (Y1\+1»
0 ••• 0
n
..:; d l (x,a l (Y2» + lim sup n-~
~
m=2
dl (91
tim_1 (Y m) ,91
0 ••• 0
0 ••• 0
9m(y m+l
»
o
+ 11m sup
dl(x, 8l (Y2»
<
n
lim
<:;
n--
;>.;
c
(-
m=l 2m
n--
6 dm(Y m,6m(Ym+1» m= 2
...
+ A- )
A-
m
m
+~.
GO
So
~
m=l projective system x E Ek •
for each
(Enon=-,9 ) ....k. But if
(x )
n
"-m
for each
~~e
have
~k
E lim
0_
Now consider the >..
~
(E
m
~., then setting n ,0n ) n~
00
lim (E , El ) 1n n
so that
dk[x, Rk(lim (E0, 0)] 5 ofn
\'
I..
m=k
A-
m
for
k
>
1.
Tne theorem is proved.
We state as a corollary the usual abstract version of the Mittag-Leffler Theorem (see for example [21], Theorem 2.14). COROLLARY 2.2.
Let
(E ,9 ) 1\
metric space and where
ao
en (Eo +l ) is dense in En in E for each m > 1.
n
be a projective system, where ~
En
for each
n
En+l
E
n
is a complete
n =: L If n (lim (E ,e» is dense m ofn n
1s cuntinuous .for each ~
I, theo
m
Denote by
PROOF:
d
the given distaoce on
n
E • n
Put
n :: 2, put
oIl (x,y) = max
oo (60 (x),
Theo
(E ,5 )
n
0
{d0 (x,y), dn- 1[8n-l(x), tin-I(Y»)'.'"
en (y»
~
for
50 +1 (x,y)
X,y E En+l , n
~
1.
It follows that
is a complete metric space, and the topology defined by
same as the topology defined immediately from Theorem 2.1.
by
d n
on
80
is the
En • The corollary follows then
MITTAG-LEFFLER METHODS COROLLARY 2.3 (The Baire Category Theorem).
(un ) n~1
space, and let
nU
Then
PROOF:
nV
Put
nu
:=
>1 n n_
>' n n:;...&.
subset of
V n
•
=
Then the sequence
m
Denote by
~
(W ,~)
n
~
1
set
d.
as
A routine well-known verification shows
we can define the topology of
with respect to which
is complete.
V
VI' hence dense in n, so that
Xl E
E.
But if
n
V , and
n
n~l
V ~
Denote by
n It follows from Corollary 2.2 that
identity map. each
W is any open
If
is a complete metric space ([11], Chapter V, Section 3), so
that for each
in
E.
is well defined, and it is clearly a distance which on
W defines the same topology that
is decreasing, and
the given metric on
d
x,y E W,
E, then for
W is open,
Since
E.
E.
f1 u • m~
E be a complete metric
be a countable family of dense open subsets of
is dense in
n~l n
Let
III
by a distance
n
Vn +1
n
.• V
n
n l (lim (V ,9
d
n
the
»
is dense n 11 (V ,e ) then Xl = x for (xn)n~l E lim -+n n n n Un = n Vn is dense in E. n~l n:::l -+-
We now give the usual Mittag-Leffler Theorem about meromorphic functions. COROLLARY 2.4.
U be an open subset of the complex plane, let
Let
be a discrete sequence of elements of of rational functions, where
=
S (z) n
each
S n
m n
A-
~
i 2n
i=1 (z - a ) n
(an ) n_ >1' such that the singular part of Denote by
s
1
positive reals with ~n
set Then
= {z
f
at
A
n
and
~ ~
E Uld(z, ~\U)
V
= WU
an
5n
is
for each
u ;.
If
f,
n
and
as
~O
Izl
< A-n }
is a relatively compact subset of
that either
U whose set of poles is n
~
1.
I, choose
and a decreasing sequence
and consider a component
v
on
f
the Riemann sphere
an increasing sequence
be a sequence
has the form
Then there exists a meromorphic function
PROOF:
(Sn)n~l
U, and let
(an)n:::l
{~}
is a bounded subset of
V
S\S~.
of
(where
c.
U
n
-~
GO
,
(en)n:::l of such that the open
is nonempty for each and
n ~ 1.
u = U Q.
Fix n ~ 1 reIn is bounded, it follows
Since ~ n n W is the unbounded component of
In the first case we see that
~\~),
V n (5\U)
or is
112
ESTERLE
nonempty. for each
V, and there exists
intersects
n
and
~
z
D(u,e )
Iz 1<
In the second case we must have
n (t:\U)
V 11 (S\U) ~ V ~
when taking
Since
V.
n(U,E) n
0.
II z I <
= {z E t:
n
i:
u E C\U
n}
for
bence
d(z, C\U)
<.
En
C
V
such that the closed disc
n gn = 0
U = t:
If
An'
u E D(U,E )
we have
n
we obtain a similar property
n;' 1.
It follows then from Runge's
Theorem [13, Chapter·S, Corollary 1.14] that
H(u)
H(~
is dense in
n
)
with respect to the topology of uniform convergence on compact subsets of THis topology can be defined by a distance 6 H(~~ )
witll respect to which ~
Then
is finite (or empty) for each
n
denote by
n
n
Now set
1.
~
n
= {m la
~
n
m
T = n
Let
the set of all meromorpllic functions
E
mEt.
~n
on
f 52 •
dn(f,g)
= 0n(f
(En,dn )
- Tn' g - Tn)
metric space. continuous. as
of elements of gp + Tn+1 E En+l
But
= 0n (f
dn (f, a n (g p + Tn+1» as
~~.
p
+-
function
9n (E n+l )
So
lim (E
that
n
When
U
H(U)
is a complete
0n(f - Tn +l , gp) -~ 0
such that
and
- Tn , gp
+ Tn+1 - Tn ) = 0n (f
is dense in
,en ) i: O. on
f
Then
an: En+l ~ En the restriction map. Then an is f ~ En. Then f - Tn+l E H(~~n)' and there exists a
Consider
p ~ -.
f, g E En.
Set
n
Denote by
(gp)p~l
sequence
for
n and
such that
n
has a holomorphic extension to the whole of
E 52 }.
S m'
~
f - T
11
n
invariant under translation
is complete.
n
52 •
En.
- Tn","'I' g p ) ~
0
Jt follows from Corollary 2.2
But this just means that there exists a meromorphic
U that sat.isfies the conditions of the theorem.
= t:, the proof becomes somewhat simpler because we can use
Taylor expansions instead of rational approximations. Incidentally,
the reader interested in
a
French way of writing a result
as clear as Corollary 2.2 in a form almost inaccessible to human mind is referred to the statement given by Bourbaki in [9, Chapter II, Section 3, Theorem 1].
3.
APPLICATIONS OF
THE
MITTAG-LEFFLER THEOREM
TO
THE THEORY OF BANACH ALGEBRAS
In this section we give a few examples of applications of the MittagLeffler Theorem.
Our first result concerns operator theory.
PROPOSITION 3.1.
Let
on
is dense in
E.
PROOF: dense in
u(E)
If
Let E.
E
n
= E,
Let
E be a Banach space and let
and
(xn)n:::l
xn = U(xn + l ), so that Xl proves the proposition.
R then
en = u
n un{E)
u
be a bounded operator
is dense in
E.
~l
for each
be an element of
n::: 1. l~m
Then
lT 1 (1im
(En,On).
= un(x n +1 > for n::: 1 and Xl E
(E ,a» n
n
We have
n un(E).
n::1
This
is
113
MITTAG-LEFFLER METHODS
to[[X]] the algebra of all formal power series in one variable with zero constant term. The following theorem is a slightly weaker Denote by
version of a basic result of G. R. Allan [I, Lemma 3).
I
n
u:::l th.ere exists a unique algebra homomorphism
= tr(x).
q>(X)
Let
PROOF:
the map n
x E A.
A be a commutative Banach algebra, and let = n x A, and den~te by tr: A -I> A/r the natural surjection. Let
THEOREM 3.2.
~
1.
u
cP
is one-to-one i f
A is not unital.
= ~ ~
xn
be an element of
'O[[X]].
f -+ ~
n
n:!l. n x + xu.
Since
= A,
[xA]
n
- A,
such that
an :
Denote by
[a (A)]-
we have
[xA1
If
cp: 'O[ [X]] -+ A/I
The map
Set
=
A -I> A
A for each
It follows from Corollary 2.2 that there exists a sequence
of elements of
A such that
un
=
9n (un +l )
n
for
~
1.
(u) n n~l A routine induction
shows that
n
Now let Then
be another element of
u
u - u1 E
such that
A
n xn A, so that
~ ~ xm E xnA for n ~ 1. m=l m So the map cp: f -+ u is well-
u -
n~l
defined, and it is clearly an algebra homomorphism from Also f -
cp(X) = tr(x).
Z ~nxn.
If CPI
n
rr(xnA)
n~l
1 (f) E
into
A/I.
is another such homomorphism, consider again n
Then
'O[[X]l
m n(x ) m=l m
Z
= tr(xm+l )
~
= {O},
and
cp
CPl(
n-m-l LX ) , so that
n::m+l n
is unique.
n~l
Ker cP :I- {Ole SineI.'! all nonzero ideals of 'O[ [X)] contain a power of X we have cp(XP) = 0 for tr(xP ) = 0 and xP E n xn A. In particular, some p ~ 1, so that NoW assume that
is not one-to-one, so that
cP
p+l xp = x u. there exists u E A such that [xPA]- = A and vxu = v for v E: A. So cp
n~1
[xA)- - A, we have
Since
is one-to-one if
A is not
unital, and the theorem is proved. Allan showed in [1] that if
A is radical and if
actually exists a one-to-one algebra homomorphism TTO~
= cpo
In particular,
(O[[X)]
~
[xA]
: to[[X]]
= A, -I>
A such that
can be given an algebra norm.
A similar idea was used by the author in [19] (with a sequence such that
[a A] n
=A
for each
n)
then there
(an)n~l
to deal with transcendental extensions
in his original construction of discontinuous ilomomorphisms from
C(K). where
K is an infinite compact space (the construction depends on the continuum hypothesis).
In fact, it is possible to construct discontinuous homomorphisms
114
ESTERLE
from
C(K)
into
A $ 'e
if
A is a commutative radical Banach algebra such
that
[xA]
=A
for some
x
E A (or more generally such that xPA E [xp+lA]-
for some
x E A which is not nilpotent).
In particular, such radical
Banach algebras do possess nonzero real semigroups.
We will not enter this
game here, and refer the interested reader to [20] and [49].
But we will
prove the following theorem, which is a part of the author's "classification of commutative radical Banach algebras" [21). Let
THEOREM 3.3.
R be a commutative radical Banach algebra. Then the
following conditions imply each other:
(1)
x E [x 2R]-
(2)
R
Assume that
-+
t
E
for each
11
n+l
Let
u
y
(a )tEQ+
C
n
[XRl ]
for each
be any element of
Then
Rl
so that
R1
1
y E R,
e En+l 11
1, and denote by
~
over the positive rationals.
R = [xR]
holds, and set
(1)
xy E [X 2R]-
x E Rl and
u
x E R,
possesses a nonzero semigroup
PROOF:
Let
for some non~ero
~
En
is radical,
=
[xRl ) •
the map
A routine verification shows that
Rl •
lXl1+lRl J- = Rl , so that there exists a sequence (up)p~l of elements of R1 n+1 such that y = lim x u . Adjoin a unit e to Rlo Since RI is radical, p--
p
P-le + up E exp (RI $ te) (v )
of elements of
>
p p ... 1
xn +l exp vp
Since
for each
n
>
1.
=
for each Rl E9 te
p ~ 1, so that there exists a sequence n+1 x exp
such that
v
as
[x exp (n ~ l»)n+l, we see that
Since
x; 0, we have
p
-+
00
0
is dense in
RI ; to}, and it follows from is dense in RIO In particular, there
(lim (E n ,9 n » -c. exists an element Y = (Yn)n~1 of l}m (E ,8) such that Yl ; 00 We n+1 (n+m)! In t for and so that Ynn =n Yn+m have Yn = Yn+1 for n ~ 1, n ~ 1 m ~ O. If r = i/j is a positive rational number, set a r _ y~(j-l)!.
Corollary 2.2 that
Tl'l
Routine verifications show that the definition does not depend on the choice r+r' r r' + 1 of i and j, and that a - a a for r,r' E Q. Since a = Yl + 0, property (2) holds in Rl , and hence in R. Now assume that U
n,m
=
{y E
II
inflla zER
(2)
1/
holds. 1
m - yzll < -} n
p > m.
Then
I for
=[ n
U +8 t R) tEQ
~
1
and
and The set
U
n,m
I which contains allp for each p > m. Fix va 1 / p = lim (ee + v)a l/p (here we again adjoin a e-.<>
is clearly an open subset of v E Rand
Let
MITTAG-LEFFLER METHODS
115
to R). Since ce + v is invertible in R e Ce for each a l/m = (ee + v)al/p(al/m -lIp (ee + v)-l) , so that
unit e we have
(ee + v)al/p E U n,m p > m, n
~
for each
1.
I
and hence
land
~
U al/PR
m ::= 1.
= n Un,m
n,m for each
a 2 m E [x 2 R]-
n
> p_m
p::=l and each
4
2
Consequently
p > m,
So
n::= 1
that the set
for
[x R]
x ~ 0
-
m
0, and
for
-U-,
and U is dense in I n,m n,m It follows from the Baire Category Theorem -
is dense in
= I.
I.
I, so that Since a r ~ 0
Choose
Then
x E 4
for each
~
x E [x 2R]-
and
C
E >
e > 0,
for some
since
x E I.
r E
r E Q+.
Q+ , we have
1
~
{O},
This completes the proof
of the theorem. follows from Bohr's theory of almost periodic functions that
It
f
~ [f2 el(Q+)]- for each nonzero element
f
of
el(Q+). This shows that
Theorem 3.3 does not extend to semisimple algebras (see the details in [21, p. 35]).
The Mittag-Leffler Theorem has also many applications to automatic continuity (see [15]).
4.
A PROJECTIVE PROOF AND SOME APPLICATIONS OF COHEN'S FACTORIZATION THEOREM In this section we give a proof of Cohen's Factorization Theorem based
upon Theorem 2.1, as well as some applications of this theorem to group algebras and operator theory which are more or less well-known. DEFINITION 4.1. ~
A Banach algebra
identity bounded
a sequence
(u n )n::=l
and such that
K if for every finite subset
of elements of
lIu a - all -,. n
DEFINITION 4.2. system
EY
A 1s said to have a left-bounded approxi-
Let
0
as
A such that n
-+ co
(A,
A there exists
II un II ::: K for n
for each
A be a Banach algebra.
6 of
~
1,
a E 4 •
A left Banach A-module is a
M is a Banach space and
is a continuous
algebra Llomomorphism. We shall write Banach A-module." a E A and
ax
instead of
<j>(a)[x] and just say that
Note that we can renorm
M so that
x E M (see for example [21, p. 9]).
"M
is a left
lIaxll::: lIallllxll
We will
for
always assume that
the above inequality holds. To avoid minor technicalities we limit ourselves to the case where the bound
K of the approximate identities equals
1.
116
ESTERLE
THEOREM 4.3 (Cohen's Factorization Theorem for Modules).
of elements of
(Yn)n2:1
I/yn -xl/
is a closed linear subspace of
AM
there exists a sequence
E AM
x
Then
as
0
-+
n
Let
x E Hi
and every
such that for every
a y n n
x, /lan 1/
=
M be a left
M, and for each A and a sequence for
S 1
n
~ 1,
and
-+ "".
Routine verifications show that for every
a E A there exists a sequence
/lun a-aI/
and Now fix
n ~ 1.
lIun x-xII
0
-+
-+
n
n -~
for
0
(u) DO
of elements of
n~l
and such that
,
x E HI' and adjoint a unit
and
> 0
€.
A
lIunll == I e
to
A.
~ on AU such that A = Ker~. We extend to
Then there exists a character
An = A e
of elements of
n~l
n
such that
H
HI = [span AM]-.
PROOF:
(a)
A be a Banach
1, and let
algebra with bounded approximate identity bounded by Banach A-module.
Let
the module action of A on M in the obvious way. Denote by o II and set the closed unit ball of A, u = {y E Mlilly-x/l < €}. Denote by B the closure in 0 x U of the set of elements of the form {(a,a
-1
(e
.
1/-1
x) 1a E D II Inv A ,a
We can define the given topology of
x E U}.
by a distance with respect to which x,y E U let (8,5) Then ~ E
that
8
is a closed subset of
P
is complete.
lIa-bll + d(y,z).
is a complete metric space.
Now set for
8
Bp
Now for
Since Pick
B( (b n , b:lX) , (a,y»
-+
0
II (e n -e)(b n ~ (b n )e) II
-+
0
lIe-A(e-e) 1/ ::: 1 n
(e-;\.(e-e» -1 n
=
m
2: A (e-e)
m~u
n
verification shows that
-+0 00
(e )
,and
and fix
p
sup 1'P(b n ) 1
lim
of elements in
n ~1
n
-+
00,
n:: 1. m
for
and sllch that
A"
n
n::
Hence
lib -1 [e-X.(e-e ) ]-lx_yll ..... 0 n n
-1
-1
IIbn as
We
is invertible in
A",
and
n
x-xli:: kllenx-x!I
n -)
00.
-1
as
such that
1
-+....
1. Since IIX.m(e-e ) mlII
Le-~(e-en)]
~
n
--+
n
0
such
1'1> (a) 1== (~)p.
<;
A of norm
jle x-xII
Also, e-;\'(e-e)
Hle-Me-en )]
is some positive constant. so that
n
D x V,
E BI !~(a) 1 s (t)p}.
(a,y) E 8
n
as for
as
and
B is closed in
= {(a,y)
p ~ 1.
a,b E D
1 1 of elements of D n Inv (3' 2)· There exists a sequence (b) n n:::l
Also there exists a sequence
have
=
6«a,y), (b,z»
U
U
S
for each -1
x-b n xii
-+
0
(2;\') m, a routine n, where as
k
n·-)o "",
We have
lim supl~«e-A(e-en»bn) 1 s (l-;\.) (t)p < (~)P+l, so that n
([e-A(e-e )]b , b-l[e-~(e-e )]-lx) E Bp+ 1 ' provided that n n n n
n
is large enough.
We have lim supl![-e\(e-e )Jb -aI/ n
n--
so that
n
=
lim sup!l[e-A(e-e )]b -b ]11 ~ 2A lim supl~(b ) 1 ~ (-43)P, n--
'n
nn
lim sup 0( ([e-A(e-en)b n ], b~l [e-.\(e-e n ) ]-lx), (a,y» n
n
n
~ (~) p.
117
MITTAG-LEFFLER METHODS
It follows from Theorem 2.1 that there exists an element TT B such that p::l p Therefore \p(a l ) = 0, a l E A, /:alll proves the factorization theorem.
of
(Up) p~l = (ap'Yp)p~
and
possible
to
obtain
y E M,
example
a similar
factorization
IIx-y II <
and with
for each
f.
x
= ay,
[8, p. 61] and [29, Theorem 2.2]).
A
for
Yl
=
e.
S
~
p
1.
This
K, it is also with
x E [span AM]
a EA (see for
Applying the factorization
theorem to the Banach A-module given by all sequences of
Yp
Ilx-Ylll
A possesses an approximate identity bound by
If and
and
(X )
n n::l
of elements
which converge to zero (this is the Johnson-Varopoulos version of the
factorization theorem --see [30] and [48]), we see in particular that if
A
poosesses a bounded approximate identity, then for each sequence A there exists
elements of
follows from this fact that
a E A such that
(xn)n~l for each n :! 1.
x
E aA n A possesses a "sequential bounded approximate
-1 (aA1 = A (by sequential bounded approximate identity we mean a bounded sequence (e)
a E A such that
identity" if and only if there
are
such that
for every
n
e u n
u
~
as
n
~ ~
of It
n~l
u E A).
Some sophisticated versions of the factorization theorem lead to the t
construction of continuous semigroups
(a )t>O
(or even of analytic semi-
t
groups (a )Re t>O~ in commutative Banach algebras with bounded approximate identities. We refer to Sinclair's monograph [45] for these topics. A formalization of these constructions along the lines of the present proof of Theorem 4.1 will hopefully be available in the near future in a forthcoming paper by Zouakia. An important example of a Banach algebra with left bounded approximate
identity is given by the group algebra group.
Ll(G), where
In this case we may choose for each compact neighborhood
a nonnegative-valued function
e V with support in
Jvev(~)ds = 1, and the family
(e V)
LlCG)
G is a locally compact
1.
bounded by
some other consequences.
f
1f
that for every
f E Ll(G) g
gives a bounded approximate identity for
= g*h,
... E U (G)
where
and
G.
00
U (G)
the set of all bounded,
It is a standard result that
g E Lm(G).
and every
f E LICG)
g,h E L1 (G), but there are also
For example, denote by
uniformly continuous functions on f*g E Uoo(G)
V such that
The factorization theorem shows that each
can be written in the form
V of unity
e. > 0
But a routine verification shows there exists a neighborhood
Ve
I:ev*g-glloo < Eo, so that Ll(G) * Uoo(G) is dense in U (G). e. We tllUS have the following consequence of Cohen's Factorization Theorem. 00
of unity such that
118
ESTERLE
CUROLLARY 4.4. 1£ G is Ll(l;) it L'l
locally compact group, then
a
One can prove similarly that
Ll(G)
*
LP(G)
= LP(G)
for 1
5:
P < .....
More gent!ra.1 resul.ts of this type are givt:!n by Gulick, Liu, and van Rooij [29 J.
But the
fUl.~torizatiull
theorem has also some nontrivial corollaries
concerning ::;omt:! uniform algebras.
Denote by
00
H
the Banach algebra of all.
fune tions bounded antI analytic on the open unit disc the usual disc algebra
= {f E Huolf(z)
<
00
e ( ZJ. = (l-z) - - lin n 2
for
determination of
u
n ::: 1
lin
is a positive real. Hl
Izi
1 with
Iz I
and
Re u :: 0
for
HI
Then
as a Banach == 1.
and
is a closed
Ml-module.
Let
Here we consider the analytic
,.rhich takes positive values when
lie n II s 1, and
Then
I}.
D
= O}
HI - {f E A(D) If(l)
Let
M], and we may consider
subalgebra of
So
z -+
0 as
-+
is the set of all continuous functions on
D are analytic).
whose restrictions to
M~
(A(D)
A(D)
D, and denote by
lie n f-fll
0
-+
for every
f E MI. DO
1, and
possesses a bounded approximate identify bounded by
u DO
MIMI
GO
is dense in
Ml,' so that we have the following result.
Our last example concerns operator theory.
Recall that a Banach space
E is said to possesB the metric approximation property if for each compact subset K of E and each E', > 0 there exists a bounded operator u g on E
of finite rank such that
x E K.
If
Ilu;;,,:::
.1
and
,Iu f; (x)-xll <
E is any Banach space, denote by
for every
g
the closure in
F(E)
K(E) the
of the set of bounded operators of finite rank, and denote by of all eompact operators on ~(E)
= F(E)
K(E)
and that
E.
L(E)
set
The metric approximation property means that
has a left approximate identity bounded by
1.
\.Je thus have the following (certainly well-known) result.
COROLLARY 4.6.
Let
E be a Banach space.
Denote by
Ll(E)
where
x
n
x E E and
u
on
E Ii: and
possesses the metric
Ll(E)
consists of all bounded
E which can be written in the form tEE' n
for
tEE', we denote by
E, where
the set of all nuclear operators on
E is any infinite dimensional Banach space. operators
E
K(E) = K(E)oK(E).
approximation property, then
REMARK 4.7.
If
n
~
x ®t
1, and where the operator
u=
E Ilx
>1 n_
y
-+
~x
>1 n n_
®t, n
!lilt II
<
n n t(y)x).
DO
(if
It follows
from a standard result in the theory of Banach spaces that there exists a sequence
(xn)n~l
of elements of
E and a sequence
(t ) n
n~l
of elements
MITTAG-LEFFLER METHODS
E'
of
such that
for
t\ ::
u •
Z.
m :: 1 •
and
1
For
n
for
n:: 1, and such that
1, let
~
y n
= n -3/2 xn
tn(xm)" 0ntm
and
u E Ll(E), and the eigenvalues of
Then
Yn ® t n •
n~l
= lI.8n ll = 1
lixnl!
119
E are given by
-3/2 (n ) n_ >1. It follows then from basic results of Grothendieck (see [28, Chapter 2, Section It Theorems 3 and 4]) that u cannot be written
the sequence as
a product of five ele~ents of
Ll(E).
Ll(E) ~ F(E)
the metric ?pproximation pr.operty, then is in fact true i f
dimensional 8anach space lip IIn- 1/2
n
n
for each
=
has the approximation property, the second equalf.ty
E
being Theorem 1.e.4 of [31]. such that
-+
This follows from the fact that any infinite possesses a sequence
E
as
I
n
~
of projections
00, and such that the rank of
5.
=
equals
P n
On the other hand, a deep recent construction of
n :: 1.
Pisier (36] gives an infinite dimensional Banach space LI(E)
E possesses K(E). This result
This shows that if
for which
E
F(E), which disproves an old outst;mding conjecture of Crothendieck.
BOUNDED APPROXIMATE IDENTITIES AND PRESERVATION OF SPECTRA If
have
A is a Banach algebra u'-ld
Sp 4>(u) c Sp u for
9: l\.
is a homomorphism, we always
B
-+
\-Je give here an example for which the
u E A.
existence of a bounded approximate identity in some subalgebra of that some elements of
Sp u
belong
Sp <.p(u).
to
A implies
This theorem is an extension 00
of a theorem of Foias-Mlak about representations of
H
on a Hilbert space
[241).
(the last theorem given in
go
Let
THEOREM 5.1. let
~
where
: f
-+
T
f(T)
A be a closed subalgebra of
be a continuous representation of
is the image of the position function
11)..11 = 1, and i f
with
H
= lim
h(A.)
h(z)
containing
A(D), and
A on a Banach space
a: z
exists, then
-+
z.
E,
If A. E Sp l'
h()..) E Sp h(T).
Z~A
~z 1<1
e (z) n
=
(l-z)l/n 2
for
Then that
1
lien Ii
= 1,
D - {I}.
Let
J
Ilg(T)[en(T)-JJIl
=
Then
on compact subsets of IIge -gil n
lim
-+
n--
0, so that
and
nth
root is
converges uniformly and
g
lig(T)en(T)-g(T)1I
= h-h(l)l. -+
O.
Assume
n-
Then for some
infllen(T)-JIl < 1.
the unital Banach algebra
= 2(en (T)]n
e n (z)
= 4>(1)
n, e (T) n
n.......
J-T
Let
Izl:=: 1, where the determination of the
the same as in Section 4. to
\. = 1.
It is sufficient to prove the result when
PROOF.
V
=
would be invertible in
[y(A)]-, whose unit element is
would possess an inverse
B in
V.
J.
We would have
So
120
ESTERLE
So lim inf lie (T)-J II n
n--
=
~-I)(-B+J-I)
(-B+J-1)(T-1)=
~
I, which contradicts the fact that
But
1.
g(T)e (T)-g(T) = lh(T)-h(l)J] [e (T)-.J J = [h(T)-h(l)!] [e (T)-J]. n
n
IIlh(T)-h(l)IJ[e (T)-J]II n
h(T)-h(l)I
1 E Sp T.
lim inf!len('r)-JI: ~ 1, and consequently
with
0
-+
Thus
11
n""'-
n~
cannot be invertible, as we were to prove.
In the proof we essentially used the fact that if a Banach algebra possesses a sequential bounded approximate identity Ily(e-e n ) II
0, where
-+
e
(e) n
11..::
l' then
A, So that each
is a unit added to
A
yEA
is a
n .......
topological divisur of zero in
if
A
In fact, if
has no unit.
A
possesses a sequential bounded approximate identity and does not possess any nonzero idempotent, then
lim
2
n
n~
n
In~
in f II [(p (e ) I -
lim
1 , so that inf Ile 2 -t! II :! -4
n
if
n
is any (nonzero) con t inuous homomo rphism
'P
such that
has no nonzero idempotents.
using the fact that
~
(un) n~l
o ::: arg
A
S
possesses a sequential bounded approximate identity
II (l-AJ un +A.II
such that
Using this observation, and
A.
for each
S 1
€ D
such that
n, we can prove the following theo rem (which is also an extension
tu general Banach spaces of a result of FoLlS, Nagy and Pearcy [25]). THEOREM 5.2.
Sp h(T) = i5 Banach space
00
H
of norm
for every continuous representation
h -,. h(T)
fhere exists an element
E such that
1 € Sp T
h
of
and such that
Im(I-T)
We refer to [22, Sections 4,5,6] ,Iud [lOJ for details.
1 of
such that H""
on a
is not closed. These results
are related to the construction of the "canonical pseudo-Banach algebra"
~
A
associated with each commutative Banach algebra which possesses dense principal ideals. space of A
is
a
t
It follows in particular from Theorem 5.2 that the carrier
can always be mapped continuously onto the carrier space of
HC«> if
commutative radical Banach algebra with sequential bounded approximate
identity, despite the fact that the
~arrier
space of the multiplier algebra of
A might be a singleton. We refer to [22J for further information.
6.
MICHAEL'S PROBLE}1 AND BIEBERBACH FUNCT lONS
It is a famous and long-standing open problem whether or not each character
Oil
a commutative fr~chet algebra is necessarily continuous.
This
problem was explicitly raised by Michael in LJ4]. and is often called "Michael's problem".
In discussing Michael's problem in this section, the
ideas presented in the preceding sections come back, in some sense, to their origin.
121
MITTAG-LEFFLER METHODS
The abstract version of the Mittag-Leffler Theorem appeared for the first time in the literature about topological algebras in a basic paper by R. Arens [3].
It was a key tool in the proof of the following theorem. which
as we will show in Corollary 6.3, yields information about the continuity of characters on a co~nutative Freehet algebra.
THEOREM 6.1
(Arens, [3).
al, ••• ,a p E A.
If
COROLLARY 6.2 let
is dense in
Let
alA + ••• + apA • A.
A, then
A be a commutative Frechet algebra, and
be a finite family of elements
of
A.
Then for each
X on A there exists a continuous character
character
= X(x i )
PROOF:
alA +.•• + apA
(Arens, [3]).
xl, ••• ,x p
~(xi)
A be a u~ital Frechet algebra, and let
Let
for
I ~ i ~ p.
We may assume that
A is unital.
ylA +••• + Y AcKer X F A, so that p
A/J
-
ylA + ••• + ypA.
closure of
A such that
on
~
~
Let
Yi
= xi -
x(xi)e.
Then
{OJ, where we denote by
Since each commutative unital Frlchet algebra
possesses a nonzero character, there exists a continuous character whose kernel contains J. 1 ~ i S p, as required.
Since
COROLLARY 6.3 (Area;;, [3] ) •
Yi E Ker~, we have
Let
A, then all characters on To say that
for each in
p
as
n
x
Now let
~
on
for
{Pn (Xl"'.'X p » n_>1
A such that
x E At and thus
Denote by
i
~
~
=
H(~P"q)
of
x
= X(x)
p.
But there exists only one continuous character for
1
x
~
i
~
p, so
and
= X{x)
X is continuous.
~. i ~l···
. .) (~l""'~p
i1
¢P
into
,q.
ip
zl ••. zp
E
P for every
p
-+
x E A there exists
the set of entire functions from
Then
of polynomials
Pn (al, ••• ,a p )
For each
such that
Hence
Z
of
of elements
A
~
].
X
If
on
= X(a.)
~(a.)
Let
family
A.
X be any character on
1
for
(al, ••• ,a p > means that
variables with complex coefficients such that
(a i ) - X(a.)
for
(al, ••• ,a p )
A is polynomially generated by
x E A there exists a sequence
-to co •.
X{X i )
A
A are continuous.
a continuous character ~
~(xi)·
on
~
A be a commutative Frichet algebra.
A is polynomially generated by a finite family
PROOF:
the
J
,
R > 0, so for every
elements of a unital Frechet algebra, the series
122
ESTERLE
ip
i l
r.
Ai
(ll, ••• ,i p )
1··· P
A.
converges in
a l ••• a p
i
We can denote by
the sum of this series, and the map (al, ••• ,a p ) ~ f(al,···,a p } is a continuous map from AP into A. Also, if F = (fl, ••• ,f q ) is an f(al, ••• ,a p )
entire function from
[p
into
¢q, then the map
F(al, ••• ,a p > = (fl(al, ••• ,ap), •.• ,fq(al, ••• ,ap» is a continuous map from AP into Aq • Finally, if X is any character on a Fr~chet (al, ••• ,a p >
al~ebra
-~
Xp (al, ••• ,ap )
=
the map defined by the formula
(X{al}, ••• ,X(a p
».
We obtain the following proposition.
Let
PROPOSITION 6 .l••
r E H«(P,C q ). Then character X on A. PROOF:
xp
A, denote by
A be a commutative Freehet algebra, and let X (F(u» = F(X (u» for each u E AP and for each q
p
for each
f
E H(CP,C).
generated by
Denote by
(al, ••• ,a p )'
= X{f(al, ••• ,a p »
f(X(al), ••• ,X(ap »
We just have to prove that
B the closed unital subalgebra of
Since
XIB
A
is continuous by Corollary 6.3,
the result follows from the definition of
(a 1 , ••• ,a p )'
f
We now present the basic step in a fairly new approach to Michael's problem. THEOREM 6.5.
If there exists a discvntinuous character on a commutative
~
Freehet algebra
A, then for every projective system
p (a:
II
, F) n ' where
F
n
E
H«(
p
p
n+l,a: n)
n ~ 1, the projective limit
for
p
lim (a: n,F) n +PROOf':
is nonempty.
We may assume that
character on
At and set
A is unitol.
r.1 = Ker
for ~quipped
with the discrete topology.
Now consider the map
e l\
(al, •••• a
Pn+l
en
,xl'.'.'x
qn+l
En +l ) =
Then Let
n:! 2.
complete topological space, so that n.
X.
E
n
En
X be a discontinuous
M is dense in
E
n
Then ~
Let
Pn
=A
A.
qn
x M ,where
Put ql M is
=0
M is homeomorphic to a metrizable
is metrizable and complete for defined by the formula
MITTAG-LEFFLER METHODS Since n
M
~ 1,
n ~
M
is dense in
on (E n +l )
A,
It follows from Corollary 2.2 that of lim (E ~
Pn
n
;(.
Pn
EA , (u)
n
x
n
,en ).
For each
is dense in n
III
n
we can write
for each
E n
Pick an element
.
= (un ,xn ), where
U n
qn
EM, and
n
= XP
(F (u +1»
n
n
n
=
belongs to
Then
F (X n
Pn+l
lim ~
(u +1». n
Pn
«(
Let
z
n
=X
n
into itself such that
Pn
(u)
for
n
n
1.
:.>0
,F), and the theorem is proved. n sequence
If there exists a
COROLLARY 6.6. from
n
is continuous for each
lim(E ,0 ) ; -t-
u
en
is equipped with the discrete topology,
and since
1.
123
of entire
= Ill,
Flo •••
function~
then all
n~l
characters on all commutative Fr~chet algebras are continuous.
n
PROOF:
Fl
Fn «(2)C lim (t 2 ' Fn ) •
0 ••• 0
n~l
~
Most of the credit for thit; approach to
~lichael' s
problem belongs
to P. G. Dixon, who mentioned a result similar to Corollary 6.b during a
discust;lon with the author over a cup of coffee during a NBFAS :;eminar at Edinburgh in June 1978.
The above formulation and the Mittag-Leffler proof
of Theorem 6.5 were obt ained by the author in November 1982, just after obtaining with rio G. Dales a short Hittag-Leffler type proof of Shah's theorem on continuity of positive linear forms on So-algebras with continuous involution [43].
Proposition 6.4, and hence Theorem 6.5, can be extended to
nonco~nutative rrechet algebras by using some algebras of formal power series
of noncommuting variables, but we will no do this here. Despite the simplicity of its statement, the question of the existence of
n
a
sequence
Flo •••
0
n~l
(F) n
F
II
of elements of
~l
(~2)
-
III
H«2,C 2 )
such that
seems to be a difficult problem.
Note that it
follows immediately from the big Picard theorem that the complement of o f «() contains at most one point for any sequence ( f ) of il f1 0 n n n~l n:::l nonconstant entire functions on t. But it follows from constructions made
by Fatou [26J and Bieberbach [6] in the twenties that there exists an entire one-to-one function dense in
~2.
F: t'
'}
-+
to.
of Jacobian One such that
2
F(t)
The literature about these functions is rather sparse,
known constructions are based on the following idea.
is not but all
124
ESTERLE
e
Take an analytic automorphism at
O.
and
of
,p
with a repulsive fixed point
Then there exists an analytic function ~ll
invertible, where
F' (0)
S~2
and
F:
wi.th
S~1 -+ Q 2
F(O) = 0
are open neighborhoods of
0
OaF = FoB. where B is a suitable analytic automorphism of B-n(y) -+ u as n -+..., for y E ,p. In the case where the
such that satisfying
F'(O)
eigenvalues of relation
• ••
do not satisfy any
m ""I\. p = 1
with
P
(ml, •••• Rl p )
:1
,p
and
(0, ••• ,0), one can just take
B
9'(0), and the treatment of
this case is related to the 1878 thesis of H. Poin(,.llr~ [37], in the context of partial differential equations. and to another paper of Poincare' L38] when
e
When p = 2 one can always take B to be of the ('1.X,AZY + ux q ), as shown in 1911 by Latt~s L321. A discussion
is a polynomial map.
form
(x.y)
-+
of the general case
was given by Reich in 1969 [391, [40J, but Reich's work
might overlap some results d'le to
,
.
Po~ncare
and Dulac, mentioned in Arnold's
book [4).
Anyway, \-1hat happens is that the solution of the equation
e~
can be extended to an entire function from
=
FoB
is one-to-one.
F(t 2 ) = {z E ,2
Also
Ie-n(z)
-+
,2
O}.
into itself which
But i f
has another
F
n~'"
P.
repulsive fixed point
then
e-n(z)
r-
n>GO .
some open neighborhood of
(:3, so that
F(G: 2 )
V is
z E V. where
for all
-+ ::l
is not dense in
,-. ?
Bieberbach's original example corresponds to the automorphism
for which both
and
(0,0)
(1,1)
are repu.l.sive fixed points.
Other con-
structions USing different automorphisms can be found in Sadu11aev [42], Kodaira [31], and Nishimura
lJ5J~
and a very clear exposition of Bierberbach's /
) original constru«.:tion is given by Stehle [46]. ::Ii.ow that, [or each
IF f. (z) I = 0 (exp(
> 0
E.
Izl~».
there exists a
Sibony and Pit Mann Wong [44)
Bierberbach function
F
such that
F-
We will not try to give comprehensive references
here, but just present another approach. Denote by
Aut1(t P)
the set of all analytic automorphisms of
,p
of
Jacobian identically equal to i, and denote by B(~P) the closure of Aut l (f.:p) in H(t P .t P) with respect to the topology cr of uniform convergence on compact subsets of J(F)(z) = 1
for every
fCP. It follows from Cauchy's inequalities that F E 8(fC P ) antl every z E (p, and it follows from
[7, Chapter 8, Theorem 9J that all elements of
B(~P)
idea just consists of finding a convergent sequence 2
Autl(C)
whose limit
F
avoids a suitable set.
elements of
B(¢P)
can be found in
elements of
B(G: P )
is always a Runge domain.)
[23].
are one-to-one.
(en)n~1
Our
of elements of
(Further properties of
In particular, the range of
MITTAG-LEFFLER METHODS
n~ 0, let 6n1 =
Fix
=
lI!
{z E
II Re z I !:
t
Riemann sphere
U {~}.
0:
Then
n
= {z
A = Al U A2 U 63•
and
n+ ;},
IRe z ~ n+l}, lI2
{z E 0:
125
n
S\F
n
z
E eiRe
Denote by
n
-n-l},
!:
the
S
is connected and locally connected at
infinity, and it follows from a deep theorem of Arakelian (see [2], or Theorem 1 on p. 11 of [27 J) that each function ~
analytic on
can be uniformly approximated on (fp)p~1
particular, there exists a sequence as
f (z) .... 0
p
(z) .... -2
P ....
00
uniformly on
2
((x,y) E re 2 Isup(l x l, For each
LEMMA 6.7. 2
Iyl)
!:
n
0
~
uniformly on
6,1
and
n'
n ~
~
-a, Re y
-a},
~}. We have the following lemma. there exists a sequence
(ep)p~l
of elements in
satisfying the following conditions:
Autl(C)
-1
(1)
en (U n+1 U Vn+l ) c Un+2 IJ Vn+2 '
(2)
ep (z)
PROOF:
In
of entire functions such that
p .... ~ uniformly on ~o Now for a E R, let p 2 2 u = {(z,y) E C IRe x ~ a. Re y ~ a} and V = {(x,y) E £ IRe x a a and for P > 0 denote by D~ the closed polydisc f
F and
F by entire functions.
{j,3, f (z) .... 2 n p
as
continuous on
f
-+
z
(fp)p~l
Let
a-pl ( z )
and
.... z
as
p ....
00
uniformly on Dn°
be the sequence of entire functions described above.
Taking away some terms of the sequence if necessary, we may assume that for p
~
1, Re fp(z)
Next we let
~
23
for
9p (x,y)
automorphism of
n+l, and
,2, and 6;1 is the map ~
Re(x+fp(Y»
and
n +
5 2'
Similarly we have
Po
Now there exists !Re (Y-fp(x)
that
~
p
-+
~
J(e) p
I ~ 21
~
1
such that
for each
uniformly for
=1
Re fp(z)
(x-fp(y-fp(x», y-fp(x».
=
If (x,y) E un+1 ' then
as
Re z
IRe
P ~ Po
zl
~
~
Then
for
9p
~
Re z
-n-l.
is an analytic
(x,y) .... (X+fp(Y), y+fp(x+fp(Y»). so that
Re(y+fp(Y»
-1
8p (Vn+l ) c Vn+2 •
IRe (x+fp(Y» I ~ and each
21
1
follows from a routine computation.
~
n +
25
Thus (1)
and
holds.
and
(x, y) ED. n
n +2' we see that
We now obtain the following theorem.
3 - 2
(2)
Since holds.
f
P
(z)
-+
0
The fact
126
ESTERLE F · There exists an ent i re one-co-one f unet10n
l'HEORE:-t 6.8.
J(F) :: L and such that F(G: 2 ) U1 = {(K,y) ~ C2 /Re x ";> 1, Re y ;;> I} that
,2
-+ ... ... 2
sue h
avoids the sets and
= ({x,y)
V1
,2 IRe x
E
< -1,
Re y < -I}. PROOF:
= Un
Wn
Let
=
d(C,H)
n::: 1. Also, if C, H E H(G: 2 .~ 2 ), set
for
UV n
1: 2-n inf (l, p (G-ll» n
n=1
= sup IG(z)-H(z) I (here we use the notation
p (G-H)
where
n
I (x,y) I
Izl~
(Ix I, Iy I».
= sup
Then
d
H(1E 2 ,re 2 ) ,
is a distance on
which 2
is complete with respect to this distance.
2
H«( ,re )
defines the topology of uniform convergence on compact se"s, and Next, let
E = {G ~ 0(re 2 ) iG(~2\W
1) c [2\W l }. Then El ~ 0, and each En is a closed n+ subset of 8([). Now let. G E En and set Gp = GOS p ' where (9 p )p:!1 is the sequence given by the lemma. Since 8p (z) -+ Z as p uniformly for Iz! ::: n, we have lim sup d(C,C p ) ::: 2- n . Also e;1(wn+1) C Wn+2 ' so that 2'
n
-)0
2
p-"'"
2
\1 p tIE ~.Jn+2) C II: \W n +1
and
n
Theorem 2.1 that
E
n:!l
£
Gp E En+1
0.
'"
n
Pick
F E
z
n
E.
n~l n 2
rE.
COROLLARY 6.9.
There exists an entire function
element
(x,y)
PROOF:
Denote by
G = HoF.
of
Wl
2
z E: q: , and such that
H the map
(::,y)
REMARK 6.10.
H«(2,(2) with
n
n::1 Dl Z 2
G: [2
case where
z E ,2.
If
(2
-+
is
such that for each
(u,v) E C(,2)
(x,y) E F(e 2 ).
2
0
then
Since
(F) .... l
=
0
F2
of el...'ments
2
n
(Flo F2)(C ) D2 = 0, etc •• 2 -1 i f and only if F2 (e ) n Fl (02) = 0, and
is the function
nonconstant function
n~
n
0, and then F2 with
n DZ
n
F ([ ) ~ 0, on~ could try to construct
02( at least i f the Jacobian of
Fl
F
(exp(x-l), exp(y-l», and set
is certainly unbounded if the boundary of interior of
J(F) s 1,
1, the corollary follows.
n Flo •••
o F2 ) (t )
Then
inf (!x!,ly!)::: 1
To construct the desired sequence
with 2
Fl «()
~
-+
for every
(u,v) = (exp(x-l), exp(y-l», where ini (Re x, Re y)
It follows then from
G([2).
J(G)(z) ~ 0
Then
~
p.
This prnves the tht!orem.
r'(z)
for each
for each
for each
one-to-one and
J(G)(z) ~ 0
...
F1
F1 (t 2 ) meets the never vanishes). In the
G g:l.ven in Corollary 6.9, then no such
call exist.
Indeed, i f
(GDH)(,2)
n 02
-
0, then
MITTAG-LEFFLER METHG)DS
2
(GOH)(a:)
~G2
c :~l
U g2' where
Ql = {(x,y) E
127
a: 2 Ilxl:!
2,
iyl ::
I}
and
{(x,y) E c2 11x I :: 1, Iy I ::: 2}. Since (GoH)(a: 2 ) is connected, then either (GoH) (a: 2 ) is contained in Ql or (GoH) (a: 2 ) is contained in 2 We have GoR = (f l ,f 2), where fl and H(G; ,IE). But f2 belong to =
and and
are not dense in
H is constant since
a:,
so that
and
£2
are constant,
G is locally one-to-one.
Some other aspects of the theory of Bieberbach functions can be found in [23], and a full discussion of this new approach to Michael's problem, with a comprehensive presentation of the theory of Bierberbach functions, will be given in a forthcoming joint paper by P. G. Dixon and the author.
REFERENCES 1.
G. R. Allan, Embedding the algebra of all formal power series in a Banach algebra, Proc. London Math. Soc. (3) 2S (1972), 129-340.
2.
N. Arakelian, Uniform approximation on closed sets by entire functions, lzv. Akad. Nauk. SSSR 28 (1964), 1187-1206 (Russian).
3.
R. Arens, Dense inverse limit rings,
,
~lichigan
:-tath. J. 5 (1958), 169-182.
...
4.
V. Arnold, Chapitres supplementaires a 1a theorie des equations differentielles ordinaires, Editions de Moscou, 1980.
5.
W. G. Bade, Recent results in the ideal theory of radical convolution algebras, these proceedings, 63-69.
6.
7.
L. Bieberbach, Beispiel zweier ganzen Functionen zweier komplexer Variablen, welche eine sch1ichte volumtreue Abbi1dung des Rn auf einem Teil seiner selbst vermitten, S. B. Preuss. Akad. Wis9 (1933), 476-479. S. Bochner and W. Martin, Several Complex Variables, Princeton University Press, 1948.
8.
F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, 1973.
9.
N. Bourbaki, Iopologie Generale, Chapitre II, Hermann, 1960.
10.
B. Chevreau and J. Esterle, Banach algebras methods in operator theory, Proceedings of the 7th Conference in Operator Theory, Timisoara (June, 1983), to appear.
11.
G. Choquet, Cours d' Analyse - Topologie, Masson, 1964.
12.
p. J. Cohen, Factorization in group algebras, Duke Math. J. 26 (1959), 199-206.
13.
J. B. Conway, Functions of One Complex Variable, 2nd Edition, SpringerVerlag, 1978.
14.
H. G. Dales, A discontinuous homomorphism from (1979), 647-734.
15.
H. G. Dales, Automatic continuity: (1978), 129-183.
16.
P. G. Dixon, private discussion.
C(X), Amer. J. Math. 101
a survey, Bull. London Math. Soc. 10
128
ESTERLE
17.
Y. OOl11ar, A solution of the translation invariant subspace problem for weighted LP on R,R+ or Z, Radic<.ll Banach Algebras and Automatic Continuity, (ed. J. Bachar et al), T.ec.ture Notes in Mathematics 975, Springer-Verlag, 1983, 214-226.
lB.
J. Esterle. Sur l'existence d'un homo!llorphisme discontinu de Proc. ].lmdon t-tath. Soc. (3) 36 (l97ii). 46-58.
19.
J. Esterle, Injection de sehligroupcs divisibles dans des algebres de convolution et construction d'homomorphismes discontinus de C(K), Proc. London ~tath. Sl'lC. (3) 36 (1978), 59-85 •
20.
.1. Esterle, LJniver5[-\1 properties of some commutative radical Banach nlgebras, J. flir die Reine Ilnd ang. l'1ath. 321 (1981), 1-24.
LI.
J. Esterle, Elements for a classification of commutative radical Banach algebras, Radh~a1 Banach Algebras and Automatic Continuity, (ed • .I. Bachar et al), Lecture Notes in Mathematics 975, Springer-Verlag, 1983, 4-65.
22.
J. Esterle, Quasimultipliers, representations of H, and the closed ideal problem for commutative Banach .3lgebras, Radical Banach Algebras and Automatic Continuity, (ed. J. Bachar et a1), Lecture Notes in Mathematics 975, Springer-Verlag, 198J, 66-162.
23.
J. Esterle, variables complexes et , Fonctions , enti~res de plu$ieurs , continuite des caractercs sur lea a1gebres de Frechet, Actes de la Conference de Toulouse, 1983, Lecture Notes in ~~thematics, SpringerVerlag, to appear.
24.
C. Foias and W. Miak, The extended spectrum of completely nonunitary contractions and the spectral mapping theorem, Studia Hath. 26 (1966), 239-245.
25.
C. Foias, C. M. Pearcy and B. Sz-Nagy, Contractions with spectral radius one and invariant subspaccs, Acta. Sci. J>.tath. 43 (1981), 273-280.
26.
P. Fatoll, Sur certaines fonctions uniformes de deux variables, C. R. Acad. Sci. Paris 175 (1922), 1030-1033.
2/.
W. Fuchs, The'orie de l'appruximation des tonctions d'une variable
C(K),
,
00
,
cump1exe, Presses de l'Universit6 de ~~ntrJal, 1968. 28.
A Grothendieck, Produits tensoriels tupologiques et espaCl:!S nuc1~aires, ~1emo irs ArneI'. Math. Soc. 16 (1966).
2.9.
S. L. Gulick, T. S. Liu, and A. C. M. van Rooij, Group algebra modules II, Canadian J. Math. 19 (1967), 151-173.
30.
Johnson, Continuity of centralizl.:rs on Banach algebras, J. London r-tath. Soc. 41 (1966), 639-040.
31.
K. Kodaira, HololDorphic mappings of polydiscs into compact complex manifolds, J. DiH. Geometry 6 (1971), 33-46 •
32.
S. Lattes, Sur les formes reduites des transformations ponctuelles a deux variables, C. R. Acad. Sci. Paris 152 (1911), 1566-1569.
11.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, 1,
B. E.
..
Ergebnisse Series
Vol. 92, Springer-Verlag, 1977.
34.
E. A. Michael, Locally multiplicatively convex topological algebras, Mem. Amer. Math. Soc. 11 (1953, third printing 1971).
35.
Y. Nishimura, Automorphismes analytiques admettant des sous-vari~t~s de points fixes attractives dans une direction transversale, J. Math. Kyoto LJniv. (2) 23 (1983), 289-299.
MITTAG-LEFFLER METHODS
129
36.
G. Pisier, Countere:,amples to a conj ecture of Grothendieck, to appear.
37.
H. Poincar:, Th~se, Gauthier Villars, 1879 (Oeuvres, tome 1).
38.
H. Poincare, Sur une classe nouvelle de transcendantes uniformes,
,
Journal de Mathlmatiques 6 (1890), 313-355 (Oeuvres, tome 4). 39.
L. Reich, Das Typenproblem bei furmal-biholomorphen Abbildungen mit anziehendem Fixpunkt, Math. Ann. 179 (1969), 227-250.
40.
L. Reich, Normalformen biholomorpher Abbildungen mit anziehendem Fixpunkt, Math. Ann. 180 (1969), 233-255.
41. 42.
c.
E. Rickart, General Theory of Banach Algebras, Van Nostrand, 1960.
A. Sadul1aev, On Fatou's example, Mat. Zametki 6 (1969),437-441.
43.
Shah Dao-Shing, On semi-normed rings with involution, Izv. Akad. Nauk. SSSR 23 (1959), 509-528 (Russian).
·44.
N. Sibony and Pit Mann Wong, Some remarks on the Casorati-Weierstrass theorem, Ann. Pomn. Math. 39 (1981), 165-174.
45.
A. M. Sinclair, Continuous semigroups in Banach algebras, London Math. Soc. Lecture Note Series 63, Cambridge University Press, 1982.
46.
J. L. Stehl~, P10ngements du disque dans ,2, S~minaire P. Le10ng 1970-1971, Springer Lecture Notes 275, 119-130.
47.
M. Thomas, A nonstandard ideal of a radical Banach algebra of power series, Acta Math., to appear.
48.
,
N. Var~pou1os, Continuite" des formes lineaires positives sur une algebre de Banach avec involution, C. R. Acad. Sci. Paris. Serie A-B 258 (1964), 1121-1124. ,#
49.
F. Zouakia, Semigroupes reels dans les algebres de Banach commutatives ,
to appear.
MATHEMATI9UE ET INFORMATIQUE UNIVERSITE DE BORDEAUX I 351, COURS DE LA LIBiRATION 33405 TALENCE FRANCE
Contemporary Mathematics
Volume 32,1984
ORTHOGONAL AND REPRESENTING MEASURES Irving Glicksberg
•
••
The title is more or less accurate since I do want to survey some of the known relationships and developments in this area of uniform algebra theory at least one new one), but really I want to illustrate the advantages of the simplicity of the uniform setting. There are a few items I'll use: the ~oting
straightforward norm, the familiar dual of
C(X),
and, of course the viewpoint.
One disadvantage, the lack of symmetry. of course provides the challenge, and some problems do finally in fact turn out to be self-adjoint. as we shall see.
A will denote a closed separating sub algebra of C(X) containing the constants, where for convenience the compact Hausdorff space X As usual,
will be assumed metrizab1e.
M.. MA,
spectrum
X as a subspace of the
Naturally we can view
the space of multiplicative linear functionBls
alternatively we can take
A as a subset of C(M)
and
~
of
A;
X any closed boundary,
i.e., with supI6<X)I
= sup I 6{M) I
..
11611.6 EA.
y
X c M can be any superset of the Silov boundary. just as in the prototypical example, the disc algebra A(V) (for V .. {z:lzl ~ l}), defined by
Thus
A(V) =
{n
E
6 analytic
C(V):
can be any closed set between
av
Here
X
A
is defined in exactly the same way.)
and
on DO}. D.
(For K c
t
compact,
I'll begin with a vanishing lemma I found in connaction with a question raised by Kenneth Hoffman [17]; Professor Rickart has a more refined version which he had found when my preprint of [18] arrived, and which he has put to good use in his study of natural function algebras [29].
Although
the lemma
was originally proved using representing measures, the elementary elegant proof due to Wermer [32] I shall give shows clearly it could have been found at any ttme by workers in function theory, had the notion of a peak set been of as Supported in part by NSF Grant Mes 61-6811. 'Ie'll
The day after the handwritten version of this paper was completed, Irving Glicksberg passed away. The editors of these proceedings would like to thank Professors Donald E. Marshall and Edgar Lee Stout for their subsequent help in the preparation and proofreading of this manuscript.
131
GLICKSBERG
132
much interest there as in uniform a1gehra theory. DEFINITION. P (LX) and
I9 I '-.
1
is a peak set (in
elsewhere (in
)()
i f for some
g
E
-1
A,
P= 9
(1),
X).
9 will denote the peaking function in what follows. Such a set arises precisely as the set on which some non-zero a. E A assumes a value Usually
t
9
of maximum modulus, since then
=
1 2/ t+a)
peaks there.
Our lemma is
simply a first use of the notion.
II P is a peak set in X and 6 E A vanishes on a neighborhood p l!!. X , then 6 = 0 near P !!l M.
V of
(1)
Wermer's proof runs as follows.
!J
16(~) gn(cp) I :::: sup I 6l'(X\V) nth
n
roots and letting
I 9 «p) I So
6= 0
E
M has
6(~)
~
O.
Then for
P,
peaking on
Taking
~
Suppose
where
I9 I
:>
a,
~
"",
I
s II 611sup I g(X\V) In.
we have
~ sup I g(XW)
I=
a < 1.
i. e., on a neighborhood of
M.
P in
In the context of function theory, this immediately implies the following
K c« with dense connected interior
fact for a compact
u: 6
on
U and vanishing on a non-void relatively open subset
on
K.
(One has only to choose a
assumes its maximum modulus over
v
20 I. K near
aK only in
v,
E C(K)
V of
analytic
aK vanishes
z ..... -1 ~-zO A = A(K) of course.
for which with
(For a proof from the more standard function theory viewpoint, see [34, p.l20].) Wermer's maximality theorem and (1) combine to yield a simple proof [l4,P.40] of Rado's theorem (that
6
E C(K)
and analytic on
U\{lCO)
is in
A(K). To further emphasize the notion of a peak set we note (2)
if
Indeed with
r
is a peak set then Iz.P =
{6
E A: 6(P)
= o}
AI P
and
is closed in 9
C( j.") •
peaking on
P we can observe
that for the quotient norm,
so
116 + kP11 ~
1I(6Ip)ll oo ;
for the reverse inequality, note that
vides a mul tipliC'.ative linear functional on
AlkP,
so
16(1/» I
~
P pro-
116 + flpll.
One of the most important and fundamental results on peak sets is (3)
[8, p.63l], [16, Lemma 4.5]. If P is a i2eak 1s a peak set for A. Po c P is a peak set for Alp, then Po
BISHOP'S PEAK SET LEMMA
set and
Suppose
9
peaks on
P
and
90 lp peaks on
PO·
We can assume
ORTHOGONAL AND REPRESENTING MEASURES
1190"
< 3/2 t
fz.n E?l+
choose
90+Jr. Since go by 1+1t for Jr. > 0 large, i f necessary. Igo(x.) I :ii 1 + 2·- n } is compact and disjoint from P we can kn -2n
replacing
l+2-~1
Kn· {x:
133
:ii
I9 I
so that
< 2
on
Now
fz.
00
h=
Kn'
-n n 2 9 go E A
~
1
will provide our peaking function for
Igol
Ihl
~
Igi
< I
PO:
and
For clearly
on P\PO' while Ihl < 1 off p where there. But if Igo(x) I > 1 then x E Km for some
since
< I
m,
so
Ihex) I
as well, and we are done. As a consequence of (3), each peak set contains a peak point, and thus the
~
Ei ill.
peak points (in this metric setting) forms !. minimal boundary
M (which is ~ always closed, ~.!. GcS ~ Bishop showed ~ another !!.1 3 ~rkable lemma, the 4 - 4 criterion; ~ closure of M is, of course, the Silov boundary). From these facts and the Cauchy transform, Bishop proved a strengthened Hartogs-Rosenthal theorem [8], [14, p. 47]. Orthogonal measures arise if we want to use dual arguments.
AIF is closed, for F c X That closure is equivalent to
for example, the question of when
Ac C(X)
a closed subspace.
A~ AIF
Consider,
closed and
has closed range
which holds iff the adjoint, sending M(F)/(AIF)~ ~ M(X)/A~,
(AIF)i denotes the set of measures on F orthogonal by well known functional analysis. But our last condition is equi-
has closed range (where to
A),
valent to the equivalence of the two norms by the open mapping theorem. so
AIF is closed (for F closed) iff for some k
(4)
Ilv
+ (AI F)~II ~ kllv + AJ. 11 , v E M(F).
X = aD, Rudin (30) and Car1eson [9] indepen-
For the disc algebra and
dently determined these sets, although their question was:
Fe aV is
A(V)
by the first
Since any F.
A,
F with
A closed
closed.
F c aV
closed
of positive Lebesgue measure has
and M. Riesz theorem, if
is equivalent to that of
A(V)IF
~~
IF = C(F)? As they showed, these are just the compact sets of
Lebesgue measure zero.
kF = 0
~ 1,
so
F = aV Fe X
AIF is closed its norm
and this is the only other compact is called an interpolation!!! for
A if AIF. C(F), and Bishop [6] next provided a general sufficient condition:
134
GLICKSBERG ~F
(5)
= 0 for all
AI F = Co-).
1.l..J. A implies
This yielded the Rudin-Carleson result by the classical
F.
and
M.
Riesz theorem, and so provided an abstract version.
Actually a necessary and
sufficient condition for interpolation sets follows
from (4), again for
Ac
a closed subspace:
C{X)
(6)
(cf. [16]; both (4) and (6) were obtained independently by the writer and P.C. Curtis, Jr. and
In fact the hypothesis of (5) is actually the
F to be a peak
condition for of Bishop
~Hoffman).
interpolat1~n ~
(1.e., both).
Another lemma
(the 1/4 - 3/4 criterion) is needed to get a characterization of
peak sets via orthogonal measures: F closed in
(7)
X
is
a
~F J.
peak set iff
A.
This has the subtle and trivial consequence that a closed set which is a countable union of peak
~
is
~
peak
~
(which is elementary for a finite
union). A remarkable non-trivial application was made by Varopoulos [31] J (20) : An interpolation
~
consisting only of peak points is .!. peak
~.
Note that measures do not figure at all in the statements of these results.
In this regard they are not typical, for usually one needs to know
some facts about orthogonal measures, some of which arise from their relation to representing measures. Every
~
E M is represented by a probability measure
Hahn-Banach and Riesz Representation Theorems;
M~
Trivially a peak point
has only the point mass
forming
(j)
point has some representing properties can be added:
M, q>
on
denotes the set
X by the
of all
(necessarily in X) and Bishop showed every non-peak
such representing measures. 5
~
A with no mass on
{~}
~
[8],[14].
MOre specific
in considering the so--cal1ed big disc algebra Arens
and Singer [3] proved the existence of representing measures, now called ArensSinger measures, for which
and subsequently Arens [2] proved the existence of Jensen measures in this context, both providing much of the initial impetus to the intensive study of uniform algebras.
Finally Bishop [7) proved the existence in general of
Jensen (representing) measures
A (for which
10gI6(~)1 ~ flogl6ldA,
all
6 E A). Representing measures easily give rise to orthogonal measures, since 1 1 6A .J. A if 6(~)· 0; thus HO(A).A.J. A, where HO(\) is the closure in l of tft-l(O). Th ese are rare1y all the orthogonal measures, but as in L (,) ~ ~
ORTHOGONAL AND REPRESENTING MEASURES
135
the case of the disc algebra there is a relationship, given by a version of the F. and M. Riesz theorem. For later use, we denote by HP()..) and RIf()..) (0), respect i LP('\) ( except f or A and tn-I t he norm c 1 osure 0 f y veI y, in I\.
p.
00,
where we use the weak
* closure) •
Helson and Lowdenslager, in a land-
mark paper [26], had extended the classical theorem to the closed span in of a half space of lattice points in
C(72)
A=span(e
and showed lJ.a,IJ..6.L
A.
~ ~
Z 2,
e,e in't"lY):m+na>O}
im
A implies the absolutely continuous and singular components ~1I
(In general, however,
It was then observed, by Bochner
'# 0.)
and others, that their proof extended to essentially the context of what Gleason had earlier termed a dirichlet algebra.
A c COO is dirichlet on X i f (Re IV are no non-zero real measures orthogonal to IV. DEFINITION.
=
'it (X)
(i.e., there
For such algebras representing measures are unique (since ~a
provides real orthogonal measures), and if one takes ~
continuous component
of
~
~ .L
(8)
as is
!-L' = 11 -
fJ"A.
M
then the argument showed ~.L
A implies
A.,
Later Forelli [12] gave a new proof of the classical
result, which became the basis for all further generalizations, and also yielded a modification of convergence result. THEOREM (Hoffman-Wermer [14, p. 42]).
!a A converges in 11 bn II 00 ~ ~1611 co
L2 (A)
6 E Hoo(A)
to
(b )
and such that
!! A
n
is dirichlet and a sequence
bn E A converges pOintwise a.e. to 6. then there are
From (8) one has a decomposition of orthogonal measures.
(Bishop [14, p. 143]).
singular, or, for each
k
~
A E Mcp' A' E M",
).. E Mcp
there is a
imply
)..
)..' E M", with
and
)..'
A~
are mutually
kA',
for some
1.
The alternatives correspond to Gleason parts [14]. (11)
We need one
A.
more fact. which holds for all (10)
~
~
IJ..L
lying in the same or distinct
Because of (10), (8) implies
A there is a sequence
measures and a measure
a
{)..n}
of mutually singular representing
singular with respect to all representing measures
for which + cr,
tJ. - LI"h,
with
cr,~
n (One has only to choose
)..1
.L
A.
n
so that
II
II~
1
> 1/2 suplI~1I
and
so
136
GLICKSBERG II~
that !~).II
\I
>
).
1/2 sup {II~II:
:). and
\
and
are mutually singular} = 1/2 sup
\
are mutually singular}, etc. •
The measure
C1
{lie \.1-
is called
"completely singular.") As I mentioned earlier, Bishop [8] used his minimal boundary
M (of all
peak points) and the Cauchy transform to give an improvement of the Hartogs-
A = R~K), rational functions with poles off K. Rosenthal theorem, viz.:
Suppose
(K\M)
If area
(12)
=
0
e(K)
the uniform closure in
of
R(K) = C(K).
then
Later Bishop [5] gave a complete proof of Mergelyan's celebrated theorem using the same application of his minimal boundary, along with a precursor of (11) and various classical facts.
Still later, with the arrival of
Forelli's
argument and so with (8) and (9) in hand, Wermer and I [24] obtained a function algebra proof relying on essentially only the Lebesgue-Walsh Theorem from classical analysis [14]:
1Io1ERGELYAN'S THEOREM [14, p. 48]:
R(K)
=
~
C\K
when
If
connected,
C\K
P(K)
is dirichlet on
is connected for a compact
aK.
K c
e,
A(K). P(K),
Here we should really write
R(K)
which coincides with
~ (z:)
a general
=
Ii11~)
Moreover, an argument of Bishop using the
had been used by Wilken [33] to show that for
~~~-~~pletelx
K there
Thus (9) and (11) are avail-
by Runge's theorem.
able by the Lebesgue-Walsh theorem. Cauchy transform
the uniform closure of polynomials,
singular measures orthogonal
~
R(K) • So to obtain Mergelyan's theorem and conclude from (9) and (11) that
6 E A(K) "'An on
lies in
P(K) = R(K),
one only has to show
n
is orthogonal to each
for!-L a measure on aK orthogonal to A = P(K) and ). =). yz a measure aK representing some point Z in K, the spectrum of both P(K) and
A(K),
as is easily seen.
Indeed, the uniqueness of our representing measures
A(K)laK
for the dirichlet algebra both algebras.
Since
).
~
P(K)laK implies ). represents z on
is multiplicative on either algebra, using either
A = P(K) or A(K) we have orthogonality of the first two summands in the decomposition
L2o...~
= H2(A) f9
2 (H 0..)
H~('X.) (~E. where the bar denotes conjugation
u2
E is simply
Er.
!lO ().»
J.
. But in fact E = {O} since each of its elements yields a measure orthogonal to Re A, so its real and imaginary and
parts are real orthogonal measures. Hence A ... P(K) or A(K), and since H2 O.) and
L2 (A) = H2 (A) €a H.~ ().) H21'\) O'~
using
b oth can on I y increase as
we pass from a smaller to a larger algebra, we conclude each yields exactly t h e same space Thus our
H21'\). \'~
6 E A( K)
lies in
H2 ().)
for
A
= P (K) ,
and so we have
ORTHOGONAL AND REPRESENTING MEASURES an E PC to
6
which converge to
L2 (>V •
in
137
Applying the Hof fman-Wermer
bn E P( to uniformly bounded by 11611 converging a. e. to so 16 c4tx. = lim Ibn ~ = 0, as desired. The F. and M. Riesz theorem (8) for dirichlet algebras was extended by
theorem (11) we get
6,
Ahern [1] to the case where the set of representing measures
Mtp are all absolutely continuous with respect to a single element, and later without
restriction by the writer [19], using the following notion, since
Mtp
in
general is large.
DEFINITION. XF
=0
Miff tp
J.l.«
for all
vanishes on all (common)
J.l.
and
J... EM), (!)
M -null sets tp
is M -singular if it is carried
!J.
M -null set. tp
One immediately has a corresponding Lebesgue decomposition ~x\F
set
where F is Mtp-null and J.l.X\F = J.l.tp « f so that !llJ.f!l is a maximum. Now our general analogue of (8) is
(13)
I.l.
1.1.
tp
by an
J.l. - J.l.F
+
we simply choose a Mtp -null
Mtp:
A implies
.L
(so
F
.L
A.
(Later KOnig and Seever [27] gave another approach which utilized a seemingly different decomposition, with IJ.~
=
-!-Lx
J.l.q>
=
~
for a
X E Mtp
II~II
with
a
X, E Mtp; the two decompositions were shown in fact to be the same by Rainwater [14J, (28).)
maximum, and thus
IJ.
singular with respect to each
In the general context the Hoffman-Wermer theorem required simultaneous 2
H (X), X EM, tp
uniform approximation in every for
6 E C(X),
which resulted, remarkably,
from individual approximation because of von Neumann's Minimax
Theorem of Game Theory [14], [19].
6E n
(14)
H2 (J...)
n C(X)
tp
sup f XEM,,I)
and
in
libnII
A such that
in
implies there is a sequence
J...EM
16 - a. 12 cO.. 00
~
n 11611
GO
-+ 0
bn
and
-+
6 a.e.
J...,
all
Quite as before there is a decomposition analogous to (11), so obtains a general fact about rational approximation in (15)
For any compact
H2 (R(K),J...)
for every
Gleason part for
R(K).
K c t,
XE
6 E A(K)
Mz(R(K»
lies in
for one
z
R(K)
C iff
6
lies in
in each (non-peak-point)
Consequently
X multiplicative on
R(K)
one
[19].
R(K) • A(K)
for all
A with
(or just those as above), or iff
138
GLICKSBERG (Re R(lO)
(Re AUO) -,
=
(Re R( K)J. = (Re A
_o_r_i_f_f
Thus, surprisingly, the question of when
R(
10
=
A(
to
is really self-
Just after chis, Garnett and I [15] found the equality of
adjointl
(~)IS
could be replaced by coincidence of multiplicative measures: R( I() = A( 10
(16) ..QJl
A( 10
iff all representing measures for
are multiplicative
R( J()
•
In fact, in many instances the coincidence
of the
~ condition 1n (15)
can be replaced by finite defect, because of an improvement by Gamelin and Garnett of a lemma of mine [13, 3.7]. (17)
aK is the minimal boundary for
If
(a)
R(K)
(This is our promised new remark.)
R( J()
=
if for one
A( K)
z
R( K) ,
in each non-peak point Gleason part for
one has
~(R(K),\) of finite codimension in HZ(A(K),X), !!! \ !n
(b)
M (R( K) ) •
z
Indeed, under our proviso, for any single representing measure R(I() ,
implies
(b)
H2(R(':~),j._) =
i2I
\
H2 (A(K),\).
(One has only to observe that for the last assertion we need only consi-
z,
der a non-peak point 2
(H )'s,
equality of
finite defect (b).
\
z
so
E
o
and in the proof in [13, 3.7] of the
K,
need not be multiplicative under the nypothesis of
Indeed this is the sole use of the hypothesis that
\
is
harmonic measure made there except for the use of [13, Cor. 3.3], which is irrelevant if part as
zl
(there) lies in hence in
Z,
KO;
but
Zl
mURt lie in the same Gleason
KO since aK consists of peak points.
assertions of (17) now follow from (15).
The remaining
We might note that for the last
assertion alone we really only need to know
\
z 1n an
represents a point
open part.) There is a deeper approach to the abstract
F. and M. Riesz theorem due
to Cole (cf. [10]) (i.n part it was also obtained by Seever (cf. [20))), which I won I t attempt to state in general.
There Is a preliminary version I'll
state because of a consequence for rational approximation. (18)
(Cole-Konig-Seever).
n
L1 (m)
~
M(I' (m)
a
il~1I
is a maXimum, then
M
rf>.
Suppose
If
m is a probability measure for which
~ E A~ n
~ ~
L1 (m)
A.
Thus we confine ourselves entirely to
(19)
If
then for
U is a component of zl,z2 E U
Moreover. any
~L.L
~
R(K)
and we choose
KO
1
L (m).
A E M (m)
Our consequence is [21]:
which is also a Gleason part for
""1 EM, there is a zl ~ aK which is « \
so that
R(K)
\2 E M equivalent to AI. z2 E M is equivalent to a
Zl
ORTHOGONAL AND REPRESENTING MEASURES
139
A' EM.
~
~g
the comRonents
{Un}
Ei.
J!l
yield all
.Eh! ~-peak-point
Gleason parts (we have what amounts to the parallel of the full classical F. and M. Riesz theorem) valent to
~
every
1-l..2!l
R( to
aK orthogonal!.2.
.!!.!. ~
.2f mutually singular .Q!thogonal measures with !Jon equirepresenting measur! 1£! Zn E Un0 (As before, zn can be any
~
convergent
~
If..1. n
point of Un') A further interconnection between orthogonal and representing measures for
is due to Gamelin.
R( K)
(20)
If every representing measure carried by
is dirichlet on
aK
R(K)
is Arens-Singer, then
aK.
We should also note that a remarkable new proof of both classical F. and M. Riesz theorems and the Rudin-Carleson result combined has recently been given by Doss [11].
While this did not involve uniform algebras, it suggested
a uniform algebra result to Don Marshall and the writer which needed orthogonal and representing measures in its proof. imply (21)
A is dirichlet on [23].
Suppose
(Note that the hypotheses immediately
X. )
{Uj}Q is a sequence of unimodular eleme~ts~of a uniform
algebra A c C(X) with 1.1. .1.1.. 1 E A for j ~ I and span {u .,1.1. .}O is dense j JJ J _in C(X) • Then X contains an at most countable set of circles S. which -' J meet pairwise in at most single points, while A/S j coincides with the disc algebra (with some
u~.,
as the coordinate function
all continuous functioJs on (A~
z),
and
A consists of
X lying in A/S. for each j. J
is also identified as the (direct) sum of measures orthogonal to the
various disc algebras, etc.) Finally, in the (l.c.a.) group setting the strong conclusion of the original F. and M. Riesz theorem (that the Haar singular component
f..1.~
= 0)
for measures orthogonal to half a totally ordered dual has been shown by Hewitt and Koshi [25] to hold precisely when the group crete or (22)
Tx
[22].
discret~.
Let
~
is
f..1.
R
x
dis-
This holds more generally.
G be a locally compact abelian group and S c G a closed
proper generating subsemigroup.
S
G carrying
Then there are Haar Singular
f..1.
orthogonal to
G is either R x discrete or T x discrete, in which case all such are Haar absolutely continuous. (Her~ S can be replaced by a separating
un~ess
proper closed translation-invariant subalgebra
A of
CO(G).)
The proof of the second (positive) half interestingly uses both the classical F. and M. Riesz theorem and the abstract version in an essential way.
140
GLICKSBERG REFERENCES
1.
P.R. Ahern, On the generalized F. and M. Riesz theorem, Pacific J. Math. 15 (1965), 373-376.
2.
R. Arens, The boundary integral of log I~I for generalized analytic functions, Trans. Amer. Math. Soc. 86 (1957), 57-69.
3.
R. Arens and LM. Singer, Function values as boundary integrals, Proc. Amer. Math. Soc. 5 (1954), 735-745.
4.
E. Bishop, Representing measures for points in a uniform algebra, Bull. Amer. Math. Soc. 70 (1964), 121-122. .
5.
, ~oundary measures of analytic differentials, Duke Math. J. 27 (1960), 331-340.
6.
, A general Rudin-Carleson theorem, Proc. Amer. Math. Soc. 13 (1962), 140-143.
7.
, Holomorphic completions, analytic continuation, and the interpolation of seminorms, Ann. Math. (2) 78 (1963), 468-500.
8.
, A minimal boundary for function algebras, Pacific J. Math. 9 (1959), 629-642.
9.
L. Carleson. Representations of continuous functions, Math. Z. 66 (1957), 447-451. B.J. Cole and T.W. Gamelin, Weak-star continuous homomorphisms and decompositions of orthogonal measures, to appear.
10.
11.
R. Doss, Elementary proof of the Rudin-Carle$on and F. and M. Riesz theorems, Proc. Amer. Math. Soc. 82 (1981), 599-602.
12.
F. Forelli, Analytic measures, Pacific J. Math. 13 (1963), 571-578.
13.
J. Garnett, On a theorem of Mergelyan, Pacific J. Math. 26 (1968), 461-467.
14.
T.W. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, 1969.
15.
J. Garnett and I. Clicksberg, Algebras with the same multiplicative measures, J. Funct. Anal. 1 (1967), 331-341.
16.
I. Glicksberg. Measures orthogonal to algebras and sets of antisymmetry, Trans. Amer. Math. Soc. 105 (1962), 415-435.
17.
• A remark on the analyticity of function algebras, Pacific J. Math. 13 (1963). 1181-1185.
18.
, Maximal algebras and a theorem of Rad6, Pacific J. Math. 14 (1964), 919-941. Errata,Pacific J. Math. 19 (1966), 587.
19.
, The abstract F. and M. Riesz theorem, J. Funct. Anal. 1 (1967), 109-122.
20.
, Recent Results on Function Algebras, C.B.M.S. Regional Conference Series, #11, Amer. Math. Soc., Providence, 1972.
21.
, Equivalence of certain representing measures, Proc. Amer. Math. Soc. 82 (1981), 374-376.
22.
, The strong conclusion of the F. and M. Riesz theorem. Trans. Amer. Math. Soc., to appear.
23.
I. Glicksberg and D.E. Marshall, Algebras containing a divisible unimodular sequence with dense self-adjoint span, J. Funct. Anal. 47 (1982), 165-179.
ORTHOGONAL AND
REPRESENTl~;I~
:"IEASURES
141
24.
I. Glicksberg and J. Wenner, Measures orth(l:.~"llal to a Dirichlet algebra, Duke Math. J. 30 (1963), 661-666. Errata Duke Math. J. 31 (1964), 717.
25.
E. Hewitt and S. Koshi, Orderings in locally compact groups and the theorem of F. and M. Riesz, Math. Proc. Camb. Phil. Soc. 93 (1983), 441-457.
26.
H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99 (1958), 165-202.
27.
H. Ktlnig and G.L. Seever, the abstract F. and M. Riesz theorem, Duke Math. J. 36 (1969), 791-797. J. Rainwater, A note on the preceding paper, Duke Math. J. 36 (1969), 799-800.
28. 29.
C.E. Rickart, Natural Function
30.
W. Rudin, Boundary values of continuous analytic functions, Proc. Amer. Math. Soc. 7 (1956), 808-811.
31.
N. Th. Varopoulos, Ensembles pies et ensembles d'interpolation pour 1es algebres uniformes, C.R. Acad. Sci. Paris, Sere A-B 272 (1971), A866A867.
32.
J. Wermer, On two theorems of classical analysis, Function Algebras (Proc. Intern. Symp. Function Algebras, Tulane Univ., 1965), Scott, Foresman, Chicago, 1966, 84-87. D.R. Wilken, Representing measures for harmonic functions, Duke Math. J. 35 (1968), 383-389.
33. 34.
A~gebras,
Springer-Verlag, N.Y., 1979.
L. Zalcman, Real proofs of complex theorems (and vice versa), Amer. Math. MOnthly 81 (1974), 115-137.
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WASHINGTON SEATTLE, WASHINGTON 98195
Contemporary Mathematics Volume 32, 1984
NUCLEARITY (resp. SCHWARTZITY) HELPS TO EMBED HOLOMORPHIC STRUCTURE INTO SPECTRA
- A SURVEY Bruno Kramm'*
o.
INTRODUCTION
We consider the following classical problem, which can be stated for (uB)-algebras (= uniform Banach algebras) or general algebras like Rickart's "natural systems" [14]: When does the algebra
B
~~
point
~
E oB (= spectrum of
B)
exhibit holomorphic structure? Classical results, starting with Gleason's famous theorem, are collected in Stout's
~ook
[16].
It contains results up to the late sixties.
An important
breakthrough appeared in 1973/74 in independent work by Basener (1) and Sibony [15].
They were able to extend the deep one-dimensional Bishop-Wermer
theorem to the multi-dimensional case by a bold generalization of the Shllov boundary.
It was later improved by Aupetit, and given a "final" short proof
by Kumagai [12).
We will apply this theorem in Section 4, but for now we
start with the original Basener-Sibony version. In the preface to his book [14], Rickart emphasizes two of the author's early papers (not yet published at that time).
He meant that they
would indicate the right path to follow, but much remained to be done.
Since
then, a considerable part of that work has been carried out and is published [71-[10].
Let me briefly explain what really new ideas are set forth in these papers:
*Shortly
after submitting this article, Bruno Kramm passed away. The Buitors have assumed responsibility for final preparation of the article.
143
144 1)
KRAMM A
(uB)-algebra
B has holomorphic structure at a point
aB
iff
BU
is a Stein algebra (Theorem 2.2».
[Here
there is an open
(uB)
nbd
2)
Since
aB
of
such that the localization
IP
= uniform Banach algebra; BU = completion of B lu with respect
to compact convergence on but a
(J C
of its spectrum
U.
Of course,
BU
is no longer a
(uB)-algebra,
(uF)-algebra (= uniform Frechet algebra).] 1)
allows an equivalent reformulation on holomorphic structure in
spectra, it might be a good idea to characterize Stein algebras by intrinsic properties within the class of
(uF)-algebras.
properties used should be "as natural as possiblE:!".
Of course, the
(Since Stein algebras
play an important role in complex analysis, such a project is also very interesting in that field.)
When sllch a characterization is achieved one
tries to apply the characterization theorem to the algebra as in
1), obtaining a solution to our old problem.
B of interest
We carry out this plan
in two independent ways, in Sections 4 and 5. '3)
But what would be good properties for
2)?
Three or four years before I
started the above project, Kiehl and Verdier introduced nuclearity into complex analysis in order to provide a valid proof for Grauert's "Bildgarben-Satz". Around that time some functional analysts started studying nuclear F-spaces extensively.
Unfortunately, they did not consider
the algebraic structure which most of the naturally occurring spaces carry, so I started to study nuclear (uF)-algebras with locally compact spectrum. To my great surprise it turned out that the behavior of these algebras is very close to that of Stein algebras. see [10].
For a survey on the above algebras
It also turned out that in some contexts the weaker and much
simpler Schwartz property leads to the same conclusions.
For the moment
we will only r.emark that the Schwartz property for a (uF)-algebra a little bit stronger than Montel's theorem on definitions we refer the reader to
S~ction
aA.
A is
For the precise
3.
In Sections 4 and 5 we give new solutions tv our old problem (Theorems 4.3 and 5.4). problem:
~len
Moreover, we give solutions to an old, somewhat forgotten is a Gleason part
variety (and the elements of
Bn
~ C
aB
for a (uB)-algebra
holomorphic on
rr)?
In all these theorems the nuclearity or "Schwartzity" of play a crucial role. structure theorems.
B an analytic
BU (resp. Bn)
The theorems of Sections 4 and 5 have the flavor of They may be improved exactly to the extent that the
theorems characterizing Stein algebras can be improved.
HOLOMORPHIC STRUCTURE
The background for Section 5 is
145
~-dimensional
holomorphy.
But I
formulate the whole section in such a way that a reader without any knowledge of
~-dimensional
holomorphy can understand it.
The only exception: (5.3),
where I sketch the proof of Theorem 5.2. If you find Sections 4 and 5 a bit abstract, then you may gain some pleasure from the very concrete Section 6, where we discuss the question: Given a eGO-smooth submanifold when is
M
~
M
in ~ domain of holomorphy
complex manifold'?
The Schwartz property helps again!
In fact, let uS take
O(M), the algebra of
1M with respect to compact convergence on M, calling the resulting (uF)-algebra A(M). If M is holomorphically convex then M ~ oA(M). Conversely, if we aSSume that M ~ oA(M) , then the holomorphic germs on
M, complete
O(M)
nice thing is that in many cases a Schwartz property for
A(M)
forces
M
to be
a complex submanifold and hence a Stein manifold!
It is the author's modest hope that the results of this survey might encourage some functions algebraists to study nuclearity and Schwartzity [10]. PRELIMINARIES
1.
(1.1)
u~FINITION.
A pair
(A,X)
is called a uniform Frechet algebra
(=(uF)-algebra) if a)
X is a hemi-compact topological space
~
0 (see the definition of
"herni-compact" below); b)
A is a closed sub algebra of
C(kX) (see the definition of
kX
below); c)
A contains the constants
~
and separates the points of
If we drop the assumption "herni-compact" in
of a function algebra is obtained. algebra). (1.2)
If
(In this case
X is compact we call
then the most general notion A remains a complete lrnc-
A a uniform Banach (-(uB)-algebra).
hemicompact" means that there exists an exhaustion of by compact subsets ••• c Kn C Kn+l c ••• , n E N, such that for any compact K to
C
REMARKS.
a)
X.
X there is
"X
n 0 E N with
X, that is, a subset
open in
kX)
and
kX
denotes the k-space associated
0
is open with respect to
K c X.
Hence
C(kX) (=
kX
~-valued
iff
U
n K is
continuous
A become Frechet algebras with respect to the topology
of compact convergence on norms
n
U C X
K for all compact
functions on
Kc K
X
X.
This topology is given by the series of semi-
KRANM
146
Ifex)l.
sup
E C{kX)
f
xEKn nr, In other words,
=~
A
~. n
n
[,1
parti.:ular, C(kX)
= lim
+---
i·!t·)re
-
=
C(kX)K
C(K ).
lim
n
n
n
n
st.ands for the separ.ated completion of
\;,
with respect to
A
n
The \..:ell-Knu\"n Gl!1fand theory for uniform Banach algebras generalizes
(1..3).
to
thE' function algebras
introduced
abov<~.
In parLicular, via the Gelfand
representat ion [
A
we may idcnt ify
A
-+
C (koA), f
~
rCA)
I'(A) C C(kaA).
X (resp. on
(.,A = continuous &pe.;; l rum of
A
~A,X)
By this ident.it ication we sCt:! tklL
algebr::t
QCf),
::
vIi th tht:.' closeJ subalg\,.'i.>ra
t0pology of c.)mp:lct convergent:e all A .....
f, f«(p)
induces the homeomorphism
aA)
endowed with the Gelfand topol'Jgy). !IlUY lJe
interpreted as a function
(m its "natural" l:<.lrr icr space
(r(A), 01\)
Note that the
aA; both spaces are linked
by the l:llilLinuous evaluation map j
: X
-+
with
(fA,
We shall often suppress {l(A), aA).
~()te
n~ed
that
(:;:"(A) ,uA)
We
by
the completion of
( 'I. j'
bllt
=
f(x), Vf E A.
AIM
~
need not iJe a
All complex an:llyt I,: spaces
(X,O)
they may possess s il1gular points.
s~ction algebra of
0
over
U by
respect to compact convergence on elegant proof see [2, p. 881).
as the pair
For
~t C
X
denote
with respect tu the semi-norm system
Of course
)
An
is a n;,lural system in the oense of Rickart.
more general localizat:1ons than above.
• .1 K KCr-! compac.t •
(1. 5) •
x
X, and consider thl' "function algebra
\1.4).
AM
toP (f)
For
O(U).
(uF)-algebra.
arl..! 30sumed to be reduced [5], ;.111
open
O(U)
U is a
C
X
we denote the
(uF)-algebra with
U (Theorem of Grauert and Remmert; for an
HOLOMORPHIC STRUCTURE (1.6).
A word on the "interaction" of
147
(uF)-algebras and
(uB)-algebras.
It is too·-simplistic to think that the former are merely generalizations of
the latter.
Instead, the reader s!lOuld consider them as "living in each other".
(A,X) gives rise to (uB)-algebras (~,K) whenever you choose a compact K C X. On the other hand, a (uB)algebra (B,Y) with Y metrizable generates (uF)-algebras (BU'U) whenever you choose open subsets U C Y. On the one hand,
2.
(uF)-algebra
HOLOMORPHIC STRUCTURE IN SPECTRA
(2.0). q>
a
Recall the following basic definition.
A function algebra E crA if (i) (ii)
there exists a neighborhood
fo'p
U C aA of
there exists an analytic subset
GC (iii)
A is said to carry holomorphic structure at a point
q>,
Y 1n some domain of holomorphy
a: n ,
tuere exists a homeomorphism
is holomorphic for all
f
in
~
Y ~ U, such that in the diagram
A.
REMARKS:
1)
Formally, this seems to be a rather complicated notion.
In (2.2) we shall
transform this notion into a very simple but equivalent one. The price to be paid for this is the introduction of the involved notion of a Stein algebra. 2)
If definition
(2.0) is satisfied then it follows easi:y (e.g. from (2.2»
that all the components of (2.1).
-1
cp
: U -+ Y satisfy
The classical case occurs, of course, when
algebra.
-1-1
{9
)l'···'(cI>
)n E
Au.
A is a uniform Banach
We want to consider: Problem I.
Give necessary and sufficient conditions for
A to exhibit holomorphic structure at a point of
aA.
There is a related classical problem which seems to have lost some of its attraction.
Gleason introduced his notion of parts because it looked
likely at that time that the parts would be the most natural pieces of exhibiting holomorphic structure. case, too.
aA
We shall contribute two solutions to this
More precisely, we will examine:
148
KRAMM
Problem II:
Let
n
C
a A be a Gleason part. this time endowed
with the metric topolOgy of conditions that
A'.
Give necessary and sufficient
exhibit holomorphic structure at all points
An
\pErr. In order to make this meaningful, carry Definitivn 2.1 over to obvious way.
(It doesn't matter that
U
= n,
in the
need not be Gelfand open.)
TI
It Is possible to reformulate the results below for Problem II
RE~~RK.
a local version for
into
rr, but you will see that the version just posed will be
more convenient. (2.2).
The following theorem will be fundamental for the sequel.
THEOREM.
Let
~
B be a (uB)-algebra and let
E aB
be given.
The following conditions are equivalent: (i) (ii)
B has holomorphic structure at There is
Au
that
~
cp E aB;
open (A-convex) neighborhood
is
~
U c aB
of
Ij)
such
Stein algebra.
To the best of my knowledge, this theorem does not appear in the literature. So we shall give a full proof here.
But first we fix some notation and provide
some preparation. (2.3). For the notion of Stein analytic space we refer to
[5].
We'll use the
following nice function algebraic characterizat:ion of Stein spaces.
This is
the famous Igusa-Remmert-Forster theorem [4]:
An analytic space j
with
Ij)
x
(2.4) DEFINITION.
REMARK.
x
•
A is called a Stein algebra if there
such that
A
If you aLe given a Stein algebra
space associated with spectrum
(X,O)
-+ ~
is a homeomorphism.
A (uF)-algebra
exists a Stein space
is a Stein space iff the natural
: X ... C"'J(X), x
= f(x),
(f)
(X,O)
aA
A.
~
O(X)
A, then there must be some
How to find it? aA
Stein
Well, by (2.3) we know that the
must be the carrier space (up to a
obtain the "right" sheaf on
as topological algebras.
~.omeomorphism).
In order to
let us momentarily forget our situation.
We introduce a most natural and simple sheaf
Ax
for quite general function
HOLOMORPHIC STRUCTURE algebras (A,X). Consider the family subsets of X; this family is a presheaf on
X.
associated with this presheaf.
(In the case
A=
U
That's it.
149
running through all open Let Ax be the sheaf X .. CIA we set
Ax.) Now let's go back to the above.
structure sheaf for
A, triat is,
(CIA,A)
A (unique up to biholomorphisms). [3].
In [11] we show that
A is the
is the Stein space associated to
A different approach is given by Forster
It is more complicated, but includes also the non-reduced case, which
we exclude. (2.5).
We need the following permanence property of Stein algebras.
THEOREM.
Let
A be a Stein algebra and
Ao
C
A a closed suba1gebra such
that the adjoint spectral map aA ~ CIAo is proper. Then Ao is a Stein algebra, too. Its spectrum is obtained by identifying those points in A
Ao •
which cannot be separated by For a proof see [9, p. 202]. (2.6)
Proof of "(1) - (ii)" of Theorem (2.2) :
Let
to Definition 2.0
Y) we may is holomorph1c for all f E~. Note that (U,~,Y) remains a Stein space after the above shrinking; this follows from the A-convexity. Thus there is a sheaf o on U defining this analytic structure. By Theorem 2.3 we know that j
then ... foq,
: U ~ oO(U)
is a homeomorphism. Using (1.5), we see that
Au
is a closed subalgebra of
O(U).
The
adjoint spectral map aO(U)
-+
crAu
:!!!! U
is a homeomorphism (by (w», hence proper.
Thus, by Theorem 2.5,
~
is a
Stein algebra. (2.7). of (2.2»
The following theorem (which is also needed for the reverse implication showa the
IIdouble-faced characterll of Stein algebras.
Stein algebras as an excellent subclass of
(uF)-algebras.
It exhibits
KRMtM
150
THEOREM.
Let
A be a (uF)-algebra.
(i)
A is a Stein algebra;
(ii)
such that
(X,O)
A
A has holomorphic structure at each
(i) ~ (ii):
PROOF.
= O(X).
Let
A be a Stein algebra.
~ O(X).
A
..i
IP E crA.
'rhere exists
Stein space
.'i
IHthout loss of ;.!cnerality we may assume that
Theorem 2.3 yieldt; th~ homeomorphism .~
Hence
The following are equivalent:
carries the
we obviously havt!
an:'tlytl\~
f =
: X ....... tJA.
stn,.:i.ure of
X oVer
to
aA.
For all
f
E A
li1us JJcfln.itio:l 2.0 can be applied to all points
[(Ij.
EGA.
q>
(li)
:::0
(i) :
The Implication of Theorem 2.2
that there is an open cover
of
(Uj)jEJ
\"hi.ch was proved in (2.6)
.:.lA
such that the
shuws
are Stein
AU j
algebras, for all all
LI.
E
j
T
Without luss I)f .;;enerality we may assume that
J
ar.e A-convex.
So we have
J Do LilesE'. strllctl.lrl;'/:i coinciJe
from the transitivity of (A-_ )'1 . -U - li J J
So
of..
has a
r"l
I
U
k
compl~x
10<':<:113
=
(A._)U . -Uk
ion~'!
zations: k
n
Vj, k E 1 .
l'
'., J
space structure
Since the elements of
5~ICil
all
tlldt
A separate the pointR of
separable; the A-convexity of is a Stein space.
0'11.
thei r intersL-C t
Oil
Now
URe
(u.,.p.,Y.). J J J That this is so follows
collect1011 of Stein spaces
~
erA
aA
are holomurphlc.
it is ilolomorphically
implies the holomorphic convexity.
the same
(2.6) - in parti.cular, (2.5) 1s the
E: A
f
argUl:!f~nt
~ssentlal
Hence
as in the second half of
tool - and conclude thot
A
is
a Stein nlgebra. With a bit more eHort it eml be shown that
A is the full algebr
Le., A '" CXoA).
(2.8). I. p
o
::l
E:
;J
As a curullary we obtain the proof (it) A possess an
Stein algebra.
(up~n
~
) A-convex. neighburiLood
(i) of Theorem 2.2: U
C
vA
(It's no loss of generality Lo assume that
namely for each open A-convex.
V C U,
Av
Let
such that U is A-convex;
is a Stein algebra whenever
AU
is a Stein algebra.) By Theorem 2.7,
particular, A
Au
~
has holomorphic st ructurc at :ach
has holomorph:i.c stl-ucture at
too has holomorphlc structure at
'.p E
tpO E U = UA ~ cr~.
O"~,
so in
Consequently,
151
HOLOMORPHIC STRUCTURE
(2.9). Some motivation. By the fundamental Theorem 2.2 we are led to look at Problem I and Problem II from a new point of view. Problem III.
This new perspective amounts to solving Give intrinsic characterizations of Stein
algebras within the class of (uF)-algebras. After having solved Problem III we will see what it tells us about the solution of Problems I and II. By the way, there are two further reasons for studying Problem III. First, within complex analysis of several variables, (2.4) shows that the theories of Stein spaces and Stein algebras are equivalent.
So it is of great
interest to find criteria which enable us to pick out Stein algebras mnong general holomorphic algebras.
Also, there are connections to the construction.
of (Stein) envelopes of holomorphy.
Second, a more philosophical reason is
that since complex spaces are locally Stein" such characterization theorems amount to reconstructing the phenomt::l1ol1 "holomorphy" by completely functionalanalytic principles.
In other words, solutions to Problem III contribute to
meta-function theory. 3.
NUCLEAR AND SCHWARTZ FUNCTION ALGEBRAS.
0.1).
Let
(A,X)
FURTHER PRiPARATIONS
be a (uF)-algebra, and let •••
X by compacta.
admissible exhaustion of
C
Kn
C
Kn+l
C
•••
be an
A is said to be nuclear (resp. to
have the Schwartz property) if, after eventually (if need be) thinning out the exhuastion
the restr.iction maps
(K) , n 11
rn : ~
-+
n+1
t\. ' n
n E fi,
are nuclear (resp. compact) operalors. Note that both
prop~rties
algebraic properties.
are locrdly convex space properties, rather than
However, we prefer to formulate these definitions in a
more convenient way in terms of function algebras. on nuclear locally convex spaces J.s Pietsch [13].
The classical monograph But it doesn't significantly
help a reader who is interested in nuclear function algebra.
Fot' this reason
I wrote a broad survey on nuclear and Schwartz function algebras [10]. (3.2). Let
For convenience, let's recall the definition of nuclear operators.
T: E
~
F
be a continuous linear operator between Banach spaces.
T is called nuclear if
Then
152
KRAMM (i)
there exists a sequence
tn E E'
(ii)
there exists a sequence
f
E Vn (e)fn ,
T(e) =
Ve E E, and
n
E n
n
E F
IIvn ll-lI f n ll
<
such that GIl
•
It is an easy exercise to conclude from this definition that such a a compact operator.
T is
Hence any nuclear (uF)-algebra is a Schwartz (uF)-algebra
(but not vice versa)_
Observe that a Banach space which is at the same time
Schwartz must be finite-dimensional. Example:
(X,O)
Let
K C LeX with
compact subsets
is nuclear.
K
C
Then for any two
LO, the restriction map
Thus all holomorphic algebras
Schwartz). (3.3).
be a complex analytic space.
O(X}
are nuclear (and hence
For more examples from different areas see [10].
We present some properties of Schwartz (uF}-algebrao
illustration [10].
For simplicity we always regard
just for
oA as the carrier space
of the algebras. 1)
If
points, and
oA
is compact then it consists of only a £iuite number of
A is a finite dimensional vector space.
(Here
A is Schwartz
and Banach at the same time!) 2)
The Shilov boundary of
A is empty;
independent points in the sense of Rickart [14].
A does not even possess Heuristically speaking,
Schwartzity pushes the Shilov boundary out to infinity. 3)
If
principle on 4)
oA
is locally compact, then
This fact implies:
A satisfies the maximum modulus
oA.
There is a weak version of the identity theorem (under a mild
hypotilesis) • 5)
The Gelfand topology and the strong topology on
logically equivalent, hence homeomorphic if
oA
oA
are compacto-
is a k-space with respect to
the Gelfand topology. 6} f E A)
O(G)
If
A is topologically singly generated (i.e., A - IU]
then there exists a domain
Gee
such that
for some
A is isomorphic to
as a topological algebra. 7)
A is antisymmetric, i.e., f E A and
1
E A imply
f
= constant.
HOLOMORPHIC STRUCTURE
153
Of course, in [10] you will find further properties whose proofs
(3.4).
need full nuclearity. ~
Is
(3.5).
A
But the following problem 1s open:
~ ~
Schwartz (uF)-algebra
locally compact
aA.
automatically nuclear?
A
A function algebra
if for all
~
A is called strongly uniform (for short:
u*A)
AIM endowed The Shilov
uniform ideals (= kernel ideals) MeA, the algebra
with the natural quotient topology is a function algebra again.
boundary for Banach function algebras often turns out to be an obstacle to strong wliformity.
Fot example, the disk algebra
H(A)
is not strongly
uniform (see [11] for a proof communicated to me by Gamelin).
But nuclear
or Schwartz function algebras seem to be "often" strongly uniform. algebras are always (u "" F)-algebras.
A = O(X), and
(Proof:
MeA a uniform ideal.
V(M)
as
Each
h E O(Y)
Let
X be a Stein space,
By (2.3) we may identify the z~ro-set
aA and identify analytic sets
a subset of
admits an extension
Stein
h E O(X)
Y
C
such that
X with
h 1y •
j-l(V(M».
h.
So we
have the exact sequence 0-- M -
and hence
(3.6).
O(X) 1M
~
O(Y)
O(Y) -
O(X) -
0,
is a (uF)-algebra by (1.5).)
We recall the notion of (complex) Chevalley dimension for (uF)-algebras,
which was introduced in [7]. d(~)
consider the integer there exists
fl, •••• f
A be a (uF)-algebra.
Let
defined as the minimum of all
~
For any
E aA
n E N such that
and a neighborhood U C crA of ~ such that ~ the fibers of the mapping (fl, ••• ,fn ) : U ~ t n are finite sets. If this minimum does not exist, set d(~) = ~. The dimension of ~ in aA is den
E
~
~
fined by dim r.p
crA
={
o,
if
d(\.p) ,
otherwise.
~
is an isolated point in
aA
It is well known that for Stein algebras the Chevalley dimension equals the topological Krull dimension (3].
154 4.
KRAMM THE FIRST APPROACH
(4.0)
In Section 3 we collected some fairlY natural necessary conditions which
Stein algebras enjoy among (uF}-algebras: (i) (ii) (iii)
Nuclearity (or weaker, Schwartz property); Strong uniformity; Locally compact s.Jt\ctrum.
It was very surprising to me that together with one mild additional assumption the conditions become sufficient, too.
I should mention here that Defore
proving this theorem I tried to prove a different theorem.
I leave it as a
problem for the reader: If
A is
~
nuclear (uF)-algebra with locally compact spectrum
such that all closed maximal ideals are algebraically nnitelY generated, is
A
~
Stein algebra?
Such a theorem would be very satisfactory since it would provide a perfect analogue to Gleason's famous theorem. an~wer
I was only able to give an affirmative
in the special case for \vhich all elosed maximal ideals are principal Note that Theorem 4.1 below and the problem posed above differ
[6J.
only in
one hypothesis. (4.1) THEOREM [9J: pure-dimensionu_! (1) (ii)
Let aA.
A be a
(uF}-algebr~
having locally compact and
The following conditions are equivalent:
A is a (pure-dimensional) Stein algebra; A is Schwartz and strongly uniform.
REMARKS.
1)
The pure-dimensillnality may be replaced by
"dimension condition" (DC).
dim~
crA <
GO
if one adds a
This condition forces the "components" in aA
of different dimension to intersect in a "nice way".
(See [11], yet to be
published. ) 2)
In [9] a variation of the above theorem is given which characterizes
r.egular Stein algebras rather than pure-dimensional ones.
(Recall that a
Stein a1Rebra is called regular if it is associated to a Stein manifold.) (4.2). it.
The proof of Theorem 4.1 is very involved, so I won't attempt to sketch
But I want to indicate Some of its ingredients.
155
HOLOMORPHIC STRtCTURE Basener's and Slbony's famous theorem on holomorphic structure
(1)
[1], [15].
In the meantime this theorem has been improved by Aupetit,
and the proof shortened by Kumagai [12]. (2)
Hereditary maximum-modulus principles.
(3)
A semicontinuity of fiber-dimensions theorem in the following
theorem from [71: Let
=
f
n
(fl, ••• ,f n ) € A.
a neighborhood "'-I
dimcpf
A
(f(ql»
U ~
C
oA
of
such that
~
"'-1 ...
dim", f
E ~A there is
~
Then for each
(f(t», for all
'" E U.
In complex analysis this theorem is proved via the Weierstrass theorems and further local theory; since these theorems are not available in our setting, we had to develop completely new proofs.
(4)
Forster's version of the Oka-Weil-Cartan theorem [4, ?
145].
(5)
A lot of technicalities concerning the higher Basener-Sibony-
Shilov boundaries and the Cheval ley dimension. Now we are going to apply Theorem 4.1 to Problems I and II. (4.3)
THEOREM.
Let
B be a (uB)-algebra, and let
Assume there is a .leighborhood
oB
Uo C
ql
~
cp E
in which
~B
be given.
B is pure-dimen-
Then the following are equivalent:
sional.
(1)
has holomorphic structure at
B
there exists a B-convex neighborhood
(ii)
BU
such that
(u'* F)-algebra.
is a Schwartz
PROOF. "(i) - (ii)": that of
BU
By Theorem 2.2
is a Stein algebra. Then
BV
(ii)
"(il) = (1)":
Let
U
C
Uo of
Choose an open B-convex neighborhood
ql
such
VC U
But a Stein algebra satisfies the
(see Section 3). U be given as In (ii).
morphism we conclude that BU
exists a neighborhood
remains a Stein algebra.
properties of
Thus
~here
crBU
is a Stein algebra, and
Since ju : U -+ crBU
is locally compact. (i)
is a homeo-
Now apply Theorem 4.1.
follows from Theorem 2.2.
KRAMM
156 (4.4)
THEOREM.
Let
B be a (uB)-algebra and let
n C aB
sional Gleason part endowed w!th the metric topoloGY.
at most countably many components.)
be a pure-dimen-
(Assume that
TT
has
Then the following conditions are equiv-
alent: (i)
(ii)
Brr
carrit!s holomorphic
B
is a Schwartz
11
st~ucture
at all
E rr
(u *f)-algebra.
The proof implies that the theorem remains valid if you replace the
RE~\RK.
metric topology by the Gelfand topology. conditions (i)
and (ii)
This results from the fact that both
imply coincidence of these topologies on
11.
1 formulated Theorem 4.4 (and also Theorem 5.5) with respect to the strong topology for historical reasons.
Before we can prove Theorem 4.4
we need two very simple lemmas.
(4.5)
LEMMA.
Let
M C aA
A be a function algebra and
endowed with the Gelfand topology.
an A-convex subset
Then the natural map jM : M ~
0"J\t
is a homeomorphism.
PROOF.
That
is bijective and continuous is obvious since
ju
We have to show that
function algebra.
;-1
"'M
(~,M)
is continuous, too.
is
a
Consider
the restriction map A-+AM • v~ -+ aA
Its adjoint spectral map (4.6)
for any
Let
LEMMA. ~
E
11
be a (uB)-algebra and
B
and
0
K
r
is continuous and factors through
~
r < 2, the
= {t
(ql)
E
11:
11'
C
aB
a Gleason part.
set~
I/;p-l.j.'/I
~ r}
are compact in the Gelfand topology. PROOF.
The closed balls
Kr (Ii»
:
=
{t E Sf : II(~-"'II ~ r}
are weak-*-compact (Bourbaki-Alaoglu's theorem). K (
= as n Kr (ql).
Since r < 2 we have
M.
Then,
HOLOMORPHIC ,STRUCTURE But
is
oH
157
Hence the intersection, and thus
weak-~-compact.
K (,p), are r
Gelfand compact. (4.7)
PROOF
Note that
n
(ii) OF THEOREM
(i) -
is
4.4:
B-convex (cf. [16, p. 168]).
By (i),
structure of a complex space such that all elements of
B
Note that the components of
rr.
Stein space, since vex.
Bn
IT
cannot cluster within
separates the points of
By Theorem 2.5 we conclude that
fin
rr
carries the
IT
are ho10morphic.
11
Now
and since
n
is a
7T
is
Bn-con-
is a Stein algebra which satisfies
the condition of (ii). PROOF
(ii)
=>
OF THEORE}f 4.4.
(i)
We need to show that
aB 11
crBn
is locally compact.
coincide as point sets via
jn'
of
fI
and
Lemma 4.5 implies that
(n, B'-Gelfand) are homeomorphic.
As above notice that
and
(rr,B'rr-Gelfand)
Both spaces are k-spaces since the former, as a subspace
(crB, Gelfand), is a k-space.
Next the Schwartz property and property
5 of 3.3 together imply that (n,B' -strong) n are
homeom~rphic!
and
(n,B' -Gelfand) rr
Now look at the continuous linear maps (the *'s denote the
corresponding topologies on them) (H'
TT
,strong)
-+
(B' ,metric) U
U
(rr,*)
" It follows that
-+
,weak-*) U
(rr ,*)
....
(n t *)
3'
h~meomorphic
(n, B'-metric) is homeomorphic to all the topologies above.
E IT basis of closed sets. Thus for each
-+ (B'
~
the sets
Kf. (~), 0 < & < 2, form a neighborhood
K (~) are compact with respect e to (H', weak-*). But since all these topologies turn out to be homeomorphic, we find that the K (cp) are compact in crB rr , too. Hence aB tT is locally I e compact. Theorem 4.1 shows now that is a Stein algebra and Theorem 2.2 ends the proof. ~4.8).
By Lemma 4.6 the
A little discussion of Theorems 4.3 and 4.4.
Both theorems are "structure theorems".
The two crucial conditions -
Schwartzity and strong uniformity - may look a bit aostract at first sight. But there is a major difference between them.
In
often possible to check the Schwartz-property (5).
concr~te
situations it is
It is somewhat stronger
158
KRAMM
than Montel's theorem, and yet often they turn out to be a equivalent! in many cases,
(S)
reduces to the (natural) question:
satisfy Montel's theorem on
Does
BU
So, B~)
(resp.
U (resp. IT)? (u) '* is subtle.
The strong uniformity
Even in concrete situations
(u). '*
it is very hard (or impossible) to c.heck
You may reformulate
(u) '*
B.
in such a way that it looks like a very weak version of Cartan's Theorem
Up to nOW I don't know any Schwartz (uF)-algebra with locally compact spectrum which isn't automatically
'*
This problem deserves urgent attention.
(u).
An affirmative answer would make super theorems out of the above results. But I'm pessimistic.
Anyway, even partial answers to the above problem would
improve Theorems 4.3 and 4.4 considerably.
5.
THE SECOND APPROACH
(5.0).
The methods of this approach are completely different.
We are making
a "detour" through ..... dimensional holomorphy; by two natural conditions this
very weak analyticity is forced to "come down to earth", that is, to come dOW1! to ordinary holomorphy.
I wish to point out that no knowledge at all of
co-dimensional holomorphy is assumed from the reader who merely wants to understand the theorems.
cour~e
(But of
in order to understand the sketch of the
proof of Theorem 5.1 some intuition in this field helps.)
(5.1).
First we need to define a sheaf
proceed stalkwise.
A
For a given
crA associated with
A.
We
we collect in the stalk
A<.p
all
on
germs of the form n
( l: f .); c E f:, f . E i<:er <.p, j n,J n,]
such that the double series converges compactly in some Obviously this defines a sheaf
crA.
on (2.4) C
Comparison of this sheaf with the (certainly more natural) one in
Ac
yields
crA.
Whenever
A
A
stalkwise, and
is called uniform
If
alg~bras
if for all open
function algebras on
(5.1.1) •
A(U)
A(U) c A(U). for all open U dense in A(U) for all open
is a uniform sheaf we have the coincidence
F of
(Recall the a sheaf
A is a Stein algebra A
A= A aA
of germs of continuous functions on U
C
aA
the section algebras
then
,...,
A
is a uniform sheaf.
F(u)
are
MoreOVer
=A~ 0 ,
0 is the structure sheaf of the underlying Stein space.
p. 260).)
crA.
C
U.)
I'V
where
neighborhood.
A of algebras of germs of continuous functions
In many instances, we also have U
<.p
'V
(See [8,
HOLOMORPHIC STRUCTURE PROBLEM:
159
compact
Up to now no example of a nuclear (uF)-algebra aA such that '" A is not uniform is known.
REMARK.
The sheaves
A
and
A
A with locally
can be formulated naturally for the more
(A,X), with '" necessarily equal to aA. One then obtains sheaves ~ and in the sequel we need only the case X = GA.
X not
general situation of function algebra realizations
(5.2) THEOREM.
Let
Ax'
However,
A be a (uF)-algebra with locally compact
is a Stein algebra if and only if (1)
A is nuclear;
b)
'" A is a uniform sheaf.
The necessity of
for Stein algebras was remarked in (5.1.1).
b)
shall roughly sketch the main ideas of the implication (5.3)
Sketch:
ture as an
First observe that
aA
A'.
(VFN)-analytic set in
a) + b)
~ (A
~
(Bil A)
~
Stein).
inherits a very weak analytic struc«VFN)
= strong
dual of a FN-space.)
Namely it is the zero set of the holomcrphic polynomial of degree A'
Here we
2:
[,
Such a notion of analyticity was introduced originally by Douady (2] (in the case of (B)-analyticity).
A
(B)-analytic space is very complicated since it
is not determined by its structure sheaf alone; in fact it is determined by what is known as a "functored structure", that is, the family of sheaves of all vector-valued holomorphic functions. (VFN)-analytic spaces behave much better. ~
E crA
In contrast to thiS, I found out that If we choose an arbitrary ?\Jlnt
and a relatively compact neighborhood
U c crA
of
~,
we have the
restriction maps
Dualize this and observe that again by property 5 of (3.3),
U inherits the
same topologies from all these strong duals: A'
+-
A'-
U -
U
(U,(VFN» But
-
AI
U
~ ~ (U.(B»
I
(U.(VF».
U also inherits three eventually different analytic structures.
160
KRAMM
-
It 1s at this point that we apply the assumption that sheaf:
this forces the
structure on structure of
(VFN)-analytic structure on
A is a uniform
U and the
(VF)-analytic
H2 • Since the (B)-analytic U lies "in between", rJe conclude that HI aud H2 are also
U to become biholomorphic via
HI
0
Thus we find that U carries a (VFN)-analytic structure, which at the same time turns out to be (B)-analytic. But I proved that such an analytic space must be an ordinary
bihoiomorphic.
(Many technicalities are omitted here.)
analytic space a convex spaces:
(This
generaliz~s
the corresponding elementary fact for locally
a nuclear (or Schwartz) locally convex space that is at the
same time a Banach space must be finite dimensional.) structure at (5.4)
Hence
A has holomorphic
and finally Theorem 2.7 yields the desired result.
cp,
Let
THEOREM.
B
be a (uB)-algebra and
let
E aB
o
be given.
Then the following are equivalent: (i) (li)
RE~~RK.
BU
then
c.p;
There exists a neighborhood
aB of
a)
is nuclear;
b)
is a uniform sheaf.
U C
Sometimes its more convenient to use
S,
such that
c.p
because if
is uniform
is uniform, too.
(5.5) THEOREM. ~ith
B has holomorphic structure at
Let
B be a (uB)-algebra and
the metric topology.
components.) (i) (ii)
(Assume that
rr
rr C aB
a Gleason part endow-
has a most countable many
Then the following are equivalent: B1I
carries holomorphic structure at all
B
is nuclear and
TT
Br.
c.p
E rri
is a uniform sheaf.
(5.6). The proofs of Theorems 5.4 and 5.5 follow from Theorem 5.2 in exactly the same way as the proofs of Theorems 4.3 and 4.4 are derived from Theorem Theorem 4.1.
6.
You may carryover these proofs word by word.
Sl'100TH MA..~IFOLDS IN
(6.0).
tn
&~D THE SCHWARTZ PROPERTY
After the two rather abstract approaches in Sections 4 and 5, I want
to show you a concrete example in which the (S)-property helps to embed holomorphic structure.
The techniques again are completely different.
results given below are due to my student Helmut Goldmann. in !d.s doctoral thesis, which will be completed soon.
They are
All the contain~~
161
HOLOMOPRHIC STRUCTURE (6.1). Let
The Problem.
M be a C·-smooth submanifold of a domain of holomorphy OeM)
Denote by
the algebra of germs of holomorphic functions on
f
each element
G c ,no
OeM)
of
M, that is,
is holomorpilic in some neighborhood of
M, the
A(M) .. OeM) 1M' the closure of f. Now define U(M) with respect to the topology of compact convergence on M. Obviously. A(M) is a (uF)-algebra on M. Next, M is holomorphically .:onvex if anti only if neighborhood dependent of
M -.. oO(M)
j
However, it is not difficult to prove that
is a homeomorphism.
crO(M) ~ aA(M) So if we wish to guarantee that require that
M be
A(H)
holo~rphically
(canonically). has the "right" spectrum, we have to
convex.
(For an
M given in a
"reasonable" way, this usually can be checked by a naturally adapted Leviform
LM
for
M: all eigenvalues of
L~.l(x)
must vanish for all
Next, we need a localization of the (S)-property. (A,X)
locally Schwartz if for all open
convex space.
U c X,
(In our situation we have
X
Au
x E M.)
Call a (uF)-algebra
is a Schwartz locally
= cA.)
It seems natural to pose the following Conjecture:
Let
MeG
be a holomorphically convex
011
C -submanifold. Then
the following are equivalent: (i) (ii) (Originally (6.2).
M is a Stein submanifold; A(M)
is locally Schwartz.
I formulated (ii) even without "locally").
THEOREM.
Let
M be as above.
If
M is in addition m-analytic,
If
M is locally
then the conjecture is true. (6.3).
THEOREM.
Let
M be as above.
O(G)-convex, then
the conjecture is true. (6.4). for
There are lots of special results, e.g., for
M a CR manifold.
M of low dimension, or
But I guess that Theorers 6.2 and 6.3 are good examples
of what is known concerning the conjecture.
Besides using function algebraic
techniques, Goldmann looks at the subbundle
ThM of holomorphic tangents of
the tangent bundle
TM
and uses theorems of Sommer, Freeman, Wells and others.
162
KRAMM
REFERENCES 1.
R. Basener, A generalized Shilov boundary and analytic structure, Proc. Amer. Hath. Soc. 47 (1975), 98-104.
2.
A. Douady, Le probl~me des modules pour les souses paces ana1yt irlues compacts d'un espace analytique donn~, Ann. Inst. Fourier 16 (1966), 1-95,
3.
U. Forster, Zur Theorie der Steinschel1 Algebren und Modulen, Math. Z. 97 (1967), 376-405.
4.
O. Forster, Holomorphiegebiete, in "Theorie del' Funktionen mehrerer komplexer Veranderlichen" (ed. by Behnke and Thullen), Springer-Verlag, 1970, 134-147.
5.
R. C. Gunning & H. Rossi, Analytic Functions of Several Complex Variables,
Prenti.ce-Hall, 1966.
6.
B. Kranuu, A characterization of Riemann algebras, Pac. (1976), 393-397.
7.
B. Kramm, Complex analytic properties uf certain uniform Frechet-Schwartz
J. Math.
65/2
..
algebras, Stud. Math. 66 (1980), 247-259.
8.
B. Kramm, Analytische Struktur in Spektren - ein Zugang uber die dimcnsionale Holomorphie, J. Funct. ALta!. 37 (1980),249-270.
9.
B. Kramm, A functlonalanalytic characLerization of pure-dimensional and regular Stein algebras, Advances Math. 42 (1981), 196-210.
...
10.
B. Kramm, Nuclearity and function algebras - a survey, to appear in Proc. III. Paderborn Conf. on Functional Analysis, North Holland, 1984.
11.
B. Kramm, Nuclear Function Algebras and thp. Theory of Stein Algebras, North Holland, Spring 1985 (in preparation).
12.
D. Kumagai, On multi-dimensional analytic structure and uniform algebras, Proc. Amer. Math. Soc., to appear.
13.
A. Pietsch, Nuclear Locally Convex Spaces,
11..
c.
15.
N. Slbul1.Y, Hu.ltidimensional analytic structure in the spectrum of a uniform algebra, Springer Lecture ~otes 512, 1976, 139-165.
16.
E. L. Stout, The Theory uf Uniform Algebras, Bogden & Quigley, 1971.
~pringer-Verlag,
1972.
E. Rkkart, Natural Function Algebras, Springer Universitext, 1980.
FAKULTAT FUR MATH & PHYSICS UNIVERSITY OF BAYREIITH POSTFACH 3008 8580 BAYREUTH \o/EST GER..\fANY
Contemporary Mathematics Volume 32, 1984
MAXIMUM MODULUS ALGEBRAS AND MULTI-DIMENSIONAL ANALYTIC STRUCTURE Donna Kumagai INTRODUCTION The quest for the existence of analytic structure in the spectra of function algebras has generated much energy since the classic result of
E. Bishop in 1963 [4].
John Wermer has cast a new light on the subject
recently [13] by exhibiting how Bishop's theorem is analogous to the well known theorems by
Harto~
[6] and Rudin [10] when all of them are put in the
setting of maximum modulus algebras. Let
X be a locally compact Hausdorff space and let
X.
of complex valued continuous functions on
algebra on
We call
A be an algebra
A a maximum modulus
X if
(i)
A contains the constants and separates points of
(ii)
If
K is any compact subset of
X. f E A.
X then, for every
If(z)1 ~ max If I , z E K, aK where
aK
is the topological boundary of
K.
The maximim modulus algebra
version of the generalized Bishop's Theorem [1] may be stated as follows: THEOREM 1. and let f-l(K) subset
Let
A be a maximum modulus algebra on
f E A with
=
f(X)
{x E Xlf(x) E K}
E of
C
2.
X.
Assume for each compact
is compact.
{f-l(~)}
=
K
Q C
be a region,
2,
Also, assume that there exists a
~ with positive logarithmic capacity and
cardinality of the set
Let
{x E Xlf(x)
=
~},
#{f-l(~)}
,the
finite for each
~ E E.
Then
#{f-l(~)} ~ 20
k
for every
r
there exists a discrete subset £-1(2 \r )
of
Q
such that
can be given the structure of a Riemann
surface and for every
g
in
At
g
is analytic on that
Riemann surface. See [13] for the proof. © 1984 American Mathematical Society 0211-4132/84 $1.00 + $.25 per page
163
J64
KtJ1-fAGAT
The first multi-dimensional analytic structure theorem for uniform algebras is due to R. F. Basener [3] and independently to N. Sibony [12], which was extended by B. Aupctit in [2. Theorem 2.131.
K. Rusek has shown that n-dimen-
sional analytic structure can be introduced for a certain class of subalgebras of
C(X). and such algebras are maximum modulus algebras, while the converse is
condition for n-dimensional analytic structure for maximum modulus algebras which is lesR restrictive than that given hy Rusek.
Our main result is
Theorem 4. 2.
A be a maximum modulus algebra on
Let
An = {(fl, ••• ,fn)lfl, •.• ,f n E A}. n
¢.
F
(Then
Let
F E An
x
and let
is a proper mapping if for every compact subset
F-I(K) = {x E Xlf(x) E K}
w~
is compact.)
x into
be a proper mapping of K
of
a;n,
introduce a certain class of
functions which "detects" the cardinality of the images of the fibres, F-lp.. ),
\ E F(X).
The plurisubharmonicity of these functions plays a key role in the
analytic structure problems. Let
THEOREM 2.
F
A and
be as above.
assume that for each cOUlElex line A
f-I(W 1 L)
for each
{gl F-1 (W n L)
=
: g E A}
is a maximum modulus algebra.
is plurisubharmonic on
toJ.
First we introduce some notations and definitions.
nx,
subset of
Then,
kEN,
and
g E A
in
L
Let W ~ F(X) be a domain and ,n, Earallel to a coordinate axis.
the n-fold Cartesian product of
Denote by
Z the
X with itself, consisting
n
of the points
=
(xl' •.• ,x n ) such that F(xl) - F(x 2 ) = ••• = F(x n ). Define Z ~ ¢ n by n(x) = F(x l ). Clearly n is a proper mapping when a function Tf nx. Let II be the subalgebra of C(t) is given the subspace topology of n n generated by the functions of the form: x~ IT gi(x i ) , gi E A. 1=1 Let
LEMMA 1.
x
A, X, F, W,
comElex line contained in a point
a
on
T.
Choose
1T
and
W, and
U be as above.
T be an arbitrary
Let
D a disc contained in
s E n-l(a). 1~(s)1 5
Then for each
T
and centered at
~ E ll,
max I~I n-l(oD)
The proof of Lemma 1 follows from that of Lemma 2 in [8], where we prove a
MAXIML'M MODULUS ALGEBRAS
165
similar result for the case of uniform algebras, building on Senichkin's method.
LEMMA 2.
Let
U be as in Lemma 1.
W, nand
the function 4>
For each element
EU ,
~
W £l_
defined on
is plurisubharmonic. PROOF.
We must show that if
restriction of
to
4>
T
T
i.s a complex line contained in
is subharmonic.
is proved by a standard method.
Let
D
The upper semi-continuity of 4> S Re P
O.
X = n(x), i.e., A = F(x l ) = (fl, ... ,fn)(x l ). 1~(x)1
zED.
PeA) Ie
S
I.
Hence, Ie
There exists
x ....
the function
yEn
~(x)·e
Let
on ao, the boundary of x E n-l(oD) and put 10gl~(x)1 S ~(A) ~ Re
We have
-P(fl,···,f )(x l ) n "I"C(x), S 1.
(z)
D
P(X).
Now pick
such that
Note that
-P(fl(xl),···,fn(x l »
is a uniform limit on
0
of
From this fact, to·gether with Lemma 1, we conclude that
U.
functions from
-1
~IT
T be a closed disc centered at some
C
point a. Choose a polynomial P with in T. We must show that 4> ~ Re P on
So
W then the
and thus,
A E D.
for each
PROOF OF THEOREM 2.
The plurisubharmonicity of
n
Lemma 2 since
[g(xi)-g(x j )] E U,
follows directly from
'fk ,g
-1
x 1 , •.. ,xk E F
and if
(X).
then
1 S i< j S k ( Xl' ••• ,x ) E .." -1 (') '" • n
THEOREM 3. exists
G
Let C
A, F,
and
W be as in Theorem 2.
G is not pluripolar,
(2)
For each
X E G,
#{goF-l(A)}
Then there exists a positive integer
PROOF. assumes
is at most
Suppose there
values on
k
is finite. such that for each
A in
W,
k.
The condition (2) implies that i
g E A.
W such that
(1)
U{goF-l(X)}
Fix
F-l(X)}.
G
For some
=
U G., where Gi = {X E Gig iEN ~ kEN, Gk is non-pluripolar.
166
KUMAGAI
Since for each
}.. E Gk ,' g
k
assumes
IT i< j :0:: k+l
max
1
on
Hence
Gk •
non-pluripo1ar.
Wk+l,g
Thus
W=
~
on
W, by Theorem 2 and by the fact that
HgoF- l (}..) ~ k
This implies that
loss of generality take k U
= -...
toJ'i'
k
F- 1 ().),
values on
}.. E W.
for all
is
Gk
Without
\vk n w = 0 .
to be the largest integer such that
This proves Theorem 3.
1=1 In Theorem 3, the requirement that
E is non-pluripolar can be replaced
by a more general "uniqueness set". DEFINITION.
~n.
W be a region in
Let
G
We say
C
W
is a set of uniqueness
W if every plurisubharmonic function defined on W that converges to
for
at every point of COROLLARY 3.1. exists a subset (i)
is identically equal to
G
Let
on
W.
W be as in Theorem 3. W satisfying:
A. F.
G of
-..
--
and
Suppose that there
is a set of uniqueness for W, For every }.. E G, #{gOF- 1 (}..)} is finite.
G
(ii)
k
Then there exists a positive integer #(gOF- l ().)} is at most k.
A
such that for each
in
W,
The proof of Corollary 3.1 Is the same as for Theorem 3. The following is a special case of Theorem 4. Suppose that a function
LEMMA 3. Let A. F. and W be as in Theorem 2. in A is constant on F- l ().) for every ). E W.
Then
goF
-1
g
is analytic on
w. PROOF. be Let
a
We show
goF
-1
is analytic in each variable.
complex line containing ~n
a
and parallel to the
be an open polydisc about
~n
a.
=
n IT
i=l
a., J
j
Let ith
8i ;
a E W, and
Li
coordinate axis.
8 i = 8(a i ,r i );
:f: i}
-1 f. (a.)} ] J
Denote by
the restriction
is a maximum modulus algebra by hypothesis, and exists a representing measure,
= \Joi'
is compact in supported on
'" -1
£i
(08 i )
-1
F
(W
n L1).
There
representing some
MAXIMUM MODULUS ALGEBRAS N
-1
si E fi
(a i ).
Let
Vi
'" -1 gof.
on
be the projection of
normalized Lebesgue measure on
Thus
a~
Vi
is the
..
1
is a complex harmonic function.
1
... -1 gof.
This shows that
is holomorphic on
~
~i
algebra for every complex line,
goF
and hence
If we require in Lemma 3 that
REMARK.
167
-1
on
~n
n Li
.
is a maximum modulus
L, without assuming that
L
is parallel to a
coordinate axis, then a simplified proof can be obtained using a multidimensional analogue of Hartog's result in [9, p. 59].
See [11].
Let A be a maximum modulus algebra on a locally compact space X. . X on t 0 a d oma1n • W• Suppose for each Let F E An be a proper mapp1ng 0 f complex line L in It n parallel to the coordinate axis, A is a F-l(W n L) THEOREM 4.
maximum modulus algeyra. (i)
G
(ii)
!or every
Suppose there exists a subset
is a set of uniqueness for
Then there exists
~ E G.
kEN
analytic covering. and every
EA
f
such that
W,
U{gOF-l(A)}
such that the
G C W
is finite. F:F
maPE.~
-1
(W)
is holomorphic on
W is a k-sheeted
-+
F-l(W).
We need the following lemma.
LEMMA 4.
~
X be a locally cumpact Hausdorff space, let
metric locally compact space, and let fibers (i.e.
UF-l(y) <
~ for all
F:X y
(Y,d)
~
Y be a proper mapping with finite
-+
E Y).
F is open.
Then the mapping
The proof is in [11]. k PROOF OF THEOREM 4.
where goF
-1
W1'
=
By Corollary 3.1 there is a
{A E Wig
assumes
i
values on
is analytic on Wk' Let values of g on F- 1 (A). Let and
Di
F-1(A)
n Dj with
="
for
g(U) C
i" j.
kEN
U Wi' i=l
We shall show
and i
such that W ...
bl •.•.• b k ~
k)
be the distinct
be a disc centered at
There exists an open neighborhood
U of
k
U i=l
Di ,
and
F(C)
C
W.
Put
N ... F(U).
By Lemma 4.
N
is
KUMAGAI
168 open.
Set
ei
= F-1 (N)
n g -1 (D- i ).
fibre, it is dear that k).
( XE N, 1
have shown that is open.
show that
F
-1
g
Since
-+
assumes at most k-values on every
is constant on the set
In view of Lemma 3, (W k )
g
\v k
{F
-1
A-+ goF-l(A)
(X)
n ei }
is analytic on
We
is a k-sheeted covering map and also that
Using the ideas of Bishop and Basener [3, p. 103], we can easily W\Wk
is a negligible set in
Wand
ConHequent.Ly, we conclude
(F-l(W),F. W)
the sense of [5, p. 101].
This proves Theorem 4.
F-l(W k )
is dense in
F-l(W).
is a k-sheeted analytic cover in
REFERENCES 1.
B. Aupetit and J. Wermer, Capacity and uniform algebras, J. Functional Anal. 28 (1978). 386-400.
2.
B. Aupetit, Analytic multivalued functions in Banach algebras and uniform algebras. Advances in Mathematics, vol. 44, No.1 (1982), 18-60.
3.
R. f. Basener, A generalized Shilov boundary and analytic structure, Proc. Amer. Math. Soc. 47 (1975), 98-104.
4.
E. Bishop, Holomorphic completions, analytic continuations, and the interpolation of semi-norms, Ann. of Math. 78 (1963), 468-500.
5.
R. C. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, 1965.
6.
F. Hartogs, Uber die aus den singu1arcn Stellen einer analytischen Funktion mehrerer Veranderlichen bestechenden Gebilde, Acta. Math. 32 (1909), 57-79.
7.
D. Kumagai, On multi-dimensional analytie structure and uniform algebras, Proc. Amer. Math. Soc., to appear.
8.
D. Kumagai, Plurisubharmonic functions associated with uniform algebras, Proc. Amer. Math. Soc. 87 (1983), 303-308.
9.
R. Narc.lsimhan, Several complex variables, The University of Chicago Press, 1971.
10. W. Rudin, Analyticity and the maximum modulus principle, Duke. Math. J. 20 (1953), 449-457. 11. K. RURek, Analytic structure on locally compact spaces determined by algebras of continuous functions, to appear in the Annales Po1enici Mathematici, Vol. XLII. 12. N. Sibony, Multi-dimensional analytic structure in the spectrum of a uniform algebra. Lecture Notes in Mathematics No. 512. Springer-Verlag, (1976), 139-169. 13. J. Wermer, Potential theory and function algebras, Texas Tech. U. Math. Series No. 14 (1981), 113-125. DEPARTMENT OF MATHEMATICS PENNSYLVANIA STATE UNIVERSITY BERKS CAMPUS READING, PA. 19608 4410 Kohler Drive Allentown, PA. 18103
Contemporary Mathematics Volume 32, 1984
CENTRAL FACTORIZATION IN C*-ALGEBRAS AND ITS USE IN AUTOHATIC CONTINUITY K. B. Laursen
1.
INTRODUCTION Suppose A is a C*-algebra,
e:
A~ B a (not necessarily continuous) homomorphism. Suppose that we have information about the continuity of 0 when restricted to certain subalgebras of A. Can we draw conclusions about the continuity of 0 on all of A? A special case of some work of Allan Sinclair [7] will serve very well as an illustration of this: Suppose that C is a commutative C*-subalgebra of A. We may then view A and B as modules over C (and for this purpose we may assume that C has a unit); in A, C acts by multiplication: C x A: (c,a) ~ ca E A, while in B, C acts via the given homomorphism e C x B: (c,b) ~ O(c)b. Clearly A becomes a Banach C-module (since IIcali :: IIclillall), while B is merely a C-module (however, for each c E C, the map
b
~
9(c)b
is continuous on
B is a Banach algebra and
B).
One result that Sinclair obtained may be phrased this way: is an ideal
J, whose closure
property that
e
is continuous on
J
J
C
there
has finite codimension and which has the
is continuous on the subspace C then
in
is a closed ideal.
A. Moreover, if e For reference, let us call
JA
of
this Theorem O. Based on this, Sinclair then showed that the homomorphism continuous on
A precisely when
9
9: A ~ B is
is continuous on every C*-subalgebra
generated by a single hermitian element of
A.
A natural question arises from this: will a smaller class of commutative C*-subalgebras suffice to ensure continuity? Even more boldly: are there cases ~ which continuity on the center ZA is enough to force continuity Dn all of
A?
For a
surp~isingly
large class of C*-algebras (containing all
hence all von Neumann algebras) this turns out to be the case. in [3].
AW*-algebras,
This was shown
Here we shall describe some of this work, with the emphasis on several
improvements found since [3] was written. © 1984 American Mathematical Society 0211-4132/84 $1.00 + $.25 per page
169
LAURSEN
170
2.
CENTRAL FACTORIZATION AND AUTOMATIC CONTINUITY The technical concept involved in this development is very simple.
If
*
I
is a closed ideal (all ideals considered here are two-sided) in a C -algebra
A
then
r
factors centrally
there is an element so that
= zy.
x
z
I" ZI· 1.
if
This means that for every
x E I
an element
y E I
in the center
and
Trivially, an ideal with identity factors centrally; equally
trivially, an ideal
I
thus the compact operators not factor centrally.
Zr
with center K(H)
=
to}
does not factor centrally;
on an infinite dimensional Hilbert space do
We shall get more substantial examples shortly.
The result of Sinclair's that we mentioned before may Je used to prove the following. THEOREM 1.
Suppose that every maximal modular ideal of finite codimension in
the C*-algebra
A factors centrally.
Then a homomorphism
a
a Banach algebra) is continuous if and only if
A~
B
(B
is
is continuous on the
center
ZA.
PROOF.
It is knv\m [3, Remark 1] that the largest ideal of
is Continuous has a closure
9:
e
A on which
K which is of finite codimension in
A.
Con-
K 1s an intersectioll of finitely many maximal modular ideals
s~quelltl'y,
Jl' •••• J n ' each one of which factors centrally. ideals that factor centrally, then seen by first observing that factorization theorem). 11
n 12
II
n
II
12
If
II
and
are closed
n 12
factors centrally; this may be
= 1112
(this follows from the Cohen
Hence
~ 1112
= ZI
1
II ZI
2
= [ZA n (II n 1 2)]1 112 = Zr Here we used that for an ideal
I
1
we have that
be proved by Cohen factorization in
Zr = I n ZA' which may also
ZI.
A straightforward induction then establishes our claim: factors centrally.
12
the ideal
K
Thus
e:
~
ZA. Then Sinclair's result (Theorem 0) shows that 8 is continuous on (K n ZA)A, hence that e is continuous on the closed subspace K, which is of finite codimension. This Now suppose that
proves that
8
A
B
is continuous on
is continuous on
A.
AUTOMATIC CONTINUITY
171
As the other direction is trivial this completes the proof of Theorem 1. Our task now becomes that of finding situations in which cofinite maximal modular ideals do factor centrally.
3.
CONDITIONS FOR CENTRAL FACTORIZATION Recall that
Prim A denotes the set of all primitive ideals of
A,
that is, all ideals which are kernels of irreducible representations of Hilbert space.
[5, p. 92]). THEOREM 2.
We equip
Prim A with the Jacobson topology (cf. e.g.
We then have the following criterion. A is a unital C*-algebra.
Suppose
factors centrally if and only if
Then every closed ideal of
Prim A is a Hausdorff space.
Hausdorff then every closed ideal factors centrally. The space x
We claim that the functions
x E A [6,4.9.19].
are continuous for each
P E Prim At for
are continuous at a point
x E A if and only if the functions
x E P.
We show the converse.
Prim A is Hausdorff precisely when all the functions
(J) : J -+ I/x+JIl : Prim A -+IR
each
A
In [3, Theorem 2 and Corollary 4] it is shown that if Prim A is
PROOF.
A on
4'x
are continuous at
This fact was first noted by Dixmier [2}.
Here
P, for each
is a simple direct
proof. P E Prim A and suppose
Let Let
e
0
:>
be given and choose
is continuous at
P
for every
yEP
N of
= IIx+JII =:
q>
x
{Jlq> (J)
:>
x
subset of
P on which
y
q> (J) < e/2 y
[5, 4.4.4], the set
a-e}
N
(
is open. -1
x
Consequently,
(la-e ,a+e [), containing
P E Prim A, for every
P E Prim A and
Obviously
the function
chosel~.
for each
n
{J1 ar-e}
x
is an open
P. A factors centrally then we
Prim A is Hausdorff by proving that all the functions
are continuous at Let
Suppose q>y
we have
1s always lower semi-continuous
may prove that
q>
x E P.
let
x Moreover, since
Write
x
= zy,
where
x
q> (J )
x
z E
a
~
and
x E P, whenever
=: q> z (.1 a ) Ilyll, to show that q> x (.I a ) -+0 it
a q> (J ) -+ O.
z a
q>x
x E P.
q> (J) = IIx+J/I =: IIz+JlIl/y+JII =
(J). Since x z y is continuous at P if and only if
-+ P. a suffices to show that
J
y
O.
(J) < a + e.
If we suppose that every closed ideal of
yEP.
IIx+yl/ < a + e12.
:>
Ilx+y+JII + Ily+JII
=: /Ix+Y/i + Since
so that
x
J E N we then have
For any such q> x (J)
q> (P) = /Ix+plI - a
Then
yEP, hence also for the
Then there is an open neighborhood J E N.
x E A\P.
Note, however, that
z E P
and that the
172
LAURSEN ~z
Dauns-Hoffman theorem tells us that Ja. ~ P
Hence, if REMARKS.
z
a.
z
(P)
= O.
Prim A
[5, 4.4.8].
This completes the argument.
The implication proved here evidently does not require the presence
*
of a unit. Prim A
~ (J ) ~ ~
then
is coatinuous on
On the other hand, any non-unital simple C -algebra
= {to}},
which is trivially Hausdorff, yet the
not factor centrally, since With Theorem 2 at hand
improper ideal
A does
(by Dauns-Hoffman).
have a rich supply of examples of algebras in
~'ie
which ideals factor centrally.
to}
~
ZA
A has
For instance, in a unital C*-algebra with
continuous trace any closed ideal factors centrally. On the other hand, for Theorem 1 the assumption of Hausdorff 1s clearly much too strong. offer the following result.
ce~trally
if and only if
A every maximal .modular ideal factors
A is strongly semi-simple and weakly central.
The appropriate concepts are of
A is a subset of
n{MIH E .pA} = to} HI ,H 2 E .pA)
As a step in the right direction we
Terms used are defined presently.
In a unital C*-algebra
THEOREM 3.
Prim A.
The set
The algebra
and is weakly central if
implies that
PROOF OF THEOREM 3.
Prim A being
.pA
of maximal modular ideals
A is strongly semi-simple if HI n ZA = M2 n ZA
= M2 •
HI
For the proof of the fact that if
simple and weakly central then each maximal
mo~ular
A is strongly semi-
ideal of
A factors
The converse may be proved this way:
centrally we refer to [3]. First suppose that
(where
and
M2 are maximal modular and that Ml n ZA = H2 n ZA· Then ZM = Z.1 ' so that if Ml or M2 factors centrally I 2 1 then MI = H2 (by maximality). Consequently, A being weakly central is HI
necessary for central factorization in maximal modular ideals. We still have to show that if all ideals then if
A must be strongly semi-simple. R
= n{MIM E
because ZR
=
~A
R n ZA
=
A is unital [6, 2.6.5]).
= fO}, because ZA is R = R·ZR = to} and this
A is strongly semi-simple.
tion.
However,
a commutative C*-algebra is exactly the claim
This completes the proof of Theorem 3.
We are left with the task of showing that if each point of
centrally,
In the lemmas below it is proved that
n{(W1ZA>1M E .pA} Thus
factor
R factors centrally (Lemma u, which applies
is compact when
and thus semi-simple. that
then
M E ~A
F has central factorization then
F
C ~A
ker F
is compact and
has central factoriza-
AUTOMATIC CONTINUITY LEMMA 4. ideal
Suppose
Po
of
P
P E ~A has central factorization.
Then the Pedersen
is characterized as
Po - {a E pl~a PROOF.
173
vanishes on a neighborhood of
The Pedersen ideal
Po
of
P
P
in
Prim A}.
is described as the minimal dense ideal
in [5, p. 175].
If we call the ideal on the right hand side (above)
J
suppose that
is not dense in
Q
such that
J
Q
~
P
then there is a primitive ideal
O.
of
~x
NI - {I E Prim AI~ x (I) >
a2}
is open; by continuity of
~
x
at
contains an open neighborhood of and 'a - I J
P.
Let
maximal so
(see the proof of Theorem 2),
P
P.
We may then select
x E A and
~
is closed, hence, by [6, 2.6.9]
s-sy, s-ys E ker F
.!!
LEMMA 5 •
P E 4'A
for every
sEA.
factors centrally and
vanishes in a neighborhood of x
= xz
and
PROOF. ~z
x
~z
vanishes in a
If we can find
wE
x = yw (suitable
P with
E A
~
with
In particular
zp
x E P
x
= yx.
=0
a
neighborhood of with
P
in
x-yx E ker F.
This places
z E Zp
x
~x
for Wilich
Prim A.
vanishing in some neighborhood of
~w
yEP) then it is easy to see that there is
vanishing in a neighborhood of
P is yEP
is chosen so that
in Prim A, then there is
P
The ideal
there is an element
vanishes off
x
a
vanishes on some open neighborhood
x
F, this means x-yx = 0, or in the Pedersen ideal [4] and proves Lemma 4.
Since
and let
the set
F be the complement of this open neighborhood. {P}
for which
x E p\Q
Choose
on N1 [6,4.9.16]. For this element a we have tha t must be dense-in p and since is minimal dense,
Conversely, suppose of
P
'"
By lower semi-continuity
Thus
~
(in a C -algebra every closed ideal is the intersection of
J
the primitive ideals containing it [6, 4.9.6]). ~ = ~x(Q) >
and
P, for which
w
D
wz
z E Zp'
and hence
= ywz = xz. The Pedersen ideal
where
0
~
x EP
and
compactly supported in
f
Po
is generated by elements of the form
is a non-negative, real-valued continuous function,
]0,-( [5, p. 175].
ization claim for all suchelements.
If
and compactly supported real functions elements
Yl""'Yn in
f(x),
P so that
0
Suppose we have proved the factor~
x E Po
then tbere are continuous
f 1 , ••• ,f n : ]o,~[ ~ R+
and positive
174
LAURSEN
(where
and
qlz
vanishing on some neighborhood of i
E Zp with qlz vanishing on a neighborhood of P such that zi = tiz (suitable t i ) and thus x ~ (lxiti)z. The generalized polar decompositi.on [5, 1.4.5] can then be used to complete the argument. So it remains to prove that for an element f(x) (with x and f as specified before) we can find a central element z for which
By Cohen factorization we can then find
Z
continuous, non-negative real valued, compactly supported functions on suppose
f
~
and suppose
g
functional calculus
Po
0
~
g
increases on the support of
f.
E Po
g(z)
and hence
Let
x
~
0
be an element of
ker F
continuous and compactly supported on
P E F so there is an element neighborhood of P for which f(x)
every
P
= f(x)zo
J
for
j
= l, ••• ,n,
vanishes on a neighborhood of
f
be non-negative, f(x) E Po
The element
Note that
a generating element for the Pedersen ideal that
(ker F)O
factors centrally.
for
z E Zp with
and so that if
F.
Prim A.
in
many such nelJhborhoods, i.e. select central elements f(x)
from
F factors centrally, then
nnd let
]o,~[.
f(x)
Note that
vanishes on some neighborhood of
LELiMA 5. If F C ~A is compact and each point of ker F factors centrally. PROOF.
Then by the
f(x) ::: g(z), so that a central factor of
has been found (apply generalized polar decomposition).
]O,~[;
Thus
f{x)
then =
f(x}z.
(ker F)o
of
ker F • Zker F
Since ker F
is
~ense
f(x)
is
this shows in
ker F.
But by the module version of Cohen factorization we know that ker F • Zker F is closed, thus equals
4.
ker F.
This proves the lemma.
MORE AUTOMATIC CONTINUITY Evidently Theorems 2 and 3 give us classes of C*-algebras to wnich
Theorem I applies. further.
However, via the decomposition theory we can go a bit
Recall that an AW'*-algebra is a C*-algebra in which
ea~h
maximal
commutative C'*-subalgebra is monotone complete, which means that each bounded increasing net of hermitian elements in the subalgebra has a least upper bound (in the subalgebra).
[5, 3.9].
Every von Neumann algebra is an AW '*-algebra
AUTOMATIC CONTINUITY
175
We then have the following result. THEOREM 4.
A be an AW *-algebra,
Let
homomorphism.
B a Banach algebra and
6: A
B a
~
e is continuous if and only if a is continuous on the
Then
center
ZA.
PROOF.
By the decomposition theory for AW *-algebras there is a central projec-
tion
z
n
e
so that
is
continuous on
(l-z)A n
and such that
algebra which is finite of type I [1, Theorem 4.2].
z A is an
A is an AW *' -algebra of finite type
generality in assuming that
n
Thus there is not loss of
I.
Such an
algebra is strongly semi-simple lB, Theorem 2.7] and since any AW *'-algebra is weakly central [8, Theorem 2.5], Theorem 3 applies: if e is continuous on ZA
then
e is continuous on A.
COROLLARY 5.
Every homomorphism defined on a factor AW *' -algebra is continuous.
PROOF.
a
REMARK.
A is
factor if
ZA
is one-dimensional.
Garth Dales points out that there are examples of C*-algebras
discontinuous homomorphisms whose restriction to
ZA
is continuous.
A with This
illustrates the ljmitations of the above approach (and it shows caseS where non-trivial central factorization is not possible). infinite-dimensional, let C
= C([O,l])
I
In
H is
B(H), where
be the ideal of compact operators on
H and let
be the continuous functions on the unit interval, (via a
hermitian operator with spectrum C*'-subalgebra of
B(H».
is not hard to see that
Then ZA
=~.
[0,1]
we may tnink of
C as a unital
A = 1$ C is a C*-algebra [5, 1.5.8], and it On the other hand, since
C is a quotient of
A on which discontinuolls homomorphisms are definable we obtain discontinuous homomorphisms of
A [1, §2]. REFERENCES
1.
E. Albrecht, H. G. Dales, Continuity of homomorphisms from C*'-algebras
Points slpar~s dans le spectre d'une c*-alg~bre, Math. 22 (1961), 115-128.
2.
J. Dixmier,
3.
K. B. Laursen, homomorphisms,
4.
G. K. Pedersen, 131-145.
5.
G. K. Pedersen, C*-algebras and their automorphism groups, Academic Press, London, 1979.
6.
C. Rickart, Banach algebras, Van Nostrand, Princeton, 1960.
Acta
Sc.
Central factorization in C*-algebras and continuity of J. London Math. Soc., (2) 28 (1983), 123-130. Measure theory for C*-algebras,
Math. Scand. 19 (1966),
176
LAURSEN
7.
A. M. Sinclair, domomorphisms from C*-algebras, Proc. London Math. Soc. (3) 29 (1974), 435-452, Corrigendum 32 (1976), 322.
8.
F. B. Wright, A reduction for algebras of finite type, (1954), 560-570.
MATHEMATICS INSTITUTE
UNIVERSITY OF COPENHAGEN UNIVERSlTETSPARKEN 5
2100 COPENHAGEN DENMARK
Ann. of Math. 60
Contemporary Mathematics Volume 32,1984
NONSTANDARD IDEALS AND APPROXIMATIONS IN PRIMARY WEIGHTED t I_ALGEBRAS
J. P. McClure*
INTRODUCTION
Elsewhere in this volume, the development of knowledge of the ideal struc1
ture of primary weighted t -algebras has been surveyed by W. G. Bade [1]. References [3] through [10] contain work on this subject, and in some cases on its relationship to questions about the invariant subspace lattice of a weighted shift operator.
The most recent major result is the construction by
M. P. Thomas [10] of weights for which the associated tl-algebra has nonstandard ideals.
The present paper is intended to complement Bade's [1] by giving
an explicit example of a weight similar to those considered in [10]. a property of the associated
e1 -algebra
We prove
which is necessary for the existence
of nonstandard ideals, and draw attention to a general approximation question 1
about weighted t -spaces which is relevant to the nonstandard ideal problem. We do not know whether the algebra constructed here actually has nonstandard ideals; the weights in [10] are more extreme, and the estimates there more difficult, than those given here. It is a pleasure to thank Professor Sandy Grabiner for some interesting conversations about nonstandard ideals and related topics, and to thank the organizers of the Yale conference for the opportunity of presenting this work to Professor Rickart.
SOME BACKGROUND
We consider algebras t 1 (w) .. {f • (fn ) : II fll •
where tBR coefficients
*Supported
f
n
(n
= 0,1, ••• )
1:1 f n Iwn <
-} ,
are complex, and the weight
by NSERC Grant A8069 ~ 1984 American Mathematical Society 0271·4132/84 S1.00 + S.2S per page
177
178
MCCLURE
(sequence)
W
=
(wn)
is a positive sequence satisfying
m+n ::: K wmwn
W
w
lin n
0
-+
as
for some n
and all
K > 0
m, n;
-+ ....
The first of these conditions guarantees that
el(w)
is a Banach algebra with
respect to coefficientwise linear operations and the convolution product, defined by (f * g)n = l:i+j=n fig j • The second condition implies that el(w) is a primary algebra, the unique maximal ideal being {f:fO = O}. See [2]. Weights satisfying the first condition are called algebraic; those satisfying the second are radical. If we identify f = (f ) in elew) with the formal power series Lf zn n n ' then e1 (w) becomes a subalgebra of the algebra of all complex formal power
series ,GUZ]] • The order of a non-zero series f is order of the zero series is'" For n ~ 0, we write
min{ n : f
for the set of elements of
n.
that each M
n
el(w)
is a closed ideal in
with order at least t lew) .
any other closed ideal is nonstandard. in el(w)
:/: O} ; the n or just M, n
Mn (w),
It is easy to see
These are the standard ideals, and
A weight
w such that all closed ideals
are standard is called unicellular.
It is we11-knoWll that a non-zero closed ideal in tl(w) only if it contains
Zn
for some
non-zero element of order
k.
n ~ 0
[4, Lemma 3.2].
is standard if and
Thus, if
the closed ideal generated by
f
f
is a
will be stand-
ard if and only if there are a non-negative integer M, and a sequence nomia1s, such that /I p * f - zk+~1 -+ 0 as m -+ GO. (A)
{Pm}
of poly-
m
For the given series
(of order
£
k),
let
Z-k
*
f
denote the series of
Zk * (Z-k * f) = f, and let F denote the inverse of (The sequence of coefficients of F has been called the associated sequence by M. P. Thomas [8].) The first known sufficient conditions for unicellularity gave the stronger result that for each non-zero f in tl(w), ZM * F belongs to tl(w) for some M [3], [6, Sec. 3.2]. In such cases, the ideal generated algebraically by f already contains a power of Z; also, (A) will hold with p = ZM * S (F) where S (F) = ~mO F.Zj is the m m' m L. J order 0 defined by Z-k * f in £f[Z]].
partial sum of degree
m of
there are unicellular weights
ZM
*F
belonging to
that his weights
~
some subsequence of
e1(w)
F.
More recently, M. P. Thomas has shown that w such that tl(w) contains elements f with
for no
M [8], [9].
unicellular by showing that {ZM
* Sm(F)}.
In these cases, Thomas shows (A)
holds with
This is not surprising, for if
{Pm} (A)
PRIMARY WEIGHTED t1-ALGEBRAS
holds, then
p
r- * F
converges coefficientwise to
m
179 as
m -+
and r}4 * F appears to be unknown, and in itself seems an interesting
stronger relation than coefficientwise convergence between is implied by
(A)
Whether any
110.
{Pm}
question. In the next section, we construct a weight one
in
M ~ O.
"zH * Sm(F) * f
such that
t I (w)
wand an element
- ZM+l" -+
lID
as
m -+
lID
f
of order
for each
m = iM * Sm(F), or any subsequence Given a positive answer to a strong enough version of the question at
Thus
thereof.
fails, at least for
(A)
p
the end of the preceding paragraph, we would have an example of a nonstandard ideal somewhat simpler than that in [10].
THE CONSTRUCTION Suppose w is a radical, algebraic weight, and
w,
translate of
determined by
vn
~
wn + l
for
n
v
=
is the first left
0,1 ••.••
If
v
is al-
Z * F E tl(w) for any f of order one in t 1 (w). Thus, in order for some element of order one in t1(~) to generate a nonstandard ideal, it is necessary that v not be algegebraic, then it is also radical, and it follows that
braic.
Our first step will be to construct
is not algebraic, but the weight n
= 0,1, •••
defined
w
= (v) by induction so that n by Wo = I and wn+l = vn for
v
v
is algebraic.
va = 1, and put An vn = AO" .An_l for all n~ 1. Set Bn is algebraic if and only if the sequence We shall take
= Vn+1/vn ,
for
= max
.
n=O.l, ...•
so that
i + j .. n} • Then {vn/vivj B .. {B } is bounded. Also, n
v
A
n-2
w is algebraic if and only if the sequence
Thus,
Define a subsequence m(l)
Vj
= AO"
.A j _l
= 1,
and
vo" 1
Now put
{m(k)}
is defined for
m(k + 1) = 2m(k) + 2 If
A.
j ~
m(k»
J
= l, .•• ,k,
we then put
is bounded.
(k
~ 1)
has been defined for
(1)
j <:
m(k)
(hence.
so that
_ 2- im (i)
i
n n
of the natural numbers by
vm(i) for
{B A }
(2)
180
MCCLURE
Am(k) = 1, (0 !: j <
This determines
(3)
and implies
o !: n With
}
m(k».
n - m(k) - I,
(4)
!:
m(k)
(4)
gives v
m(k)-l vm(k)
= n - m(k)
while
(5)
leads to V
=
2m (k) + 1
(6)
(Vm(k») 2 •
Finally we require vm(k + 1)
_ 2-(k + l)m(k + 1) •
(7)
Then (1), (2), (6), and (7) imply
_
2-(k + 1) (2m(k) + 2)
=
2-2km(k) + (-2m(k) - 2k - 2)
= vm(k) =
Am(k + 1) _ 1
Thus
v
vm(k + 1)
2-2m(k) - 2k - 2
= 2- 2m (k) - 2k - 2
and the associated ratios
Clearly,
1
is determined:
A m(k + 1) - 1 and
2 2-2m(k) - 2k - 2
{Am(k) _ I}
{A} n
(8)
'
are constructed by induction.
is a decreasing sequence, so that (3) implies (9)
Together, (5) and (8) .how that braic.
{B} is unbounded, so that v is not algen On the other hand, the following lemma and an earlier remark show that
w is algebraic. LEMMA 1.
With
Because of (2), v,A,B
as defined,
w is also radical. Bnn A
~
1
for all
n.
181 PROOF.
S1nce BO· I
and
Ao·
-1
2
,
the claim 1s true if
n - O.
Moreover,
-1
vi+j/viVj 1 =: (~+j) whenever i = 0 or j - 0; such cases are to be excluded from the following argument. Suppose k ~ l, and the claim is true for n < m(k). Let 0 =: r =: m(k) + 1, and consider II:
Vm(k) + r Vm(i) + p Vm(j) + q where m(i) + p + o !: q !: m(j) + I, m(1) + p + m(j) + We consider three (i) Suppose
m(j) + q ... m(k) + r, m(j) + q ~ m(i) + p, 0 ~ p ~ m(i) + 1, and I!: j !: i. Clearly i =: k, and if i < k - 1, then q < 2m(k - 1) < m(k). Thus, i must be either k or k-l. cases: i = k. If p = 0, then m(j) + q - r, so (3) and (4)
iJIIP1y Vm(k)
-
+r
=
(Ar_l)-l
=
('X.m(k) + r)-l
So we can assume p > O. If also and r · m(k) + 1. Now (4) and V2m (k)
So we can assume
+1
j < k.
j - k, then necessarily (6) give
p
= 1, q. 0,
=
Then
= by (4). S1nce j ~ I, P < r, and therefore p - 1 < m(k). The required inequality now follows, either from the induction hypothesis and (3), if r - 1 < m(k) ,
or from one of cases (ii) and (iii), below.
(i1) Suppose 1 = j = k - 1. Then (1) implies p + q - r + 2. Therefore, p =: m(k-l) + 1 and q!: m(k-l) + 1 imply r!: 2m(k-1) - m(k) - 2. Note also that p> 0, since m(k-l) + m(k-l) < m(k). So
Vm(k-l) + P vm(k_l) + q
=
~
m(k)-l
vr-l v v p-l q-l
182
MCCLURE
•
v
\n(k) -1
v
r-1 Vr
r
v p _1 v q _l
\n(k)-l (Ar_1) -1 (Ar)-l uSing (5), the induction hypothesis and the fact that Since r < m(k) , holds. (iii)
Suppose
i
j <
2m(k-l) + 2 + r,
so that
m(k-1),
m(j)
so that
As before,
p >
Am(k)-l (Ar )
(9) implies
O.
~
1,
p - 1 + q - 1 • r < m(k).
and the required inequality
Now m(k-l) + p + m(j) +
k - 1.
a
-1
m(j) + q + P
= m(k-l) +
+ q + P < 2m(k-l) + 1,
2 + r.
q =
Also,
and therefore
m(k) + r m(j) + q <
r < m(k-1) - 1.
Therefore,
Vm(k) + r vm(k-1) + P vm(j) + q
=
=
=
Am(k)-l v m(k-l) v r-1 v v p-1 m(j) + q A.m(k)-1 vm(k-l) + r
v
p-l vm(j) + q Vm(k-l) + r
\n(k)-1
Vm(k_l) + r + 1 vm(k-1) + r + 1 vp _ 1 vm(j) + q
( A.m(k-l)
+
r
)-1
= because of (3), (9), and the induction hypothesis.
That completes the proof
of the lemma. Now we define
Setting
zO
$l(v)
as follows:
go
-
1
gn
=
0
'm(k)
=
- (k 2 vm(k)~-1 ,
g E
= land
y
=1
- g,
if
n ,. m(k) ,
we have
g
n
~
1
k - 1,2, ••••
=1
- y,
)
(10)
and the formal power
series inverse of g is G = 1 + Y + y2 + Note that each coefficient of G is a sum of positive terms, and the coefficient of Zm(i) + ••• + m(j) includes the term
Ym(i)"'Ym(j)
(where
Ym(k)
=
-Sm(k)
for all
k ~ 1).
PRIMARY WEIGHTED t 1-ALGEBRA5
Recall that f E t l(w), where
= wn+1 ,
= 0,1, . . . .
n
So, 1f we put
and the closed ideal generated by
f
f
=Z *
in t l(w)
g,
then
is just
Z
*
I
g' is the closed, (right) translation-invariant subspace generated by
I
g.el(v).
elements of
In fact, the map
*
h : .el(v) -+- ,el(w) is an isometry, 1 and the various arguments that follow concerning the t (v)-norms of various g
in
vn
183
el(v)
h-+- Z
transfer readily to the corresponding elements of
Recall the notation
Sm(g), Sm(G),
tl(w).
etc. for the finite partial sums of the
formal power series g, G, etc. In the sequel, we need to work with certain "tails" of g, and it will be convenient to write (k) for Zm(i) =
,,00 ~i=kgm(i)
g
g - Sm{k)_l(g).
LEMMA 2. 1/ a
*
zj
PROOF:
We shall use the following easy lemma.
Suppose a is a complex number, and j ~ m(k) + 1. Then g (k) 1/ = I a I v. 1 2 -k+1 • JIt 1s clear from the definition of g that IIg{k)1I = 2-k+ l ,
k = 1,2, ••••
Since
/!a
*
zj
g(k)"
=
lair::=k l gm{!) , vm(i) + j '
for
the lemma
follows, using (4). We now prove our main result, which shows that the polynomials are intuitively the likely candidates in order to make power of
Z in
THEOREM.
For each non-negative integer
IIZM
*
Sm(G)
PROOF:
*g
el(v) - z~1
For positive
*
g
m -+
""
p
p
which
approximate a
actually fail to do so.
= IlzM * k
*
(Sm(G)
M, g -
and non-negative p(k,M)
= m(k+M)
1)!1
-+
M,
define
co
as
+ m(k+M-l) + •••
+ m(k+l) + 2 m(k) .
(11)
Using (1), observe that p(k,m) = Now fix unique
M. k
Civen j
satisfy
m(k+~~l)
(12)
- 2(M+l).
Then, for each sufficiently large positive integer such that m,
we fix
p+m(j)
~
p(k,M) k
as above.
= p(k,M)
Clearly,
k
-+
GO
as
there i.s a m -+- "".
Suppose the non-negative integers
+ m(k+2+M).
p(k,M) + m(k+2+m) < m(k+3+M), j < k + 2 + M,
m < p(k+l,M).
m,
By (12) and (1).
so that necessarily
j ~ k + 2 + M.
then p
= p(k,M) ~
+ m(k+2+M) - m(j)
m(k+M+l) - 2(M+l) + m(k+2+M) - m(k+l+M)
= m(k+2+M) =
p(k+l,M).
- 2(M+l)
If
p
and
184
MCCLURE
Since
Sm(e) has degree m< p(k+1,K), Zp(k,H) + m(k+2ifQ can occur in S (e) m
Gp (k,H) gm(k+2+M)
• shows that for any
Sm(G)
*g
- 1
*
g - 1
is with coefficient
similar argument, with . i +.M in place of k + i ~ k + 2, the coefficient of zp(k,H) + m(i+H)
A
Gp(k,M)~(i+M)'
is
II tt
the only way a term involving
*
(Sm(G)
2
+ in
H,
Therefore,
* g . - 1)11
~ II tt *
(ep(k,K) Zp(k ,M) )
:; II G
p(k,K)
Zp(k,K) +
H
* g (k+2+M)"
* g (k+2+M)"
•
Using (12), we see that p(k,M) +
M
= m(k+1+M) - K - 2
< m(k+2+M), SO
we can use Lemma 2 to conclude that
II ZM *
(S m(e) G
>
-
*g
p(k,M)
- 1)"
v
2-(k+2~)
p(k,M) + M - 1
+ 1
(13)
Now, using (10) and Ym(j) • -gm(j),
G
p,(k.1O '"
'=
(i~l (k+1) Ym
(n ~-(k+i»)
Y;(k)
2-2k
i=l
(v2
(14)
\m(k)
Also, using (4) repeatedly, and (11), v p (k,1O + K - 1
M ) v (V!(k) i n ... 1 m(k+i)
-1
vm(k)-2
.. - - ' - < - - -
)-1
-- (~m(k)-2 ~m(k)-l
Finally, using (3), (8), (13), (l4) , and (15), we obtain "ZM
*
(S
(e) m
*
g _ 1)" ~ 2q (m) ,
where q(m)
= -(M+3) (k + 2M + 1) + m(k) + m(k-1) + 4k-6
(15)
PRIMARY WElalTED t1-ALGEBRAS (recall that k> 2
k
is determined by
(by (I)), we have
gem}
~
-
m). as
185
Since M is fixed, and m~·,
m(k) > 2k
for
and the theorem is proved.
To conclude, we remark that a slight modification of the above construction gives a weight
w with the additional property that
wn+1 I wn
~
0
as
and the conclusion of the theorem holds for a suitably chosen element. calculations involved are similar to those above, but messier.
n
~
.,
The
Since all known
sufficient conditions for unicellularity either include or imply wn+k I wn ~ 0 as n~ - for some k. it would be interesting to have an example of a nonstandard ideal for such a weight. REFERENCES 1.
W. G. Bade, Recent results in the ideal theory of radical convolution algebras, these Proceedings, 63-69.
2.
I. Gelfand, D. Raikov, G. Shilov, Commutative normed rings, Chelsea, New York, 1964.
3.
S. Grabiner, A formal power series operational calculus for quasi-nilpotent operators, Duke Math. J. 38 (1971), 641-658.
4.
S. Grabiner, A formal power series operational calculus for quasi-nilpotent operators, II, J. Math. Anal. Appl. 43 (1973), 170-192.
5.
S. Grab1ner, Weighted shifts and Banach algebras of formal power series, American J. Math. 97 (1975), 16-42.
6.
N. K. Nikolskii, Selected problems of weighted approximation and spectral analysis, Proc. Steklov Inst. Math. 120 (1974), A.M.S. Translation, Providence, R.I., 1976.
7.
M. P. Thomas, Closed ideals and biorthogonal systems in radical Banach algebras of power series, J. Edinburgh Math. Soc. 25 (l982), 245-257.
8.
M. P. Thomas, Closed ideals in tl(oo) when {oo} is star-shaped, Pacific n n J. Math., to appear. M. P. Thomas, Approximation in the radical algebra t1(OO} when {oo} is n n star-shaped, Radical Banach algebras and automatic continuity, Proceedings Long Beach Conference, 1981, Springer-Verlag, Lect. Notes in Math. No. 975.
9.
10.
M. P. Thomas, A nonstandard ideal of a radical Banach algebra of power series, preprint.
DEPARTMENT OF MATHEMATICS UNlVERS ITY OF MAWITOBA WINNIPEG, MANITOBA, CANADA R3T 2N2
Contemporary Mathematics Volume 32, 1984
HOLOMORPHIC APPROXIMATION IN LIPSCHITZ NORMS '* A.G. O'Farrell, K.J. Preskenis, and D. Walsh
1.
INTRODUCTION For basic material, see [6,7,11,18,23]. Let
X c ¢n
O(X)
be compact, and let
denote the space of complex-
valued functions, ho10morphic on a neighborhood (depending on the function) of
X.
In order that
O(X)
continuous functions on (i.e. that
be dense in
C(X), the uniform algebra of all
X, it is necessary that
X be holomorphically-convex
X coincide with the set of nonzero algebra homomorphisms of
O(X) ~ ¢), and have no interior. It is also necessary that X contain no nontrivial (i.e. positive-dimensional) analytic subvariety of Cn, and, for this reason, efforts to derive sufficient conditions have centered around the study of totally-real sets. a neighborhood of
¢n
N in
en
A set A C In such that
having no complex tangents.
is totally-real if each point has AnN is a subset of a Cl submanifold
For locally-compact
as saying that each point has a neighborhood C2
A, this is the same
N on which there is defined a
nonnegative strictly plurisubharmonic function, vanishing precisely on
AnN
[9).
Naturally, it is far from necessary that order that
O(X)
be dense in
C(X).
X be totally-real, in
Having a few complex tangents is a
long way from containing a nontrivial analytic variety.
In one variable,
where Vitushkin [20, 6] has completely solved the problem, there are examples of sets X whose Cl tangent space (the space of bounded point derivations on the quotient of the Whitney algebra 2
at each point, whereas
O(X)
Cl(X)
is dense in
by its radical) has dimension C(X).
Thus, one is led to
conjecture that not only uniform, but "better than uniform" approximation should be possible on totally-real sets, and that one ought also to be able to handle sets having modest "singular subsets", on which they are not totally-real.
'*Dedicated
to C.E. Rickart on the occasion of his retirement.
187
© 1984 American Mathematical Society 0271-4132/84 51.00 + 5.i5 per page
188
O'FARRELL, PRESKENIS AND WALSH Range and Siu [17] proved that i f
submanifold, then
o (X)
is dense in
Ck
X is a
totally-real bordered
Ck(X).
See also [5,8,10,12,19,23]. If, however, the manifold has even one complex tangent, then Cl approximation fails, for obvious reasons.
This suggests that for
X having occasional
complex tangents one could profitably look at the Lip (a ,X) norms (0 < a < 1), which interpolate between the uniform and c l norms. For Lipschitzian graphs .:2, two of us proved [141 that if th~ set of points where
X in
X has
complex tangents has (Hausdorff) area zero, then uniform polynomial approximation implies
Lip
a
polynomial approximation.
~2n,
considered polynomially-convex graphs in exceptional set
In [15], we
totally-real off a closed
E, and we showed that cloS Lip (a ,X)O (X) = lip (a ,X)
n closLip (a ,E)O (X) •
(1)
Our present purpose is to extend this result to cover general holomorphically-
X.
convex sets THEOREM 1. E c X N
In
Let the compact set
X c ~n
be holomorphically-convex.
a E X E has a neiihborhood N is a subset of a Cl submanifold having no
be closed, and suppose that each point ~n
Xn
such that
complex tangents. The space
Then (1) holds for Lip(a,X)
0 <
lip(a,X)
has the norm x,y E X, x
+y}
,
is the closed subspace in which If(x)-f(y)1 sup Ix-yla o < Ix-yl < 5
as
N
a < 1.
sup If I + sup { I f (x)-f (ylL X Ix-yla and
Let
-+
0
8 J. O. In case
X is a bordered submanifold and
E
is empty, the hypothesis
that
X be holomorphically-convex follows from the other hypothesis [8]. This case of the theorem follows from' the Range-Siu theorem, since cl convergence implies
Lip a
convergence on nice sets.
The compact sets which are intersections of (Euclidean) Stein neighborhoods form a proper subclass of the holomorphically-convex compact sets. are called holomorphic
~.
They
In general, a holomorphically-convex set is
an intersection of projections of Stein Riemann domains [2].
A sufficient
condition for
X to be holomorphic is that it be rationally-convex. Another sufficient condition [8, 10] is the existence of a C2 strictly plurisub-
harmonic function
p
on a neighborhood
W of
bdy X such that
HOLOMORPHIC APPROXIMATION
x n W=
{p < o}.
(Note that the interior of
189
X is not assumed empty in
Theorem 1.) We prove the theorem by using duality, combining the method of Berndtsson [1] with the technique of [15].
We remark in passing that
Berndtsson's method also proves the analogue of Theorem 1 for uniform approximation.
Weinstock (e.g. [22]) has proved some cases of this theorem.
The statement is as follows. THEOREM 2. subset of
Let
X be a holomorphically-convex set, let
X, and let
X - E be totally-real. =
closC(X) O(X)
C(X)
be a closed
E
Then
n cloBC(E) OeX).
(2)
This result is also implicit in the constructive work of Henkin and Leiterer [10], but the duality proof is simpler.
Of course, Theorem 2 is a
corollary of Theorem 1. 2.
PROOF OF THEOREM 1
Lip(a,X)
T E Lip(a,X)
tic
O(X). In the same way as in [15] it suffices to show that the distribution Tlc~ is supported on E. Briefly, 00 this reduction depends on three facts: (I) C functions are dense in lip(a,X), (2) there is a continuous extension operator from lip(a,E) to lip (a,X) , and (3) if a lip(a,X) function vanishes on E, then it is a Let
annihilate
lip(a,X)
limit of
functions which vanish on a neighborhood of
Thus it suffices to show that each point en such that T~ = 0 wherever ~ E Co
CD
a EX'" E has a neighborhood
E. in
U
u.
has support in
a E X - E, and choose a neighborhood N of a such that X n N is a subset of a Cl submanifold M having no complex tangents. Following Fix
Berndtsson [1], construct kernels is a neighborhood of
a
and
K(~,z)
and
...
K(~,z)
W is a neighborhood of
on
X.
u
x
W, where
U
Note the following
points:
(1) For our present purpose, the set V should be chosen a neighborhood of Cr
n M,
not
Cr
n X.
holomorphic hull
R
Next,
D should be a neighborhood of
X whose
(which is a Riemann domain) has projection nCR) c en,
Cr - V. This is possible, because a holomorphically-convex X has a sequence of neighborhoods Dn ~ X such that the projection
disjoint from set n (R ) n
of the holomorphic hulls
problem should be set up on open sets (2)
{Inl
< 2r},
{Inl
R
Rn of Dn shrink to X. Then, the Cousin instead of D, using the covering by the two
> r}.
Berndtsson refers to Ovrelid [16] for
functions on
~.
C1
dependence of the various
However, Ovrelid refers to Hormander and Bungart.
There
190
O'FARRELL, PRESKENIS AND WALSH
are (at least) three published proofs of the desired facts (solubility of Cousin and related problems with smooth dependence on a parameter) - by Bishop [3], Bungart [4], and Weinstock [21]. is the most elementary. product theory. (3)
Of the three, Bishop's method
The others use the powerful Grothendieck tensor
Berndtsson's function
H has Weinstock's "omitted sector property", Le.
for each ~ there exists 6 such that H~ .z) takes no value in the sector {w E I: : 0 < I wi < 6. I Imwl + 6 Rew < O}. (He also needs this fact, to establish the relation O
J
...
K~,z)d~z)
"J
=0
on page 125.) Once the kernels are constructed, we proceed as 1n [15].
Since
,
K is
a Cauchy-Fantappie kernel, we get T
= T
z
f K~,z)
A
aq>«:)
U
aD
whenever
€ C
o
has support in
on
lip (a,X)
of
X x X, as in [15]. we may write
U.
by means of a measure
~
T
on the set of off-diagonal elements
'l\P
as
Ix-yl-cI J {K(~,x) - K~.y)}"
! Xx X
a
U
Now there exists a constant
J
Using the DeLeeuw representation of
HI < 0
such that for all
{K(~,x) - K~ ,y)} " oo~) I
I
x,y E X,
5 MIl x-yl a
110011.
U
for all
(0,1)
forms
00
having bounded measurable coefficients, where
denotes the total variation of the in [15].
(n,n)
form
v
•
This is proved just as
This estimate allows us to apply Fubini's theorem to write
J
!
Ivl
Ix-yl-a{K(~,X)-K(~,y)} d~(x,y)"
~
as
~~).
U XxX
2n L
It remains to show that the inner integral vanishes for all"
and for this it suffices to show that for almost all
exist sequences of functions in K(~tz)
...
in an appropriate way. for
K - K for (m
O(X)
~
almost
there
approximating the coefficients of
z E X.
This is done by noting that
z E X, and (using the omitted sector property) using
a sufficiently large integer) in the denominator of
The details go through in the same "manner as in [IS].
i,
-1
H+ m
instead of
H.
HOLOMORPHIC APPROXIMATION 3.
191
EXAMPLES
(3.1) Let
A denote the truncated cone {(re
and let
i9
t re
319
X
~
r ~ 2, 0 ~ 9 ~ 2TT},
B denote the torus
Izi = Iwl = I}
{(Z,w) E.:2: If
"21
)
=AU
Bt and
(zOtWO) E
~2 - X, then at least one of the polynomials
3 3 z-zO' w-w O' zwO-zOw, z wO-zOw convex.
The set
E
=An
.
is nonvanishing on
B is the curve
X.
Thus
{(e i9,e3i9)} •
X is rationally
The set
totally-real, so Theorem 1 shows that the closure of O(X) the intersection of the polynomials in X,
Z
lip(a,X)
with the closure of
and
are dense in
O(X)
it follows that
lIz
O(X)
in in
X
Lip (a ,X) Lip(a,E).
lip(a,E), and since
and hence the rationals in
C(X)
E is
N
z
is But
+0
on
are dense in
C (X) •
More generally, let
X C ~n
be a compact holomorphically-convex set
which is totally-real off a closed to area zero in each coordinate. zero.
Then O(X)
is dense in
O(X)-convex subset For instance, take
lip(a,X), for
suffices (in view of Theorem 1 and the lip that the rationals are dense in
E, where
E with Hausdorff area
0 < a < 1.
a
E projects
To show this, it
extension theorem) to show
lip(a,F), where
F
n
= n
zjE
is the
j-1
product of the coordinate projections of theorem [13, p. 287], the rationals in each
j.
E. Zj
By the extended Hartogs-Rosenthal are dense in
Thus the closure of the rationals in
symmetric product
lip(a,zjE)t
for
Lip(a,F)
contains the ~ C·(¢) , which is well-known to be dense in C·(C n ).
Since
j
C·(¢n)
is dense in
lip(a,F), and
Cl(¢n)
convergence implies
lip(a,F)
convergence, the result follows. Obviously, area zero is the sharpest metric condition possible here, because positive area would allow possible analytic structure. point in applications is the
O(X)-convexity of
The tricky
E.
It seems plausible that the result should remain true in Lipschitzian submanifolds
X in which the (no longer necessarily closed) set
E where
there are complex tangents has all coordinate projections of area zero. (3.2)
C
p be a
C2
strictly plurisubharmonic function on a neighborhood n of the boundary of a compact set X C ( t with bdy X • {p = O} and with {p < O}
Let
D = int X.
Then
Theor~~s
1 and 2 apply, where
E
= clos
D.
192
0' FARRELL, PRESKENIS AND WALSH
hence in
,
o(X)
Furthermore,
is dense in O(E), in the usual Frechet topology, and
Lip(a,E)
norm.
for a similar argument). bounded by
+
p
Since D U {p + c(j>
D.
X.
0 (U 1)' where
< 2e}
UI = D U {p
T')«
is a subset of
Ul
e > 0,
u
contains
bdy
< T')}
e
,
D
DU{P< e} ,
+ eq>
for all
For this it suffices (by
KC U l Given such a
.
plurisubharmonic exhaustion function a < 1
«n,
It suffices to show that
the functional calculus) to show that, given K
on
U2 •
Call it
sufficiently small positive constants convex hull of
~
Then for all small constants
E.
it is a (Stein) neighborhood of is dense in
function
is strictly plurisubharmonic, vanishes only on
e~
and is negative only on
o(U 2)
Cm
Choose a nonnegative
1, vanishing only on
the function
as follows (cf. [8, Theorem 2.2(b)]
This is seen
O(U 2 )-
compact, the
K, choose a strictly U 2 , with u ~ 0 on K. Choose
for
such that
on
K - D. and let
on
L, and
L
L
= iu
~
O}
n
+ eq>
{p
is positive on
L.
U2 ' hence
so we are done.
~
Clearly,
+ c(p + F.q> -
C
n N
{u ~ O}
+
c > 0
e~
- a'Y]
~ (l-a)T')
> 0
such that
a'Y])
n
~ O}
is
{111' ~ O}
O(U 2 )-convex. C
It follows that (1) and (2) hold.
the approximation problems in
p
is a strictly plurisubharmonic exhaustion
{u::: a} K
problems on
Then
is compact, so there exists a constant '" = u
function for
~ T')}
Lip
a
But
Ul ,
Thus, for such sets
X,
and uniform norms are reduced to the
clos(int X).
Of course, Henkin and Leiterer [la, Lemma 3.5.4] have already established (2), and have gone on to show that closC(X) O(X)
=
{f E C(X):
This new proof of Lemma (3.5.4) is simpler.
f
is analytic on
int X} •
We hope to address the problem
of proving that closLip(a,X) O(X)
=
{f
E lip(a,x):
in a later paper.
f
is analytic in
,
int X}
We are grateful to Joaquim Bruna, Joan Castillo, and Jose Burgues for useful
conversations on the subject of this paper.
HOLOMORPHIC APPROXIMATION
193
REFERENCES 1.
B. Berndtsson, Integral kernels and approximation on totally real submanifolds of Cn , Math. Ann. 243(1979), 125-129.
2.
F.T. Birtel, Holomorphic approximation to boundary value algebras, Bull. AMS 84(1978), 406-416.
3.
E. Bishop, Some global problems in the theory of functions of several complex variables, Am. J. Math. 83(1961), 479-498.
4.
L. Bungart, Ho10morphic functions with values in a locally convex space, and applications to integral formulas, Transactions AMS 111(1964), 314-344. v E.M. Cirka, Approximation by holomorphic functions on smooth submanifolds in ~n, Math. USSR Sbornik 7(1969), 95-114.
.5. 6.
T.W. Gamelin,
Uniform Algebras, Prentice-Hall, 1969.
7.
R.C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, 1965.
8.
F.R. Harvey and R.O. Wells, Jr., Holomorphic approximation and hyperfunction theory on a Cl totally real submanifold of a complex manifold, Math. Ann. 197(1972), 287-318.
9.
, Zero sets of nonnegative strictly p1urisubharmonic functions, Math. Ann. 201(1973), 165-170.
10. G.M. Henkin and J. Leiterer, The theory of functions on strictly pseudoconvex sets with non-smooth boundary, Report R-Math-02!81, Institut fur Mathematik, Akad. der Wiss. der DDR, Berlin, 1981. 11. L. Hormander, An Introduction to Complex Analysis in Several Variables, North-Holland, 1973. 12. J. Nunemacher, Approximation theory on totally real submanifo1ds, Math. Ann. 224(1976), 129-141. 13. A.G. O'Farrell, Annihilators of rational modules, J. Functional Analysis 19(1975), 373-389. 14. A.G. O'Farrell and K.J. Preskenis, Approximation by polynomials in two complex variables, Math. Ann. 246(1980), 225-232. 15. A.G. O'Farrell, K.J. Preskenis and D. Walsh, Polynomial approximation on graphs, Math. Ann., 266 (1983), 73-81. 16. N. Ovrelid, Integral representation formulas and LP-estimates for the a-equation, Math. Scand. 29(1971), 137-160. 17. M. Range and Y.T. Siu, Ck approximation by ho10morphic functions and o-closed forms on Ck submanifolds of a complex manifold, Math. Ann. 210(1974), 105-122. 1& C.E. Rickart,
General Theory of Banach Algebras, Van Nostrand, 1960.
19. A. Sakai, Uniform approximation on totally real sets, Math. Ann. 253 (1980), 139-144. 20. A.G. Vitushkin, The analytic capacity of sets in problems of approximation theory, Russian Math. Surveys 22 (1967), 139-200. 21. B.M. Weinstock, Inhomogeneous Cauchy-Riemann systems with smooth dependence on parameters, Duke Math. J. 40(1973), 307-312. 22. ~n,
, Uniform approximation on the graph of a smooth map in Canadian J. Math. 32(1980), 1390-1396.
194
O'FARRELL, PRESKENIS AND WALSH
23.
J. Wermer, Banach Algebras and Several Complex Variables, Springer, 1976.
A.G. O'FARRELL AND D. WALSH DEPARTMENT OF MATHEMATICS MAYNOOTH COLLEGE CO. KILDARE IRELAND
PRESKENIS DEPARTMENT OF MATHEMATICS FRAMINGHAM STATE COLLEGE FRAMINGHAM MA 01701 USA
K.J.
Contemporary Mathematics Volume 32, 1984
USES OF
~S
IN INVARIANT MEANS
AND EXTREMELY LEFT AMENABLE SEMIGROUPS M. Rajagopalan and P. V. Ramakrishnan ABSTRACT. A semigroup S is called extremely left amenable if there is a multiplicative left invariant mean on the space m(S) of all bounded real valued functions on S. A subset A C S is called left thick if given a finite subset B C S there is an element a in A so that Ba C A. We prove the following in this paper: The following are equivalent for a semigroup S: (a) S is extremely left amenable (b) There is an ultrafilter on S so that all members of this ultrafilter are left thick. (c) The collection of all left thick subsets of S can be expressed as a union of ultrafilters. We further show that the set of all multiplicative left invariant means on S is either finite or has cardinality greater than or equal to 2c • If a semigroup has a unique multiplicative left invariant mean then it is also extremely right amenable. We investigate the relationship between continuous extensions of the semigroup operation in S to ~S and extreme amenability, and we discuss when the collection of left thick subsets of S will form a filter. INTRODUCTION.
One of the important problems in mathematics is to find when a
given collection of self maps of a set
X will have a common fixed point.
This
led to the study of existence of invariant means on certain function spaces associated with a semigroup
A.
We mention in this connection the works of
Day [5], Rickert [20], Furstenberg [7], A. T. Lau [22], among many others.
In
1966 Mitchell [17, 18] introduced the class of semigroups which have the fixed point property on compacta and showed that they coincide with the class of semigroups m(S)
S which admit a multiplicative left invariant mean on the space
of bounded real valued functions on
S.
Granirer [9] gave an algebraic
characterization of extremely left amenable semigroups and proved that it is a large class of semigroups.
So far no topological characterization of extreme
amenability of a semigroup is known. (or right) amenability of a semigroup ~S,
In this paper we study the extreme left S
.,
from its Stone-Cech compactification
or equivalently in terms of ultrafilters on
us two dividends.
S.
This point of view gives
First of all we are able to study the cardinality of the set
of all multiplicative left invariant means on
S.
We remark that the parallel
study of the dimension of the set of all left invariant means on a semigroup
S
was a difficult problem and formed the subject matter of investigation of the © 1984 American Mathematical Society 0271-4132/84 $1.00 + $.25 per page
195
196
RAJAGOPALAN AND RAMAKRISHNAN
doctoral dissertations of I. S. Luthar [14] and E. Granirer [8]. direction we get the interesting result that if
S
In another
is an Arens regular semi-
group or a weakly almost periodic semigroup, then the uniqueness of multiplicative left invariant means on amenable as well, and SECTION 1.
S
S
implies that
S
is uniquely extremely right
has a unique multiplicative two sided invariant mean.
NOTATIONS AND PRELIMINARY RESULTS ./
First, for purely topological notions and Stone-Cech compactifications of completely regular spaces we follow [21]. in [13].
S
The notions on Banach spaces are as
denotes a semigroup with discrete topology.
Banach space of all bounded real valued functions on
m(S)
denotes the
S with usual vector
space operations and supremum norm. If X is a Hausdorff, completely re$ular ./ space then ~X denotes its Stone-Cech compactification. If X is a T2
./
~X
completely regular space,
its Stone-Cech compactification,
f:X ~ Y a continuous function, then
space, and
unique continuous extension of the map
x
~
ax
on
Sand
f
ra
We put
Banach space, E* by
cPA
and
or
= ax
(x) a
=0
if
xES - A.
cPS
DEFINITION 1.1.
a E Sand
~
f
a
Let
a
~
A mean on
S
A mean
on
a
If
r f) a
f
~S
and
a
a E S.
S and a subset
If
E
is a
A C S, we denote
A.
Thus
cPA (x)
=1
IJ. E (m(S»*
then
IJ.(A)
denotes
if
x EA
1.
a complex valued function of
the function
f
0
t
(f
a
0
r) a
on
S.
We
S.
The map
IJ.
r f)
is a positive linear functional S
S
IJ.
on
m(S)
so that
is called left invariant (right invariant, two-sided
(r * IJ. = IJ., t * IJ. = IJ. = r * IJ.) for all a E S. A semia a a is called left (right, two-sided) amenable if there exists a left
invariant) if
* t IJ. a
= IJ.
(right, two-sided) invariant mean on DEFINITION 1.2. lJ.(fg)
~ S denotes '" t a , '"r a are x ~ xa on S. respectively to maps from ~S
t:S
then
on m(S) defined by an element a E S is also denoted by a This will not cause confusion with the earlier meaning of t a (r a ). denotes the adjoint operator of ta(r a ) on (m(S»* for all a E S.
(r*)
group
r
x E
Given
is denoted by
(similarly
t f a
t f (f a
t a (r a ). t
for all
denotes its dual.
The constant function
denote by
and
a
the characteristic function of
XA
cPA(x)
t
t
a compact
f:~X ~ Y denotes the
a ES
If
denotes the map
the unique continuous extensions of into itself.
~X.
to
Y
= lJ.(f)lJ.(g)
S.
A multiplicative mean on for all
f,g E m(S).
S
is a mean
A semigroup
S
IJ.
on
S
so that
is called extremely
left (right, two-sided) amenable if there is a multiplicative left (right, twosided) invariant mean on DEFINITION 1.3.
A subset
S. A
thick) if given a finite set
C
S
of a semigroup
S
is called left thick (right
of elements of
S
there is an
197
EXTREMELY LEFT AMENABLE SUBGROUPS g E A so that
element
a.g E A (ga.1 E A) 1
for all
= 1.2.3 •.•.• n.
i
The
A is called a left ideal (right ideal, two-sided ideal) if xa E A (ax E A, both xa and ax E A) for all xES and a E A. subset
The ideas 9f left thick sets and extremely left amenable semigroups were introduced by T. Mitchell [17].
Algebraic characterizations of left extreme of E. Granirer [9].
amenability are given in an article
We approach the
problem of characterizing extreme amenability from a topological point of view. For this we investigate how to find means on S
are evaluations at points of
S.
The easiest means to find on
Thus it is important to know when a mean
~S.
~S
defined by evaluation at an element of
will be left or right invariant.
This leads us to consider the extended multiplication a ES
and to find out when
~S
will have a right zero.
to viewing the non-isolated elements of
~S
the existence of special ultrafilters on of
DEFINITION 1.4. f
S.
a
on
~S
for all
This brings us back
as free ultrafilters and studying So we give below some properties
which we will need in the future.
~S
Then
t
Let
S be a semigroup with discrete topology.
is the mean defined on
ea
S
Let
ea (f) - f(a)
by the formula
a E
~S.
for all
E m(S). The following is an easy consequence of the fact that every bounded complex
valued function
f
on
S extends continuously to
~S
and a theorem on
characterization of multiplicative linear functionals on the space of all continuous complex-valued functions on a compact Hausdorff space (proved in [13]) • LEMMA 1.5.
a E
Every multiplicative mean on
s is of the form
for some
~S.
LEMMA 1.6.
Then
Let
a E
~S
and
Va
is an ultrafilter in
V
a
=
{V
S and
n
S I V is a neighborhood of a n W. In addition, {a}" WEV
in
~S}.
a
U Q!l S is of the form Va for some a E ~S, and free if and only if U = V for some a E ~S/S. Further. let a E ~S a E c S. Then E E V if and only if a E E. a every ultrafilter
PROOF.
U is and
Lemma 1.6 is well-known and can be found for example in [4].
e (f) = f(a) for all f E m(S). Thus a for all f E m(S) e (f) = e (tb(f» e a is left invariant if and only i f a a 1--/ '" '" and b E S. Now e a (f) = (f) (a) and ea(tbf) = (tbf)(a) = f(ba) for all b E S and f E m(S). So e (f) = ea(tbf) - '" f(a) = '" f(ba). Comb ining this a with Lemma 1.5 we get: Now we note that if
LEMMA 1.7.
A semigroup
a E
~S
then
S has a multiplicative left invariant mean, or in
RAJA GOP ALAN AND RAMAKRISHNAN
198 other words a E
S
is extremely left amenable, if and only if there is an element
so that
~S
ba
=a
b E 8.
for all
8
Likewise
amenable if and only if there is an element
a E
~8
is extremely right so that
ab
=a
for all
b E S.
DEFINITION 1.8.
Let
S be a semigroup. ~S
zero (right zero) of
An element of
An
if and only if
element
ab
right zero and a left zero of
is called a left
~S
= a)
(ba
for all
b E 8.
if and only if it is both a
~S
is said to be a zero of
~s
a
=
a E
~S.
The Lemma 1.7 can be restated as: COROLLARY 1.9.
li..
A semigroup
is extremely left (right) amenable if and only
8
has a right (left) zero.
~S
has a two sided zero if and only if
~S
S
is two-sided extremely amenable. Now i f
is a continuous function from
then
b E S
a E ~S is a right zero of then we have that given a neighborhood W of
into
80 if
~s.
so that
~S
Then choosing an element for all
i
1,2, ... ,n.
~
of the ultrafilter Let
LEMMA 1.10.
U
in
S
x E W n VI Wn S
So
and
~s
a in '" tb (Vi)
b l ,b 2 , ••• ,b n E 8 ~S there are neighborC
W for
i
n ... n Vn
S
i
= 1,2, ... ,n.
bix E W n S
we deduce that
is left thick in
~s
and also is a member
V. Thus an easy application of Lemma 1.6 gives us: a S be extremely left amenable. Then there is an ultrafilter
so that all members of
U are left thick in S.
~
similar result
holds for extremely right or two-sided amenable instead of extremely left amenable. Note that Lemma 1.10 and Lemma 1.6 imply that if amenable then there is an element ~
mean (Take
on ~,.
S
so that
E
a E
~S
S
is extremely left
and a multiplicative left invariant
is left thick and
=1
~(E)
for all
E E Va •
e .) a
The converse also holds as is shown in the following lemma. LEMMA 1.11.
Let
left thick.
Then there exists a multiplicative left invariant mean
that
9(;E)
=
1.
S be an extremely left amenable semigroup.
U are left thick subsets of
Ec S
be
e
S
U on S so that
Moreover there is an ultrafilter
and all members of
Let
S.
on
so
E EU
A similar result holds
if left is replaced by right throughout in the lemma. PROOF. F
C
Let
S let
a E
~S
be a right zero of
x F be an element of
be a cluster point of the net
S under containment.
all
and let
xES
rb
E so that
x F when
subsets of
~S.
Let
For each finite non-empty set y~
E E for all y E F.
Let
b
F varies over the filter of all finite rb:S
~ ~S
be the map
rb (x) - xb be its unique continuous extension to ~S. Let
for
EXTREMELY LEFT AMENABLE SEMIGROUPS
=
Ya be a net in S converging to a. For a fixed a, yax F E E eventually when F varies over the filter of all finite subsets of S. So '" rb(y a ) = Yab = It yax F belongs to E. Now Ya ~ a. So z = rb(a) = It rb(y a ) E i. Now ~ is left invariant.", To see
z
Then
this let
f
z E E.
199
E m(S)
and
To see this let
t E S.
50 it is enough to show that
e (f) = fez) and e (t f) • f(tz). z z t Now z = '" rb(a) = It '" rb(y a ) = It Yab for
Now
tz = z.
Ya ~ a. So tz = It tyab = rb(lt (ty a » = rb(ta) = rb(a) ~ z since ta = a for all t E S. Put 9 = e. Since z E E we get that E E V and z z 9(~E) = 1 by Lemma 1.6. Moreover Vz is an ultrafilter all of whose members are left thick. Thus we get the lemma in the left invariant case. The right some net
invariant case is proved similarly. T. Mitchell [18] proved that a subset 5
E
C
5
of a left amenable semigroup
is left thick if and only if there is a left invariant mean
= 1.
~(~E)
that
~
on
S so
We note that Lemma 1.11 is an analogue of this theorem of
Mitchell in the multiplicative case. We also get the larger part of the following characterization of extreme left amenability by an easy application of Lemmas 1.11, 1.10 and 1.6. THEOREM 1.12. (a)
Let
S be a semigroup.
Then the following are eguivalent:
There is at least one ultrafilter are left thick subsets of
(b)
U
~
S.
expressed as a union of ultrafilters of If
A, B are subsets of
5
then
only if at least one of the sets (d)
~S
(e)
S. is extremely left amenable.
PROOF.
all of whose members
W of all left thick subsets of S can be
The collection
(c)
S
S.
A U B is left thick if and A
~
B 1s left thick.
has a right zero.
The equivalence of (d) and (e) has been proved in Corollary 1.9.
implication (e)
~
ultrafilter in
S all of whose members are left thick subsets of
a E
~S
(a) is Lemma 1.10.
be such that
To see that (a)
~
(e)
let
The
U be an S.
Let
U = V as in Lemma 1.6. Let b E S. It is enough to a For that it is enough to show that ba E E, where E E U
show that
ba = a.
Now given
FEU, choose
xF E E
n F so that
b~
EE
n F.
(This can be done
n FEU and each member of U is left thick.) If t is a cluster point of the net (xF ) then t E F for all FEU. So t = a by Lemma 1.6.
because So
xF
bXF
~
(b)
~
1.11.
E
~
a.
ba. (a).
Likewise So
ba
bXF
= a.
To see that
Therefore we get
Assume that
(b), that
~
a.
'" tb
However, the continuity of
Thus we get
(a)
~
(d)
~
(e).
implies that
It is easy to see that
(a) ~ (b), apply the equivalence (a) ... (e) and Lemma (e)
~
(d) ... a
-
(b).
A, B are subsets of
We now prove S, and that
(b) .. (c).
A U B is left thick.
200
RAJAGOPALAN AND RAMAKRISHNAN
A U B belongs to an ultrafilter
Now from (b) we know that members are left thick. or
V •
belongs to
B
To prove (c) subsets of
(b)
~
S.
F so that
filters
V is ultrafilter, it follows that either
Since
So either
assume (c).
Let
E E L
Vall of whose
.
A or L
Let
B
is left thick.
So (b)
~
A
(c).
be the collection of all left thick
Apply Zorn's Lemma to the collection of all E E F and Fc L Then we get a filter H so that the
.
following hold: (i) Hc L (ii)
E E
(iii)
H H
(iv)
H is a filter
is maximal with respect to the properties (i), (ii) and (iii)
above. H is an ultrafilter of subsets of
We claim that show that if
A
C
S
then either
A or
S.
For this it is enough to
S/A belongs to
H.
If
AnM
is
H then it follows that A E L and A n M :/: tI> and left thick and belongs to H for all M E H. So the maximality of H implies that A E H. Suppose that there is a set K E H so that A n K is not left thick. Let M E H. Then A n K n M is not left thick. However, left thick for all
we know that
K
(S/A)
nKn~
(S/A)
nM
(c)
(b).
~
M E
nM
is left thick.
So an application of (c) gives us that
is left thick for all
is left thick for all
ME
MEH
H
This in turn implies that
and hence
S/A E
H.
Therefore
Thus we have the theorem.
Finally we remark that the statements in Theorem 1.12 hold when left is replaced by right, and can be proved in a similar fashion. Now we come to the cardinality problems related to multiplicative invariant means. SECTION 2.
CARDINALITY OF MULTIPLICATIVE INVARIANT MEANS
In this section we study the cardinality of multiplicative left invariant means.
An
analogous problem of uniqueness and dimensionality of left invariant
means on a left amenable semigroup was considered to be an important problem and formed the subject matter of investigations by I. S. Luthar [13] and E. Granirer [8].
Our results on the multiplicative invariant means are sharp.
We get the surprising result that if a semigroup has a unique multiplicative left invariant mean then it also has a multiplicative right invariant mean. We need the following results for future discussion. IAI
If
A is a set let
denote its cardinality.
LEMMA 2.1.
Let
X be a discrete space and
Then every closed subset
F
C
~X
~X
ttl
its Stone-eech compactification. is either finite or IFI ~ 2c , where c is
EXTREMELY LEFT AMENABLE SUBGROUPS For a proof see [9], [6] or [4].
the cardinality of the continuum. LEMMA 2.2.
Let
S
two among
a,b~c
are defined, and PROOF.
(ab)c
S.
=
Let
'" t b a
Then all the products
= It
be a net in a
ab. a
= ~c (ab) = It a
and assume that at least (ab), (bc), a(bc), (ab)c
= a(bc).
(ab)c
b
a,b,c E (3S
Let
We prove only the case in which
similarly. (ab)
be a semigroup.
are in
201
a,c E S.
such that
S
The other cases are proved
ba
b.
4
Then
Moreover
= It a
(ab )c a
a(bac)
= I: (It (b c» = I: (~ (b ) a
ac
a
=
1a (bc)
.-
a(bc). Corollary 1.9 gave the relationship between a left zero of fixed points of the maps DEFINITION 2.3. point set
~
Let of
rb,
b E S.
where
to be the set
right fixed point set of
b
b E S.
{xix E (3S
Define the left fixed
and
bx = x}.
{xix E ~S
to be the set
and
Motivated by this we put:
S be a semigroup and
b
(3S
and
~
Define
the
xb = x}.
Lemmas 1.S and 1.7 tell us that the set of all multiplicative right invariant means on
S
is exactly
invariant means on
S
is
~s,
~.
and the set of multiplicative left ~
Since each of
and
~
is closed in
Lemma 2.1, the following.
we get using
THEOREM 2.4.
n
bE S-b
n
L, bE S 0
The set of all multiplicative left invariant means as well as
the set of all multiplicative right invariant means are closed subsets of Hence a semi group
~S.
can have either only finitely many multiplicative left invariant means or has at least 2 c multiplicative left invariant means. A S
similar result holds for the set of multiplicative right invariant means. Note that the simplest of all infinite cardinals, namely
~,
does not
appear as a value in many natural cardinality problems in functional analysis.
~
Theorem 2.4 is one instance where we see that
is not the cardinality of
S. Analogously of a Banach space satisfies
the set of multiplicative invariant means on any semigroup Bhaskara Rao [3] proved that the cardinality
y
the equation there is no
m
o
"" m,
and S. Janakiraman and M. Rajagopalan [12] proved that
interval of cardinality
on any Abelian group.
m
t\r J·o
of locally compact group topologies
It would be interesting to know what cardinals can
appear as the cardinals of the set of all multiplicative left invariant means on a semigroup
S.
It is easy to see that every integer
n ~ 0
as well as
2c
can be such
cardinalities due to the following examples: EXAMPLE 2.S.
Let
Z be the set of all integers under usual addition.
Z has no multiplicative left invariant mean.
Then
202
RAJAGOPALAN AND RAMAKRISHNAN
PROOF.
This follows from the algebraic characterization of extreme left
amenability due to Granirer [9].
Granirer showed that a semigroup
extremely left amenable if and only if given that
= ya
xa
a.
a
EXAMPLE 2.6.
n
0
>
be a given integer.
elements with right zero multiplication. ~X
EXAMPLE 2.7. ax
2c
have that
a E S
X be a set with
That is,
ab
=b
so
n
for all a,b E X.
a
Let
=x
Let
is
Z.
X, and each element of X is a right zero if multiplicative left invariant means.
multiplication. that
there is an
This obviously is not satisfied by
Let
Then clearly has exactly n
x,y E S
S
X.
So
X
N be the set of positive integers with right zero Thus
ab
for all
=
b
for all
a,b E N.
x E ~N.
a E N and
Then it is easily checked Since I~NI = 2c (see [18]) we
is a possible value of the cardinality set of all multiplicative
left invariant means on a semigroup. We remark that if
n > 1
even right amenable though
in Example 2.6 then that semigroup
X is extremely left amenable.
X is not
So it is interesting
that this cannot happen if the multiplicative left invariant mean is unique, as the following theorem shows: THEOREM 2.8.
Let
S be a semigroup.
left invariant mean. PROOF.
Let
bead) - (ba)d
a E
a · ad.
zero of
~S.
S
be a right zero of
~S
S have a unique multiplicative
is also extremely right amenable.
by Lemma 2.2, so
right zero of that
Then
Let
=
bead)
Let
~S.
(ba)d
=
b, dES.
(ad).
Thus
Then
ad
is also a
The uniqueness of multiplicative left invariant mean implies
~S.
Since Then
dES ea
is arbitrary it follows that
a
is also a left
is also a multiplicative right invariant mean
S.
o~
Thus we have proved the theorem. The above theorem does not imply that if
S has a unique multiplicative
left invariant mean then it also has a unique multiplicative right invariant mean.
It would be interesting to know whether a semigroup is uniquely
extremely left amenable if and only if it is uniquely extremely right amenable. We can settle this problem in the affirmative in one particular case.
But
then we have to consider continuous extensions of the semigroup operation of S
~S
to
and their relation to extreme amenability.
THEOREM 2.9. on
~S
Let
S be a semigroup.
G>t
so that the following holds:
(a)
~S
(b)
S
is a semigroup under is a subsemigroup of
given operation on (c)
Then there is a binary operation
x
at
y
at . ~S
at
under
and
G>t
agrees with the
S.
is continuous in
y
~
~S
for any fixed
x
of
~S.
EXTREMELY LEFT AMENABLE SUBGROUPS
er
Similarly, an extension defined in
~S
PROOF.
xES
r
continuous extension of
r
~
of the semigroup operation on
S
can be
which is continuous in the left variable only. let
x
For
203
be the map
x
~S.
to x xES.
y
~
If
yx
a E
on ~S
S, let
and let
r
x be the map
t
be the
a r (a) - ax for all Then t is a continuous map from S into x a and hence has a unique continuous extension ta to ~S. Finally if
'V
'V
~S
a, b E
a 0 b
that
Let
ba
be a net in
bc~ = lat(bac~).
5!
... l~t[(lat(aba»C~]
that
~t
b
=
0t
~
converging to
c~
Now
c~ E S
=
~S
50
lt ~
and
Gt
ba
a net in bc~.
50
~
50
~t(bac~)]
b)c~]
b.
5
~S
ta
~S.
S
l~t[~t(abac~)]
=
Gt
It is clear
•
So we get the theorem. It is not true
is a semi group then the multiplication in
to a semigroup operation in
'"
from definition of
is a semigroup under
a, b E 5.
for all
0t
=
a,b.c E
Let c~
and
E S,
l~t[a
l~t[(a
~S.
= tb(c)
(b (!)t c)
(b 0 t c) =
c.
a E
for all
in the theory of numbers is given in [4].
in general that if separately.
b
et(bc~).
0t
ab
~
A use of
a
Then it is clear from the definition of
Now
since
= (a Gt b) a
S
~5.
in
c
St (b 0 t c) = lt a
a <:>t b
'V
= ta(b).
G)t b
is continuous in
converging to a
a
put
~5,
S
extends
which is continuous in each variable
Needless to say, the multiplication need not necessarily extend to
a jointly continuous multiplication in multiplications from
~S
to
5
~S.
Continuous extensions of
is a very fascinating and difficult subject,
and some partial results have been obtained by H. Mankowitz [15], T. Macri [16], R.P. Hunter and L. W. Anderson [1], Aravamudan [2], and others.
The following
gives us an interesting class of semigroups: DEFINITION 2.10. in
5
A semigroup
S
is called R-semigroup if the multiplication
extends jointly continuously to a semigroup operation on
semigroup
S
is called a V-semigroup if the multiplication in
semigroup operation on
~S
~5.
S
The
extends to a
which is continuous in each variable separately.
Now we are ready to improve our Theorem 2.9 for the class of V-semigroups. THEOREM 2.11.
Let
S be a semigroup.
Then the following are equivalent:
(i)
S
has a unique multiplicative left invariant mean.
(ii)
The collection of all left thick subsets of
S
in
S.
(iii)
~S
has a unique right zero under the operation
(iv)
Given
f E m(S)
is an ultrafilter
or .
there exists a unique constant function in
is the weak * -closure of the set
{r f I a E S}. a A similar theorem holds if left is interchanged with right throughout. where
kef)
kef),
204
RAJAGOPALAN AND RAMAKRISHNAN
PROOF.
The equivalence (i) - (ii)
follows from Theorem 1.12.
The
equivalence of (ii) and (iii) follows from Theorem 1.12 and Lemmas 1.7 and 1.10.
We now show (iv)
right zero of in
~S,
in
then
r
a
f
~
(iii).
and a r a converges in weak <:)
a Thus we have that given
'V
f(a).
constant function The implication
f(a) (iii)
(iv)
*
is a net in
S
~S
is a
which converges to
a
topology to the constant function
f E m(S)
belongs to ~
a E
To see that, observe that if
and a right zero
a
~S,
of
the
So it is clear that (iv) ~ (iii).
kef).
follows easily by using Theorem 1.1 of [10].
Thus we have the theorem. THEOREM 2.12.
Let
operations
S
rulll Or
or
!i. S is a V-semigroup then the
be a semigroup. then
S
Conversely if
~8.
coincide on
G>t
cQincides
is a V-semigroup.
The proof is straightforward and hence is omitted. THEOREM 2.13.
Let
S
be a V-semigroup.
Then the following are equivalent:
(1)
S
has a unique multiplicative left invariant mean.
(2)
S
has a unique multiplicative right invariant mean.
(3)
S
has a unique two-sided multiplicative invariant mean.
(4)
has a unique right zero under
(5)
has a unigue left zero under
(6)
has a zero under
e
as well as
G)t
t
(7)
The collection of right thick subsets of
(8)
The collection of left thick subsets of
(9)
Given kf
f E m(S)
<:)
•
r
G. r
as well as
(!)
or .
as well as
~
is an ultrafilter.
S
8
is an ultrafilter.
there is a unique constant function in
k f , where
is as in Lemma 2.11.
(10) Given Lf
f E m(S)
is the
there is a unique constant function in
*weak closure
of the set
{t f
a
Ia
Lf , where
E S}.
The proof is an easy application of Theorems 2.11 and 2.12.
Theorems 2.11
and 2.13 show the relationship between uniqueness of multiplicative left invariant means and the collection of all left thick subsets forming an ultrafilter.
In fact, the collection of all left thick subsets of a semigroup need
not even form a filter, as is the case of the semigroup of strictly positive integers under the usual addition, or the semigroups in Example 2.7.
A nice
characterization of semigroups for which the collection of all left thick subsets of
S
THEOREM 2.14. collection
is a filter is not known. Let
S
be an extremely left amenable semigroup.
Then the
L of all left thick subsets of S is a filter if and only if for all left thick subsets
PROOF.
However, we have a partial result.
The "only if " part is clear.
A, B of
S.
80 we have to only show that if
EXTREMELY LEFT AMENABLE SEHIGROUPS C
nDI
C
=
~
for all
(C
n D)
either
C
n D')
U (C
nD
then
C, DEL
or
Let
C is left thick.
and
C
n DEL.
C
n D'
is left thick.
So
n DEL.
C
sID.
D' -
Now
So Theorem 1.12 implies that
However
D is left thick and
any two left thick subsets have non-empty intersection. left thick.
205
So
C
n D'
cannot be
Thus the theorem.
We are able to conclude that the collection of left thick subsets is a filter for a class of semigroups other than the one given in Theorem 2.14. First we give a definition. DEFINITION 2.15.
Let
S be a semigroup.
A left thick subset
M of
S is
called the smallest left thick subset if A ~ M whenever A is a left thick subset of S. A left thick subset B of M is called a minimal left thick subset of
S
if
THEOREM 2.16.
B contains no left thick subset of
of
S.
S
form a filter.
right ideal of
S.
direct product
G x F, where
Further, such a right ideal
left zero multiplication. Let
(That is, xy
=x
thick subset of L
Ka
S.
K.
To see that K=K
So
x, y E F). Let H
L.
S.
be the
H is
Since
where
THEOREM 2.17.
K is. since
and
IKal
x, y E K, a E S,
direct product
K is a minimal left
So
K
n Ka
is finite. and
Then
xa = ya, then
Then
G is a group and Let
F
A of
G x E,
Then where
L
Thus
= K. So we IKal = IKI for Ka
get all
x - y.
K cannot have a proper left K can be expressed as
is the left zero semigroup.
S be a right cancellative semigroup and S.
a ES
is left thick because
K is minimal left thick.
By a theorem in [11] it follows that
collection of all left thick subsets of thick subset
Clearly
K is a right ideal notice that if
K is right cancellative.
ideal of itself. G x F,
K is a
is a finite set with
K is a right ideal and also that
Therefore if
Consequently
n Ka
IKal ~ IKI
However
at the same time that a E S.
for all
F
T which are left thick in
is also left thick since
is a filter. J
so that
K can be expressed as a
T be a finite subset which belongs to
H is again in L and hence is left thick. Ka
S
L is a filter we have that the intersection K of all members in
finite and
then
K of
G is a finite group and
collection of all subsets of
of all left
Let there be a finite left thick subset
Then there is a minimal left thick subset
PROOF.
other than itself.
S be a semigroup and let the collection L
Let
thick subsets of
S
S.
L
the
Let there be a smallest left
is a filter, and
G is a group and
A can be expressed as a E
is a semigroup with left
zero multiplication. We omit the proof since it is similar to that of Theorem 2.16.
206
RAJAGOPALAN AND RAMAKRISHNAN
PROBLEMS: (1)
Let
S be a semigroup with a unique multiplicative left invariant
mean.
Is the right multiplicative invariant mean on
S always
unique? (2)
What are all the cardinal numbers
a
for which there is a semigroup
S such that the set of all multiplicative invariant means on
S has
cardinality a ? (3)
What are all the ordered pairs
(a,~)
so that there is a semigroup
S whose set of all left invariant means has cardinality set of all right invariant means has cardinality
~
a
and the
?
(4)
Find the algebraic structure of R-semigroups.
(5)
Find the algebraic structure of V-semigroups.
(6)
Find good necessary and sufficient conditions on a semigroup
S
so
that the set of all its left thick subsets forms a filter. REFERENCES 1.
L.W. Anderson and R.P. Hunter, On the compactification of certain semigroups, Proceedings of the symposium on the extension theory o~ topological structures, Springer-Verlag (1969), 21-27.
2.
R. Aravamudan, Arens' regular semigroups, Madurai Univ. Technical Report, (1977).
3.
M. Bhaskara Rao and K.P.S. Bhaskara Rao, Cardinalities of Banach spaces, J. Ind. Math. Soc. 37 (1973), 347-349.
4.
W.W. Comfort and S. Negrepontis, The theory of u1trafi1ters, SpringerVerlag, Berlin, 1976.
5.
M.M. Day, Fixed point theorems for compact convex sets, Ill. J. Math. (1961), 585-590.
6.
J. Dugundji, Topology, Allyn and Bacon, 1965.
7.
H. Furstenberg, A Poisson formula for semigroup Lie groups, Ann. of Math. 77 (1963), 335-386.
8.
E. Granirer, On amenable semdgroups with a finite dimensional set of invariant means I and II, Ill. J. Math. 7 (1963), 31-48, 49-58.
9.
E. Granirer, Extremely amenable semigroups, Math. Scand. 93-113.
5
20 (1967),
10.
E. Granirer, Functional analytic properties of extremely left amenable semigroups, Trans. Amer. Math. Soc. 137 (1969), 53-76.
11.
K.H. Hoffman and P.S. Mostert, Theory of compact semigroups, Charles Merrill Co., 1966.
12.
S. Janajiraman and M. Rajagopa1an, Topologies in locally compact groups II, Ill. J. Math. 17 (1973), 177-197.
13.
L.H. Loomis, Abstract harmonic analysis, Van Nostrand, 1953.
14.
I.S, Luthar, Uniqueness of invariant mean on an abelian semigroup, Ill. J. Math. 3 (1959), 28-44.
15.
H. Mankowitz, Continuity of products in
S, to appear.
EXTREMELY LEFT AMENABLE SUBGROUPS 16.
207
., T. Macri, The continuity of Arens' product on the Stone-Cech compactification of semigroups, Trans. Amer. Math. Soc. 191 (1974), 184-193.
17.
T. Mitchell, Fixed point theorems and multiplicative invariant means on semigroups, Trans. Amer. Math. Soc. 122 (1966), 195-202.
18.
T. Mitchell, Constant functions and multiplicative left invariant means on semigroups, Trans. Amer. Math. Soc. 119 (1966), 244-261. S. Mrowka, On the potency of subsets of ~N, Colloq. Math. 7 (1959), 23-25. N.W. Rickert, Amenable groups and groups with the fixed point property, Trans. Amer. Math. Soc. 123 (1967), 221-232.
19. 20. 21.
R. Vaidyanathaswamy, Set theoretic topology, Chelsea, 1960.
22.
Anthony To-Ming Lau, Invariant means on almost periodic functions and fixed point properties, Rocky Mount. J. 3 (1973), 69-81.
(The results of this paper have been announced in 1970 and have been circulating in a preprint form.) DEPARTMENT OF MATHEMATICS UNlVERS ITY OF TOLEDO TOLEDO, OH 43606 U.S.A.
DEPARTMENT OF MATHEMATICS MADURAI UNIVERS ITY MADURAI-2, (625021) INDIA
Cont.eiuporary Mathematics
Volume 32, 1984
DEFORMATION THEORY FOR UNIFORM ALGEBRAS: Richard Rochberg I.
AN INTRODUCTION
..
INTRODUCTION This is intended as an informal introduction to a theory that is starting
to develop.
The emphasis on general themes has perhaps been at some cost of
precision and no effort has been made to be sel£-contained. For a uniform algebra
(that is, a commutative Banach algebra with unit in which every element satisfies lIa 2 11 = lIaIl 2 ), we denote the spectrum by
M(A)
A
and the Shilov boundary by
we denote by
C(X)
For a compact Hausdorff space
the algebra of continuous functions on
smoothly bounded domain in we denote by
aA.
~,
R.
For
R a
or more generally a bordered Riemann surface,
A(R). the algebra of holomorphic functions on
continuously to
X.
X
R which extend
To minimize technical issues we assume throughout that
all algebras considered are uniform algebras which have their Choquet boundaries equal to their Shilov boundaries. A fundamental question in Banach algebra theory is the relation between
A (as
the structure of a Banach algebra
space, etc.) and the structure of M(A) ture~
etc.).
a
TVS,
normed algebra, Banach
(topological structure, analytic struc-
Here are two results which bear on this question and will be of
interest to us: THEOREM 1 (Banach-Stone).
1&! X,Y be compact Hausdorff spaces. The follow-
ing are equivalent. a)
C(X)
and
C(Y)
are isometric Banach spaces
b)
C(X)
~
C(Y)
are isometrically isomorphic
c)
X
~
Y (i.e., M(C(X»
THEOREM 2 (Nagasawa).
Let
R,S
~
M(C(Y))
Ban~ch
algebras
are homeomorphic.
be finite bordered Riemann surfaces.
The fol-
lowing are equivalent. a)
A(R)
.!lli! A(S) are isometric Banach spaces
b)
A(R)
and
c)
R
~
S
A(S) (i.e.
are isometrically isomorphic Banach algebras M(A(R»
~
M(A(S»)
are conformal1y equivalent •
..Supported in part by a NSF Grant. © 1984 American Mathematical Society 0271.4132/84 $1.00 + $.25 per page
209
210
ROCHBERG
One focus of deformation (also called perturbation) theory for uniform algebras is on understanding the relationship between small changes in the structure of
A and small changes in the structure of
this precise we recall two results from function theory. D - {z E a:;
Iz I <
l}, Dr, 1
=
{z E C; r <
THEOREM (Riemann Mapping Theorem).
r
~
t
~
If
Iz I <
r',
Before making
Let
I}.
R is the inside of the Jordan curve
R is conformally equivalent to
D.
This result shows a rigidity of structure; if by curve
M(A).
r
is replaced by a near-
the new inside is conformally equivalent to the old.
The situa-
tion is different tor multiply connected domains (and for topologically trivial domains in higher dimension). THEOREM.
Su:e:eose
For instance:
R is the region between two Jordan curves,
r l 2!ll!. r 2 ,
ltllh r 2 inside r l • Then there is an r such that R is conformally equivalent to Dr,l; Dr,l !!!S! Ds,l are conformally eqUivalent i f and only if r
= s.
In this case a small change can produce a conformally inequivalent domain (but one in the same family).
Furthermore,
domains, depends continuously on
rl
and
r,
r 2•
the natural parameter for such This pair of results (together
with their analogs for compact Riemann surfaces of genus zero and one) are seminal in the theory of deformation of analytic structure (which asks what new structures are obtained by small changes) and in the theory of moduli (which asks how the new structures can be parametrized). The work in this paper gives the uniform algebraic analogs of these two results.
Small changes in the algebra
Small changes in the algebra
A(D)
produce equivalent algebras.
A(D
r, 1) produce A(D s, 1) with s near r. Questions of deformation and moduli go back to Riemann and are now analyzed with a huge range of techniques -- see II], [4], [5], [11] and [18] for some indication.
II.
DEFORMATION THEORY FOR UNIFORM ALGEBRAS
A.
DEFINITIONS There are several possible ways to make precise the notion of a small
change in a uniform algebra. 1. and
Here are three:
(Banach space oriented definition)
Bt
Given two uniform algebras
A
we can measure the similarity between them by the Banach-Mazur dist1 -1 ance, given by dl{A,B) = 2' log inf{IITIIIIT II; T is an invertible linear map of
A
to
B}.
DEFORMATION THEORY Thus
=
dl(A,B)
(1)
211
e if there is a linear map
1 - O(e)
S
II~fll"
1 + O(e)
S
T of
A onto
for all
f
E
B with
A
This quantity.plays an important role in general Banach space theory. relation to uniform algebra theory is suggested by Theorems 1 and 2. d l (A,B) = 0
is not quite the same as
A and
Suppose
"x"
is a new multiplication on IIfxg - fgll
(2)
Then we say that
S
eilfllllgll,
(However
B being isometric.)
(Normed algebra oriented definition)
2.
Its
Start with a uniform algebra
A.
A which satisfies for all
f,g
in
A with this new multiplication is an
A. ~deformation
of
A.
This new algebra is, in fact, a Banach algebra after renorming with its spectral norm (we regard that as done).
For an algebra
B we set
d 2 (A,B) = inf{e; B is isometrically isomorphic to an
e -deformation of
A}.
This is a normed version of the algebraic notion of deformation of an algebra given in 14]. 3. (Gelfand theoretic definition) A to a subalgebra of algebra of d3 (A,B)
=
C(aA)
C(aA)
III - Til is small. Then T(A)
and
which is close to
inf{ilI-TII;
T maps
trically isomorphic to
B}.
Suppose there is a linear map
A.
For a uniform algebra
A onto a subalgebra of A and
If
are clearly close in the first sense.
C(aA)
T of
is a subB set
which is isome-
B are close in this sense then they By setting fxg = T-l(TfTg) one checks
that they are also close in the second sense. B.
EXAMPLES For
A 1 = A(D 1) • Fix r, p with a < r t P < 1. Any r, r, noon can be written as fez) = L_ooanz. Set Tf = LOanz +
a< r
< 1
let
00
in
Ar 1 -1 n' n L (rIp) a n z. f
_00
Then
T maps
Ar, 1
to
AP. 1.
It is an extended exercise in
the use of the Schwarz lemma to check that (1) is satisfied with and hence
d1 (Ar, 1,A p, 1)
= O(log(r/p».
multiplication which satisfies (2) with
Ae
=
A be the disk algebra,
{f; f E A,f(O)
are in Set
D.
If
f
=
fee)}
is in
and
A
Tf = f(O) + z(z-e')g.
e
= O(e-e') for e,e', d3 (A ,A ,) = O(e-e'). e e
lennna,
III-Til
e
fxg
= T-l(Tf
= O(log(r/p».
Tg)
Thus
= log(r/p) gives a new
d2
satisfies
cl 1 •
the same estimate as Let
Setting
e
AO
then Then e,e'
= A(D). For e in D, = {f; f E A,f'(O) = a}.
A
f = f(O) + z(z-e)g T maps
A
let
Suppose
for some
g
e,e' in
A.
A, and, using the Schwarz e in any compact subset of D. Thus, for such e
to
e # 0,
212
ROCHBERG
c.
RESULTS
1.
General Results One of the basic general results is that the three definitions of defor-
mation are equivalent.
In other words, when anyone of
eli (A,B) , i
1,2,3 is
=
small then so are the other two, and the three quantities are comparable. is more than just a chase through the definitions.
This
Note, for instance, that
the equivalence yields the following strengthening of the Banach-Stone theorem. If
C(X)
and
C(y)
d3
is small also.
are almost isometric (i.e., dl(C(X),C(Y» It is easy to check that if
isometrically isomorphic to
C(Y)
and
d3
d3
is zero.
is small) then
is small then
C(X)
is
(This improvement of
Theorem I was first obtained by Cambern [3].) If
A is a uniform algebra of the type being considered (i.e.,
aA =
Choquet boundary) then any sufficiently small deformation of A (in, say, the 2nd sense) is also a uniform algebra of the same type. Furthermore aA and aB
are homeomorphic.
(The algebras
However
and
Aa
A
some stability.
and
M(B)
need not be homeomorphic.
of the second example give a counterexample.)
f.
Although the topology of
M(A)
M(A)
can change when
A changes a bit, there is
For example, the number of connected components of
M(A)
is
unchanged under small deformation (a continuity theorem for the dimension of the a th cohomology group of the spectrum.) Under additional hypotheses the dimension of the 1 st cohomology group can be shown to be semicontinuous. (Such semicontinuity results are common in other approaches to deformation theory. ) The metric
dl
can actually be shown to be complete on the appropriate
space of uniform algebras.
The algebras obtained by taking limits of dl-Cauchy
sequences of common examples can be surprising.
The sequence
{Ar ,l}
is not
n
dl-Cauchy if
r
n
~
1.
If
r
n
~
0
the sequence is Cauchy and the limit is a
direct sum of two copies of the disk algebra with their constant functions identified.
This (and other) examples are related to the obtaining of singu-
lar algebraic curves by continuous deformating of non-singular curves. 2.
Rigidity The proof outlined for Cambern's extension of the Banach-Stone theorem
shows that
C(X)
has no non-trivial small deformations.
Such algebras are nd In fact any Banach algebra with vanishing 2 and
called rigid (or stable). 3 rd Banach algebra cohomology groups is rigid (see [ 61, [1T]). the case of
C(X)
different reasons).
for metric
X.)
(This includes
The disk algebra is also rigid (but for
This is the precise form of the uniform algebraic compan-
ion piece to the Riemann mapping theorenl.
It is not known if
00
H
is rigid.
DEFORMATION THEORY 3.
213
Deformation Spaces Consider the metric
dlC·,·)
on uniform algebras (actually on uniform
algebras modulo the ill-understood equivalence relation is not rigid then
dl(A,B) = 0).
A.
If
is known that the nearby points consist exactly of algebras
A
= A(R)
A(R')
with
R in terms of the classical moduli for bordered Riemann surfaces.
the collection
A
A is not an isolated point in this metric space and one
would like to describe the points of this space near near
If
then it R' Thus
r, 1; 0 < r < I} is an open set in the space of uniform algebras (with the natural topology). The algebra AO is more complicated. Every neighborhood of AO contains algebras Ae for small e and also algebras of analytic functions on non-planar Riemann surfaces. Roughly here is how this happens. The ideal I in AO of functions which vanish at the origin is 2 3 2 3 generated by x(z) = z and y(z) = z which satisfy the relation y = x • Under small deformation of the multiplicative structure this can be replaced by 2 3 2 y = x + ax + ~x + y for small numbers a,~,y. This is the equation of a Riemann surface of genus 1. (Recall that the Weierstrass p function satisfies (p') 2 = p3 + ap2 + bp + c.) This observat ion suggests that AO can be obtained as a limit of algebras of functions defined on a torus with a disk removed -- and that is in fact correct. These results center on analytic functions of one complex variable. Essentially nothing is known about the several variable theory.
1.11.
{A
VARIATIONS The proofs which we have been omitting make crucial use of the full power
of the Gelfand transform in the uniform algebra arena.
The arguments do not
even extend directly to general commutative semi-simple Banach algebras.
How-
ever similar problems have been studied for general Banach algebras and for various classes of
op~rator
algebras.
The perturbation theory for general Banach algebras is closely related to questions involving Banach algebra cohomology -- see [ 7] for an introduction and further references.
(The close formal relation between a moduli space for
deformation of an algebra and algebra cohomology is presented in [4].) Purturbation theory for various classes of operator algebras has been studied extensively.
The theory is similar in philosophy to that for uniform
algebras but the details are very different.
A good introduction is given in
12] •
IV.
PROOFS We now outline very briefly some of the steps used in establishing the
214
ROCHBERG
results just described. Suppose map of
A and
A to
B are uniform algebras and
B with
(3)
T(l) = 1 1 -
T
is an invertible linear
and which satisfies, for some small
e :5 l!!11 IIf II
:5 1
+ e
for all
f E A.
Ap = {f E A; IIf II fairly straightforward to show that there is a unique point
Let
be a peak point in M(A)
p
which all functions in
T(A)
-
and let
..
If (p) I }. It is p' in M(B) at
are large; Le., ITf(pf) I ::: (1-3 e) IITfll.
p
then possible to show that the map of each such is a homeomorphism between
e.
OA
and
aB,
It is
to the corresponding
p
and that for all
f
in
A,
p'
p
in
OA, tTf)(p')
(4)
= f(p)
+ small error.
This idea is basic in establishing the equivalence of the three types of deformation. Suppose now that then, by (4),
IT(z)
A
I~1
= A(D), on
i.e.,
aBo
A is the disk algebra.
Hence
(Tz)B
which is easily checked to be of codimension consequence of (4», ideal in
B.
Tz
A similar analysis holds for
is a closed subspace of
1.
Since
is not invertible in
to show that if
~
(Tz)B
T(fg)
M(B)\aB
is the center of an analytic disk.
uses the fact that all the maximal ideals in
).•
This idea can be extended
B is a small deformation of an algebra of the form
then every point of
(another
(T(z»-l({w; Iwl < c})
c,
(Tz)B.
B
is a maximal
for small numbers
T(z+\)
near
TfTg
Thus
B.
Using this information one checks that for small gives an analytic structure in M(B)
If (3) holds,
M(A)\aA
A(R)
This analysis
are principal.
The
techniques can be extended to deal with ideals with powers contained in princi2 2 pal ideals (for example, I ~ z AO in the example in II 3).
v.
QUESTIONS Deformation theory for uniform algebras is still in its infancy, and most
natural questions are still open.
Here are two I think are especially worth-
while. 1. tions
Let
B- be the unit ball in
continuous on
mations of
A(B)?
B
(2
and analytic on
and B.
A(B)
the ball algebra, func-
What are the possible small defor-
One reason I think this is a worthwhile question is that it
will require techniques which are essentially different from those used so far (which are oriented toward analytic structure of one complex dimension). ther reason is that the deformation theory for open sets in
€n
Ano-
is both very
hard and relatively rudimentary [ 5] -- perhaps a function algebra point of view can make a contribution.
DEFORMATION THEORY 2.
215
When can a small deformation of an algebra
one parameter family of deformations?
If
(A,x)
is an e-deformation of
is there a one parameter family of multiplications fxeg with the
Ri
= fg +
bilinear maps and
A be put in a smooth A,
xe with
2
eRl (f,g) + e R2 (f,g) ••• Xl
=
x
(and with appropriate continuity and
convergence conditions?) When this can be done then the diverse approaches of [2 ], ["4], [6J, [13 J and
n. 7]
can be unif ted substantially.
The
0
pe.rators
Ri
are related to interesting operators in classical analysis [15] and can be viewed as elements of algebra cohomology groups [4].
Finally, existence of
such families is related to the possibility of putting analytic disks in the moduli space.
VI.
BACKGROUND AND REFERENCES MOst of the uniform algebra results which were mentioned are proved in
[8-10] and [12-l6J, and the references given there.
REFERENCES 1.
Bers, L., Uniformization, Moduli and Kleinian groups, Bull. Lond. Math. Soc. 4 (1972) 257-300.
2.
Christensen, E., Derivations and perturbations, Proc. Symp. Pure Math. 38 Part 2 (1982), 261-273.
3.
Cambern, M., On isomorphisms with small bounds, Proc. Amer. Math. Soc. 18 (1967),1062-1066.
4.
Gerstenhaber, M., On the deformation of rings and algebras, Ann. of Math. 79 (1964),59-103.
5.
Greene, R.E. and Krantz, S.G., Deformation of complex structures, estimates for the equation, and stability of the Bergman kernel, Advances in Math, 43 (1982), 1-86.
6.
Johnson, B.E., Perturbation of Banach algebras, Proc. Lond. Math. Soc. (3) 35 (19771 439-458 •
. 7.
, Low dimensional cohomology of Banach algebras, Proc. Symp. Pure Math, 38 Part 2 (1982), 253-259.
8.
Jarosz, K~, Perturbations of uniform algebras, Bull. London Math. Soc. 15 (1983~ 133-138.
9.
, The uniqueness of multiplication in function algebras, manuscript,1982.
10. 11. 12.
a
Metric and algebraic perturbations of function algebras, Proc. Edinburgh Math. Soc. 26 (1983), 383-392. Pa1amodov, V.P., Deformation of complex spaces, Russian Math. Surveys, 31.3 (19761 129-197. Rochberg, R., The Banach-Mazur distance between function algebras on degenerating Riemann surfaces, Lecture Notes in Mathematics 604 (1977),82-94.
216
ROCHBERG
13.
, Deformation of uniform algebras, Proc. Lond. Math. Soc. (3) 39 (1979),93-118.
14.
, The disk algebra is rigid, Proc. Lond. Math. Soc. (3) 39 (1979),119-130.
15.
, A Hankel ~pe operator arising in deformation theory, Proc. Symp. Pure Math. 35 Part I (1979), 457-458.
16.
, Deformation of uniform algebras on Riemann surfaces, manuscript, 1982.
17.
Raeburn, I. and Taylor, J.L., Hochschi1d cohomology and perturbation of Banach algebras, J. Functional Anal. 25 (1977),258-267.
18.
Sundararaman, D., Moduli, deformations and classifications of compact complex manifolds, Pitman Publishing Limited, London, 1980. DEPARTMENT OF MATHEMATICS WASHINGTON UNIVERSITY ST. LOUIS, MISSOURI 63130
Contemporary Mathematics Volume 32, 1984
NEVANLINNA'S INTERPOLATION THEOREM REVISITED Walter Rudin
*
In this lecture I shall present a strengthened version of Nevanlinna's classical interpolation theorem in the open unit disc
U c ¢.
The result
(Theorem 5) was suggested by A.B. Aleksandrov's recent solution [1] of the inner function problem in the open unit ball of
¢n(n> 1),
although its proof
uses nothing from several variables. I had several interesting conversations about this topic with David Stegenga while visiting the University of Hawaii. To state Nevanlinna's theorem, let 00
H (U),
and let
description of PROPOSITION. (i)
W denote the closed unit ball of
E be the set of extreme points of
W.
E: For
fEW,
the following are equivalent:
fEE.
fff log(l - If*(e ie )1 2 )de
(ii)
-rr
Jff log log .-rr
(iii)
1 If*(e ie )\ f*
Here, and later,
de
=
= -. -00
•
denotes the radial boundary values of
It is well known that (i)
~
(ii); see [5], [4, p. 138].
of (ii) and (iii) is an easy exercise. Recall that an inner function is an THEOREM 1 (Nevanlinna). exists an
There is a very simple
00
h E H (U)
If
f E W\E
and
f EH
00
with
If * I
f.
The equivalence =
1
a.e ••
B is a Blaschke product, then there
so that f
+ Bh
is inner. In the usual formulation of Nevanlinna's theorem (see, for instance, p.15! of [2]) the hypothesis is that two distinct members of set of
B,
on that set.
W coincide on the zero-
and the conclusion is that some inner function coincides with them Since this looks a bit different from Theorem 1, let me indicate
* This
research was partially supported by NSF Grant MCS 8100782 and by the William F. Vilas Trust Estate. © 1984 American Mathematical Society 0271-4132/8451.00 + $.25 per page 217
218
RUDIN
a proof of the latter: Since
f
E W\E,
oo
there exists
g E H , g ~ 0, with Ifl2 + Igl 2 ~ 1, hence
parallelogram law, this implies
If I + -
~lgl2 ~
f ± g E W.
By the
1.
oo
Bf + H
(on the unit circle T) contains therefore - ·12 two distinct functions of norm ~ 1, namely Bf ± ~ • The theorem of Adamyan, Arov, and Krein [2; p.1S1] implies now that laf + hi ... 1 a.e. on T, for some The coset
GO
h E H.
For this
h,
f + Bh
Q.E.D.
is inner.
The conclusion of the theorem shows, in particular, that there is an inner function that interpolates the values of
f
on the zero-set of
We now turn our attention to the open unit ball ings of
B should cause no confusion).
carries a unique rotation-invariant probability measure drov [1; p.lS4] we introduce the class A(B): I'V
(He called this class
CA(B).)
denotes the Poisson integral, in
S cr.
(the two mean-
that bounds
B
Following Aleksan-
it consists of all
B U X for some set
that have continuous extensions to
~n
B of
The unit sphere
B.
XeS
f E HOO(B) with
cr(X) = 1.
Here is one of his basic results [1; p.154]; H(B)
P
is the class of all ho10morphic functions
B:
THEOREM 2.
(Aleksandrov) •
Assume that
1 q> E L (cr), q> is lower semicontinuouson
S, and '" ~ (ii) there exists gl E A(B), gl I 0, such that Igli ~ q> a.e. on To every g2 E A(B), g2 ~ 0, corresponds then an h E H(B) such that (a) Re h ~ P[q>] in B, (b) Re h * ... q> a.e. on S, (1)
(c)
g2
divides
h;
i.e.,
h E g2H(B).
Conclusions (a) and (b) show that function in
P[q>] - Re h
B whose radial limits are
integral of a positive measure
S.
on
~
0
S
a.e.
[cr];
is a nonnegative harmonic it is thus the Poisson
that is singular with respect to
cr.
It is in this form (involving measures) that (a) and (p) are stated in [1], as well as in [7]. The next theorem is not stated in [1] but, as we shall see, it is an immediate corollary of Theorem 2, and it contains many interesting special cases that do occur in [1]; see also [3]. Smirnov class
N*(B).
the functions
10g+lf
every
Recall that r
I,
N*(B)
0 < r < 1,
e > 0 should correspond a
The proper setting for it seems to be the consists of all
YeS
with
cr(Y) < 0
for which
form a uniformly integrable family: 0 > 0
such that
fylog+lf(r~) Idcr(~) for every
f E H(B)
< e
and for every
r E (0,1).
to
NEVANLINNA'S INTERPOLATION THEOREM THEOREM 3.
219
Assume that
(i)
f
E N* (B), f
(ii)
t
~ If * I
'=
0;
s, t/ If * I
a.e. on
agrees a.e. with some lower semicont-
inuous function, and
f slog 'lr da < (iii)
00;
gl E A(B), gl ,0,
there exists
such that
Ig~ I ~ lo~ a.e. on To every (a) (b)
F 0,
g2 E A(B) , g2
S.
If I corresponds then a function
F such that
F E N*(B) , \F
* I = 'it
on S , have the same zeros 'in
a •e •
(c)
F and
f
(d)
F E f + g2·H(B).
Note that (d) implies that
F (z)
;;0
f Cz}
B,
wherever
g2 (z)
matches (interpolates) f on the zero-variety of g2' If 0 < p ~ ~ and the data f and t are in HP(B) pectively, then
F
is also in
I'l
and
Thus
O.
LP(a),
F
res-
HP(B).
To prove Theorem 3, apply Theorem 2 to ep
Note that
ep E L1 (a)
since
= log
--\- • * If \ fsloglf Ida> _00 [6, p.8S].
Let
h
be given by
Theorem 2, and put h F = fe •
The assumption
f E N*(B)
implies 10glfl ~ p[logff * I].
Hence 10giFI ~ p[loglf * I] + prep] so that
10g+IFI
= P[log t]
is dominated by the Poisson integral of
+
1
log 'it E L (a).
This
gives the required uniform integrability, and proves (a). (b) follows from Theorem 2(b); (c) is obvious, and (d) holds because g2 divides hand h divides eh - 1, hence F - f. Q,E.D. Here are some special cases of interest:
1 (1) Take 'it = 1, choose f E H00 (B), f ~ 0, If\ < 2' so that If * I agrees a.e. with some upper semicontinuous function, Theorem 3 furnishes inner functions F in B with the same zeros as f.
(2)
= zn'
Take
f E HOO (Bn _1 ), If I ~ 1,
Conclusion:
'" - 1.
There is an inner function F(zl'····zn_l'O)
Apply Theorem 3 with F
in
Bn
= f(zl,···,zn_l)·
so that
81 • 82
220
RUDIN
To see this, one has to verify (iii), but this is an easy consequence of the Schwarz lemma. (3)
V
Take
= 1.
f
1s then seen to be
Every bounded, lower semicontinuous, strictly positive
IF * I
F E H~ (8).
a.e. for some
This last application shows, incidentally, that the lower semicontinuity hypothesis cannot be dropped from Theorems 2 and 3 when
n > 1,
has then what I have called the LSC property; see [7]. ~
examples, that
"g2 E A"
cannot be replaced by
do not know whether the same is true of ~
keep
0
sufficiently far away from
gl.
when
mai~ pu~pose
,/(f I
and to keep
H (B)
One can also show, by m
"g2 E H"
The
~
because n > 1,
of
gl
away from
but I
1s to
1.
The question arises now whether these various continuity assumptions can n = 1.
be dropped when
The answer is affirmative, and the proof turns out to
be a surprisingly simple application of Theorem 1. Here is Aleksandrov's theorem: Assume that
THEOREM 4.
~ E Ll(T), ~ ~ 0,
(i)
m
g E H (U), g
To every
~
(a)
Re h
(b)
Re h *
(c)
h E g.H(U}.
PROOF:
-....
~(eie)de >
I1T log -1T
(ii)
~
P[q']
=
~
corresEonds "then an
0,
in
U,
a.e.
on
There is an F E HeU)
T,
with
in
U,
fEW, f
~
1.
= P[~].
Re F F
for some
such that
h E HeU)
Since
Re F > 0,
l+f
= 1-f
Consequently,
so that 10g(1 - If * I2 )
T.
=
log ~ + 2 10gll - f * I ~
E. Let B be the Blaschke product with the same zeros as g. By Theorem 1 there is an inner" function u in U such that B divides f - u. Thus g divides f - u. Put
a.e. on
Our first proposition shows therefore that
h = l+f _
l-f Then
f - u
divides
h.
This proves (c). Re h
Since
u
is inner,
f
.!±!!. . l-u
Also,
= P[~] _ Rel+U •
Re{(l+u)/(l-u)}
1-u
is positive in
U and has boundary values
NEVANLINNA'S INTERPOLATION THEOREM
O' a.e.
on
T.
221
Q.E.D.
This gives (a) and (b).
We now come to the announced stronger form of Theorem 1: THEOREM 5.
Assume that f E N.(U), f
(i)
If• I
t ~
t
0,
a.e. ~ T, n ~ i9 I_nlog 10&lf.l(e )de >
(ii) (iii)
g E Hm(U), g ~ 0,
To every
_me
corresponds then a function
(a)
F E N.(U) ,
(b) (c)
IF I = t a.e. on T, F and f have the same zeros in
(d)
F
F
such that
• E
U,
f + g·HCU).
This follows from Theorem 4 in precisely the way in which Theorem 3 was proved from Theorem 2. When t
=
1
and
g
is a Blaschke product, this is Theorem 1, but with
(c) as an added conclusion. 1 has no zeros in
U,
In particular, if the given function
in Theorem
then the interpolation can be done by a zero-free (i.e.,
singular) inner function. One final remark: The desired function outer function whose absolute value is formed with the zeros of satisfy (a), (b), (c).
f
f,
F must be the product of (1) the
'it on T,
(2) the Blaschke product
and (3) some singular inner function, in order to
The point of Theorem 5 is simply that the singular inner
factor can be so chosen that the interpolation property (d) holds as well. REFERENCES 1.
A.B. Aleksandrov, Existence of inner functions in the unit ball, Mat. Sb. 118 (160), N2(6) (1982), 147-163.
2.
John B. Garnett, Bounded Analytic Functions, Academic Press, 1981.
3.
Monique Hakim and Nessim Sibony, Va leurs au bord des modules de fonctions holomorphes, Math. Ann. 264 (1983), 197-210.
4.
Kenneth Hoffman, Banach Spaces of Analytic Functions, Prentice Hall, 1962.
5.
Karel de Leeuw and Walter Rudin, Extreme points and extremum problems in HI, Pacific J. Math. 8 (1958), 467-485. Walter Rudin, Function Theory in the Unit Ball of ,n , Springer Verlag, 1980. Walter Rudin, Inner functions in the unit ball of Cn , J. Functional Analysis 50 (1983), 100-126.
6. 7.
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN-MADISON MADISON, WISCONSIN 53706
Contemporary Mathematics Volume 32, 1984
UNBOUNDED DERIVATIONS IN C*-ALGEBRAS AND STATISTICAL MECHANICS (KMS states, bounded perturbations and phase transition)
1.
INTRODUCTION There is a good possibility that the theory of quantum lattice systems
in statistical mechanics may be well-developed within the theory of unbounded derivations in C*-algebras.
In fact, many theorems in the theory of quantum
lattice systems have been formulated for
l~ormal
hyperfinite C*-algebras (called UHF algebras).
*-derivations in uniformly One of the most ambitious pro-
grams in the theory of unbounded derivations is to develop statistical mechanics within the C*-frame work.
Especially the abstraction and generaliza-
tion of the phase transition theory in classical lattice systems to the C*theory, including quantum lattice systems is one of the most important subjects. This program is not so easy, because the phase transition has not been established even for the three-dimensional Heisenberg ferromagnet with nearest interaction (for the anti-ferrogmagnet it has been proved by Dyson. Lieb and it Simon). In this paper, as a step to bring the phase transition into the C theory, we shall study KMS states in detail, and as an application, we shall show the absence theorem of phase transition in lattice systems with bounded surface energy in the most general form.
This was previously done for normal
it-derivations in UHF algebras ([2], [7], [11]). eliminate the assumption of UHF algebras.
In this paper, we shall
This becomes possible, because the
set of all KMS states obtained after bounded perturbations with bound less than a fixed number is relatively weakly compact in the set of all normal states defined by a starting KMS state (Theorem 2.2, (8».
Because of the restriction
of space, most of the theorems will be stated without proof.
More details on
the matters discussed here will appear in my forthcoming book [12]. it C -DYNAMICAL SYSTEMS AND KMS STATES it Let A be a C -algebra with identity and t -+ at be a strongly continuous one-parameter group of *-autoNorphisms on A. The system {A,a} is called
2.
a C*-dynamics. generator of Let
0(5)
Let a;
a
then
t
= exp to (t E R), where 0 is the infinitesimal 5
is a well-behaved closed *-derivation in
be the domain of
5;
then
V(5)
is a dense
A.
it-subalgebra
© 1984 American Mathematical Society 0271·4132/84 $1.00 + $.25 per page
223
224
SAKAI
of
A and
~i)
e(ab)
e
satisfies the following properties:
= e(a)b +
ae(b)
(a,b E
Vee»~;
* e(a)
(ii)
= e(a) *
(a E
Vee»~.
GO
An element
a
n V(6 n )
in
n=l
is said to be analytic if there is a positive
'"'
such that
~ (a E A). Let n=O A(6) be the set of all analytic elements in A with respect to e. a(EA(e» : /Ie n (a)/1 n is said to be entire analytic if u , r < + '"' for all positive numbers
number
r
n=O
r.
n.
The set of all entire analytic elements with respect to
Al (c).
a(EAl(e»
A with
a *-subalgebra of
A2 (e). A and it is dense in A.
2.1.
Lo:!t
to
DEFINITION. on
e
is denoted by
is said to be a KMS state for
A
It is easily seen that
be a C*-dynamics.
{A,a}
the complex plane so that
for
\3
~ 0
F a,b Im(z) ~
Fa,b(t) =
and
The KMS condition gives every
~
\3
0 < Im(z) <
on the strip
\3
for
\3
in
< 0)
(resp. 0 > Im(z) > (3)
Fa ,b(t+il3) = 4>13(a t (b)a) evide~ce
is
if
{A,a} at inverse temperature (resp. 0
which is analytic on
A2 (e)
\3, a state
For a real number
a,b E A, there is a bounded continuous function
Sp = {z E tlO ~ Im(z) ~ 13}
Ma
The set of all geometric elements in
a
r~spect
is denoted by
is said to be 6eometric if there is a positive number
/Ion(a)1I ~ Mnllall (n=1,2, ••• ).
such that
e
for
t E R.
of being the abstract formulation
of the condition for equilibrium of states (cf. [3]). Let
4>
be a KMS state for {A,a} at
\3;
then
4>
is invariant under
a - i.e., 4>(a t (a» = 4>(a) (t E R, a E A). Let {TT! be ti.le GNS representation of A constructed via 4>. Put uq,(t)a
* 2 = 4>(a t (a) at(a» = 4>(a t (a*a» = q,(a*a) = la4>"' U4>(t) can be uniquely extended to a unitary operator on H(t l + t 2)aq, = (a tl + t2 (a»"
2
=
2
lIu4>(t)a4> - aq,1I = 4>«a t (a) - a)f:(at(a) - a» = ep(a*a - at(a*)a - a*at(a) + a*a) ~ 0 (t ~ 0). Hence t ~ Uep(t) is a strongly continuous one-parameter group of lluitary operators in Uep(t) TTep(a) Uep(-t)bep
=
Hep.
Moreover,
(at(aa-t(b»)ep = (at(a)b)ep
= TTep(at(a»bep
(a, b E A).
Hence Uep(t) TTep(a) Uep(-t) = TTq,(at(a» (a E A). Namely, {epcp' Uep' Hq,} is a covariant representatlon of the system {A,a}. By Stone's theorem, there is a self-adjuint operator
Hep
in
Hep
such that
Uep(t) = exp itHep •
Now we :li.lall study the relatir'n. bl:.!t:ween Y.MS states and tions.
Let
q,
let {TT4>,Uep,Hep}
be a KMS state for
{A,a} with
boun~ed
at = exp te (t E R)
be the covariant representation of
{A,a}
perturbaat
13
constructed via
and
ep
UNBOUNDED DERIVATIONS and let U~(t) = exp [13], we have
itH~.
p
CD
~
~ (-1) P p=O
p-
k(=k*) E A2 (o), by the theory of semi-groups
...
exp - t(H~+rr~(k»
•••• exp«-s +s
For
~ ~ (-1)~ (exp-sIH~)
JO;;as 1;;as 2;;a ••• ;;as p;;a t
p=O
l)H~)rr ~
~
225
(k) exp«-t+s
p
rr~(k)exp«-s2+s1)H~)rr~(k) ••••
)H~)dslds2 ••• ds ~
p
J
rr ~ (a. (k»1T (a. (k» ••• rr ~ (at" (k» (exp-tH~)dS1ds2' .ds ~ 1S1 ~ 18 2 ~ 8p ~ P O~s1~s2:a ••• s p;;at CO>
exp-t(Hcp+rrcp(k»exp tHcp = ~ (-l)P!rrcp(a is (k»rrcp(a iS (k» ••• p-O 0;;asl=s2=' < 1 •• :is p:it 2
Hence
(t E R), where
••• rr (a. (k»ds 1 ds 2 ••• ds E rr~(A) cp 15 p P ~ of ( • ).
(
)
is the closure
00
lIexp-t(H~+rr~(k»exp tH~1I
Moreover
~
~
~
~
~
p=O
flla t
51
(k)lIl1a. (k)1I 15 2
O;;a51;;a52;;a···;;a5p~t
••• lIa.1S (k) IIdslds2' •• ds P • p
Suppose that lIon(k)1I ~ Mnllkll (n=0,1,2, ••• ); then on(k) (is.) n lIa 1S .(k)1I = lI(exp is j o)(k)1I = II ~O nl J 1/ J n n n00
00
::::
~
M
n=O
Is I ,j
n.
Hls.1
IIkll = e
J
Ilk II
(j=I,2, ... p).
Hence
= i[H~+rrcp(k),rr~(a)] = rr~«O+6ik)(a» and so 1T~ ( exp
t ( o+Oik )() a)
in the strong operator top01cgy of
__ eit(H~+rr~(k»1T~(a) e-it(H~+rr~(k». ~
~
~
~
~
B(Hcp)'
226
SAKAI ~ ~
Let
k
(x)
=
(H +rr (k» tk ck
~Htk
2
2
(rr (x)e
e
tjJ
k
~(H +IT' (k»
_
1
ck
e
cp'
k
E A). Then cp (x)/cp (1) is a KMS state for k at (3. In fact, for a,b t: A2 (O+6 ik ), cp (a exp (x
-p (llcp +11 cp (k» ( 1f () a e
rr
tjJ
(b)
e
cp
e 2
~H
( e 2
e
2
[3H
:J. 2
= (e ~
~(H
-
ck
2
e
~H
cp
e
'jH
:J 2
e
e
2
1
e
2
cp ) _ ?(H~+11!l!(k»
pHck
~Hcp
-
_ PH~
2
Tlcp(b) e
e
2
2
e
p(H!l!+rrcke K»)
~Hp' 2
e
2
e
TTcp(a).
PHck
2
e 2
1
cp'
1) ;p
= elf cp (b)rr cp (a)
e
(use that
is a KMS state for
tjJ
=
~
2
e
p{H2+11 ~ (k»
e
i~(o+Oik)(b»
lq"lcp)
+11 (k» !l!
ellck +n !l! (k»
e
PH!l! e
cp
cp'
2
rrcp(a)e
r~ (Hcp +rr ~)
rr (b)e
1
P (H +rr (k» !l! P.
2
{A,exp t(o+Oik) (t E R)}
[:' (Hck+rr ck (k»
pliq,
J _ P(Hck+1T~(k»
=
tcp)
I
2
e
2
e
i3 (Hcp +rr cp (k) ) •
~) (Irq, +rrip (k) )
(
~Htk
ck·
2
e
Now we shall show that
{A, exp to}
for
cp
00
cp
~).
at
eiz(Hcp +rr cp (k» e -izH E rr (A)
'" e' )p e iz(H cp +rr cp ek» e -izHcp =11 cf> ( u 1Z
1 )
z E t.
In fact,
f
p=o
Hence e(e
e
Mizi
)
izeH +IT (k»
cp
cp
e
II k lllzl.
For
-izH
cp E rr (A)
cp
and
III. e iz(Hcp+rr cp (k» e -izHcp " ~
b E A 2 (o),
P(Hck +Tr q, (k» 1 )
cp
_
e
~eH
+11 (k»
CP!P 2
=
(e
2
~ (H
+rr (k»
ck ck 2
rr (b)e
cp
~(H
d>
cp
_ ~ (Hck +rr ck (k) )
+rr (k»
2
1 ,
e
cb
lcp' e
2
lq)
UNBOUNDED DERIVATIONS ~ (H
Since
+rr (k»
p
2
= (Tfcp(b) e
lcp) (the invariance
is dense in
A2 (5)
k
cp (a) = (Tf~(a)e
227
-~(H
k
A,
-~(H
= (rrcp(a)e
q> (a)
of KMS states).
+rr (k»
cp
cp
lcp' lcp) (a E A).
+rr (k»
cp cp
lcp' lcp)
-~ (Hcp +Tf cp (k) ) ~Hcp
(rrcp(a)e
e -~(H
=
(Tlcp (a)e
~H
+rr (k»
cp
cp
e
-~(H +rr (k»
where
cp
e
2.2. THEOREM. for
141 , lcp> CPlq,' 141 )
(a E A)
~H
41
e
41 E rrq,(A).
Let
cp
be a KMS state for
~ ~ 0 (resp.
Then we can show the following theorem.
~ ~ Im(z) ~ 0
adjoint portion of the weak closure (z,h) ~ f(z,h)
a mapping
~ < 0)
for
M of
S~ x MS
of
= {ziO
at~, s~
{A,a}
and let
rrq,(a)
in
~
Im(z)
~ ~}
M b~ _~I.!..~self S
Then there is
Hq,
into the predual
M.
of
M satisfy-
ing the following conditions: for (2) s~
x E j'.I, hEMs, f (z, h) (x)
For
If a directed set
number) converges to
h
{f(z,ha )} converges to compact subset of S~. (4) f(i~,h)
For
hEMs,
{ha.}
f(z,h)
f(O,h)
IIhali ~ M
with
f(i~,h){eit(Hcp+h)e-itHq, x)
in the norm of
=~
,where
p03i~ive
is a faithful normal
(t E R)
M.
¢(x)
h E
tl,
and
=
(xlcp,lcp)(x EM),
f(t + i~,h)(x) f(t,h){x)
2
)
and
lq,' e
2
1 41 )
M,
and
and
=
= f(O,h)(xeit(Hcp+h)e-itllcp)
f(i~,h)
~(Hp +h)
~(Hp+h)
e
2
lcp E Vee
B(Hcp)' tilen
uniformly on every
~(Hp+h)
For
(M, Uixed
linear functional on
and moreover,
t E R.
= (x
s~.
in the strong operator topology of
eit(Hcp+h)e-itHcp E M (t E R)
(5)
of s M
in
s~.
is a bounded continuous func tion on
s~
and is analytic in the interior (3)
for
z E
(x EM).
(x)
..
228
SAKAI Im(z)(H!/? +h)
f(z,h)(lH)
=
( e iRe(z) e
2
1) 4>
cJ>
for
E S~.
7.
And, if {h} converges strongly to a Im(z)(H +h ) -
cp
M, then r.,1( z) (Hp+h)
y.
2
{e
h with
2
converges to {e
Im(z) - ~)
in the norm of Let 0 be a bounded *-derivation on A and let TI (0 (a» = -- 0 4> 4> i[h,n (a)] (a E A) with hEMs (cf. [9]); then tea) = 4> -f(lp,h)(nep(a»/f(ip,h)(lH ) (a E A) is a KMS state for {A,exp t(o+OO) (6)
(t E R)}
at
(7)
For
(8)
For
I~.
where
u
h,k E MS ,
= exp to (t E R). t Iplmax{lIhll,lI k ll} lIf(z,h)-f (z,k) II ~ Ip Ie IIh-kl! (z ESp).
y > 0, let
ry
relatively a(M:,:,M)-compact in
ry
closure of
in
= {f(ip,h) I M*.
(i)
and is analytic
~
IFt,;(Z)(X) I F~(O)(xx*)
k 2
for each
F~(Z)
elPIYllxll,
IF~(t+iP)(x) I
(x EM); (ii)
and
l~tF~(t+iP)(X) I ~
and
~(x)
= F~(O)(x)
F~
Sp
be the
is
a(M*,,\I)-
is a faithful normal ~
E r , there is a bounded Y
satisfying the following Sp
Sp' and
F~(iP)(x*X)~
(iii)
and
F~(t)(X)
x E M,
yelP IYllxlla.e.;
IF~(t)(x) I ~ F~(t+iP)(x)
and
d
IdtF~(t)(x) I ~ yllxll
~(x)
=
F~(iP)(x)
a.e.,
(x E M),
(x EM).
and
a(M*,M)-closure of on
ry
1'y
is bounded continuous on
t E R, and
>
0
S2
let
f(0,h)(h2)~ ~ Y
a(M*,M)-relatively compact subset in (the
of
:::
for each
More generally, for y
f(ip,h)(h2)~ ~ Y
Sp
F~(z)(x)
x E M,
are differentiable for almost all
(9)
on
o the interior Sp
1n
ry
in -
~
M and for each
M*-yalued continuous function ~F~erties:
Moreover, let
M*; then each
positive linear functional on
Ilhl! ~ y, hEMs}; then
M*.
y
= {f(ip,h)
hEMs}; then
f(iP ,h) (LH S2
Y
) ~ y,
ep is again a
Furthermore, for each
~
-
E S2 y
S2y in M*) there is a bounded continuous function satisfying the same properties occurring in (8).
UNBOUNDED DERIVATIONS REMARK.
229
The assertions (1), (2), (3), (5) and (6) were proved by H. Araki
[1] in slightly different forms. The assertions, (8) and (9) are new. These assertions are the key lemmas to show a generalized absence theorem of phase transition. In mathematical physics, it is important to study the strong convergence of the one-parameter groups of *-automorphisms. 2.3. DEFINITION. Let an t ~ an,t (n=1,2, ••• ) and a : t ~ at be a family of strongly continuous one-parameter groups of *-automorphisms on a C*-algebra
A.
a
is said to be a strong limit of
{a} n
= strong
(denoted by
lim a or at = strong lim a ) if lI a n ,t(a) - a t (a)1I ~ 0 n n n n,t uniformly on every compact subset of R for each fixed a E A. (By using
a
the Baire's category theorem, one can easily see that (si.mple convergence) for every !la.n,t(a) - at(a)/I ~ 2.4. PROPOSITI0N.
110.n, tea)
- at(a)1I ~ 0
a E A implies the uniform convergence
0 on every compact subset of
R.)
at"" exp to and at = exp to ; then n, -1 n-=l at = strong- lim a n,t-iff (1-0) ~ (1-0) strongly in B(A), where n -is the algebra of all bounded operators on A. PROOF.
Let
By the Kato-Trotter theorem ([13]) in semi-group theory,
(1-0 )-1 ~ (1-0)-1 (strongly) is equivalent to
110.n, tea) - a t (a)1I ~
n
t ~ O.
for
For
t <
/lan,t(a) - at(a)!!
0,
,. /I (a_ t - an,_t)(a t (a» /I ~ 0 (n-+-).
Now suppose that
Hence
= strong
a
Let
n
Let
lim a.
*
on
a n,t - exp to n ; then
A and
weak closure of neAl such that [H6 (a» 'n
(exp tih n ) n(a)exp(-tih n ) (a E A). following definition.
inner if there exists a sequence
(h) n
= i[hn ,n(a)]
=
A C*-dynamics
a - strong
is a bounded *-derivation on
the well-known theorem [9], there is a sequence
2.5. DEFINITION.
q.e.d.
n
{n,H} bp. any *-representation of A on a Hilbert space
n(an, tea»~
(n-+-)
{a In-I,2, ••• } is a sequence of uniformly continuous n
n
0
""
one-parameter groups of *-automorphisms on a C -algebra lim a
B(A)
A.
H; then by
of self-adjoints in the (a E A).
Hence
This leads us to the
{A,a} is said to be weakly approximately {a} of uniformly continuous one-parameter n
groups of *-automorphisms on A such that a a sequence of bounded *-derivations (1-0 )-1 ~ (1-0)-1 (strongly), where n
R
strong lim a , i.e., there is n
{on} on A such that a - exp to. t
230
SAKAI A C*-dynamics appearing in mathematical physics usually satisfies a
stronger property than the weak approximate innerness, as follows.
A C*-dynamics
2.6. DEFINITION.
{A,a}
is said to be approximately inner
if there is a sequence
{a} of uniformly continuous one-parameter groups n of inner *-automorphisms on A such that a = strong lim a - i.e., there n
is a sequence (1-5 ih )
-1
n
(x E A).
n 1
~ (1-5)If
A such that
of self-adjoint elements in
(h)
strongly, where
at = exp t5
and
5 ih (x)
*
= i[hn,x]
n
*
is a simple C -algebra with unit (often enough for C -p;.ysics),
A
then any bounded derivation is inner ([9]), so that a weakly
approximately
inner dynamics is approximately inner in this case. In mathematical physics, we are often concerned with a C*-algebra containing an identity and an increasing sequence 1 EA
such that
n
in
=
0(5)
U An is A. In addition, n=l A satisfying the following conditions:
and the uniform closure of
n
we are given a *-derivation 5 (1)
of C*-subalgebras
{A}
00
A
...
U A; (2) n=l n A such that 5(a)
2.7. DEFINITION.
in
there is a sequence of self-adjoint elements
= i[hn ,a]
(a E A) n
{h } n
(n-l,2, ••• ).
We shall call such a *-derivation a general normal
A (we shall define normal *-derivations in a UHF-algebra
*-derivation in
more restrictively).
2.8. PROPOSITION.
(h)
i.e., there is a sequence lim 5 ih (a) = 5(a) n
n
a E D(5).
for each
If
(1
~
5)D(6)
is dense in
A,
n
then the closure
5
of
5
in particular, {At exp t5 PROOF.
5
A is approximately inner of self-adjoint elements in A such that
Suppose that a *-derivation in
Since
5
is a generator and
exp t5 = strong lim exp t5 ih
(t E R)} is approximately inner.
is well-behaved, the density of
(1
~
5)D(5)
implies that
is a generator. 11(1 ± 5 ih )-1(1 ~ 6)(a) - (1 ~ 5)-1(1 ~ 5)(a)II n
=
II (l +
6 ih ) n
-1
-1 (1 ~ 6)(a) - (1 ~ 6 ih ) (1 . n
~ 11(1 ~ 5)(a) - (1
Since
II (l ~ 5 ih ) -111 ~ 1
± 5 ih and
± 5 ih
)(a) II ~ 0 (~) (a E
)(a)II n
D(B».
n
(1
~ 6)0(5)
are dense in
A,
n
(1 ~ 5 ih ) n
-1
~
(1
+ 5)-1 (strongly). q.e.d.
n
UNBOUNDED DERIVATIONS 2.9. PROPOSITION. is a sequence
Let
{o}
be a *-derivation in
A and suppose that there
of bounded *-derivations in
n
lim 0n(a) = o(a)
0
231
a E V(o).
for
Then if
(1
A such that
± o)V(o)
is dense, then
0
is
n
a pregenerator and
{A,exp to}
is weakly approximately inner.
The proof is the same as the proof of Proposition 2.8. 2.10. uEFINITION.
Let
0
be a general normal *-derivation in
(a E A ) (n=1,2, ••• ).
0
surface energy if there is a sequence
k
o(a) = i[h ,a] n
such that
k
n
EA
n
and
n
Ilk -h n
n
A such that
is said to have bounded n
of self-adjoint elements in
A
II = 0(1) (n=1,2, ••• ).
One-dimensional quantum lattices with finite
range interaction have
bounded surface energy. 2.11. PROPOSITION ([6]).
If a general normal *-derivation 0
bounded surface energy, then
is a pregenerator and
6
in
A has
exp t6 -
strong lim exp 6 ih • n
PROOF.
n
Suppose that
IIh -k II n
~
M (n=1,2, ••• ) •
n
is not dense in
A; then there is an
6*f = 10Mf
IIfll = 1.
and
and an element
f(
00
aO(=a O*)
Since in
= f* )
(10M -o)V(o)
Suppose that
* E V(6)
U A is dense in n=l n such that
A nO For a E A , (o*f)(a) = f(o(a» = f(i[h ,a]) and so n n lo*f(a)-f(i[k n ,a))1 = If(i[h n-k n ,aDI ~ 2Mllall. Since
such that A, there is an
nO
If(ao)1 ~
21 .
and
o*f = 10Mf,
110Mf (a) - f(i(kn,aD 1= If«lOMl-o ik ) (a» I ;~ 2Mllall. n
sup If «lOMl - 0ik )(a»I n II a lI;;il aEA
Hence
:£
2M.
n
On the other hand,
II (lOMl
- 0ik )(a) II ~ 10MlIali. n
Since
(lOMl - oOk )(An ) = An , there is an element 1
b(-b*)
such
n
that
(lOMl - 0ik
)(b) = a O'
Then
1 = lIaOIl
~
10Mllbil
and so
nO 1 libii ~
10M.
Put
c =
b libiT
then aO
If«lOMl - 0ik
)(c» I = If(]bO) nO
and
I~
k
II~II ~ SM, a contradiction.
232
SAKAI
Hence
(10M! - O)V(O)
o)V(o). E V(o», exp
(lOMl + (a
is dense and analogously we have the density of
Hence to
0
is a pregenerator.
= strong
Since
~
0ih (a)
o(a)
n
lim exp t 0ih • n
2.12. PROPOSITION ([8]). inner and state
at every inverse temEerature
Since
A via
{A,a}
i~
(strongly), where
't
(0 (a»
{h}
< ~ < of-j •
A and let
{TT
,H } 't 't
at
{o} on n
A such that
exp to.
=
= i[hn ,TT 't (a)]
(a E
(1
+ 0 )-1 ~ (1 + 0)-1
-
n
-
There exists a sequence of self-adjoint
V(o».
TT
't
Put
of
(A)
tPP. (x) t',n
(A) such that 't -~h n (TT (x)e 1,1) 't 't 't TT
=
-~h
(e
Lhen
{~~,n}
point of
is a sequence of states on A.
{tP~,n}
a KMS state for for
an
for
z E C and
at
Let
~.
- in fact, let
~
at
in the state space of a
be the GNS
'to
in the weak closure
n
has a KMS
{A,a}
weakly approximately inner, there is a sequence of
bounded *-derivations elements
~(_oo
't be a tracial state on
Let
representation of
TT
is weakly aPEroximately
{A,a}
SUEEose that
A has a tracial state; then the dynamics
~~
PROOF.
q.e.d.
a,b E A; then
A.
Let
~~
(x
E A);
nl , 1 )
't
't
be an accumulation
We shall show that
tPp
is
a
t = exp to ; then tPp. is a KMS state n, n t',n Fa, b ,n (z) = tPp.t',n (aan,z (b» = tPp.t',n (a(exp zo n )(b» F
a,b,n
is entire analytic.
Moreover,
-~h ith -ith ~h -~h (a)e ne nTT (b)e ne ne nl , 1 ) -~~P.::-:h-------'t.:..---------'t.:..----='tt' (TT
F b
a, ,n
(t + i~)
=
(e
=
nl , 1 )
,
't
't
=
(because and
't
is a tracial state).
Fa, b ,11 (t)
n
I :1!
~
b
a, ,n
(t
+
i~) =
tPp, (a (b)a) t',n n, t
lIailllbll expl~llIo n II (z E S~). Therefore tP~,n is Moreover, by the theory of harmonic functions
~.
([4]), there exist kernel functions Kl
F
tPpt',n (aan, t(b».
=
Moreover IFa, b ,n (z) a KMS state for a at
such that
Hence
0
and
K2
~
o.
Kl(t,z), K2 (t,z) (z E
a
S~)
and
(t E R)
UNBOUNDED DERIVATIONS
233
!.
Fa, b ,n (z) = - -Kl(Z, t)j:l... ,n (aa.n, t(b»dt + L-K _ 2 (Z, t)ct>r.:t... ,n (a n, t(b)a)dt, for 0 0 * Z E S~, where S~ is the interior of S~. Let AO be a C -subalgebra of A generated by {p(t)(b)}; then AO is separable; hence there is a subsequence
{nj } of
ct>~,nj
{n} such that
(aat(b»
~ ct>~(aat(b»
(t E R).
~ lIaanj,t(b) - aat(b)1I + Ict>~,nj(aat(b» - ct>~(aat(b»1 ~ O(n j ~ 00),
and
t(b» I ~ ilallllbil
Ict>r.:t n (aan
and
Ict>j:l n (an
j'
""j
""j
t(b)a) I
:iii
lIalillbll •
j'
Hence by the dominated convergence theorem, there is a bounded continuous function
on the strip which is analytic in the interior of the strip
F
a,b
such that
lim y-+o
3.
q.e.u.
PHASE TRANS IT ION Let us begin with the definition of phase transition.
3.1. DEFINITION. Suppose that Then
{A,a}
{A,a.}
110
be a C*-dynamics and let
{A,a}
be a real number.
~
has at least one KMS state at the inverse temperature
is said to have phase transition at
KMS states at to have
Let
{A,a.} If phase transition at
~,
then it is said
~.
If a-general normal *-derivation then by Proposition 2.11, 6
if it has at least two
~
has only one KMS state at
~.
~.
6
has bounded surface energy,
is a pregenerator and
exp tB = strong lim exp tO ih • n
Now we shall show the following theorem. 3.2. THEOREM.
Suppose that
n
has a unique tracial state
A has a unique tracial state
(consequently, *-derivation
A (n=1,2, ••• )
0
"t).
"tn
If a general normal
in A has the bounded surface energy, then the C*-dynamics
{A, exp tB (t E R)}
has a unique KMS state at
(namely it has no phase transition at
~
~
for each real number
for each real number
~).
~
SAKAI
234 PROOF.
Since
exp t6
by Proposition 2.12
strong lim exp tO ih
=
it has a KMS state forneach
{A, exp to (t E R)}
factorial KMS states for {n~ ,u~ ,H~}
A has a tracial state,
and
~.
at
~
be the covariant representation of
Let ~1'~2 , and let
be two
{A, exp to (t E R)}
con-
III
structed via for
a
A •
~
n
~l·
(exp to )(a) n has a unique tracial state,
On = 5 + 0i(k -h ); then
Let
n
Since f (i~, n
A
(k -h »(n,j, (a» n
't(ae
'f'1
n
.. f(i~,TT
Since
4>1
'tee
(k -h »(lH ) n n ~ 1
cr(M*,M)-compact in
M*(M
= TT~
n) (a E A ) • n
-'--~~k:--
n)
IIn~l (kn-h n )II ~ 0(1), by Theorem 2.2 (8),
relatively
to' k )(a) ~ n
-~k
n ~1
= (exp
n
{f(~,n,j,(k
'f'
n
-h »} n
is.
(A)"), so that by Eberlein's 1
theorem there is a subsequence
{f(iA n
h»} of {f(i~,n,j, (k -h »} n.- n. 'f'l n n J J which converges to a normal faithful state t in cr(M*,M). Hence 1'"
4>1
(k
t(l14> (a» 1
(a
Quite similarly, we start with 11
~2
(A)"
~2;
E A).
then there is a normal state
S
on
such that
(a
E A).
)
E;.(1H
q,2 t(TT~
Hence
(a» 1
E;.(nq, (a»
2
--'~--=----
t(lH) q, 1
(a E A), and so
is quasi-equivalent
E;.(lH)
q,2
q.e.d.
UNBOUNDED DERIVATIONS
235
REFERENCES 1.
H. Araki, Relative Hamiltonian for faithful normal states, pub1. RMS, Kyoto Univ. Vol. 9 (1973), 165-209.
2.
, On the uniqueness of KMS states of one-dimensional quantum lattice system, Corom. Math. Phys. 44 (1975), 1-7.
3.
R. Haag, N. Hugenholtz and M. Winnink, On the equilibrium states in quantum statistical mechanics, Comm. Math. Phys. 5 (1967), 215-236.
4.
E. Hille, Analytic Function Theory, Vols. I, II. 1959, 1962.
5.
E. Hille and R. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloquium publ. Vol. 31, Providence, 1957.
6.
A. Kishimoto, Dissipations and derivations, Comm. Math. Phys. 47 (1976), 25-32.
7.
, On uniqueness of KMS states of one-dimensional quantum lattice systems. Comm. Math. Phys. 47 (1976), 167-170.
8.
R. T. Powers and S. Sakai, Existence of ground states and KMS states for approximately inner dynamics, Comm. Math. Phys. 39 (1975), 273-288.
9.
S. Sakai, C*-algebras and W*-algebras, Springer-Verlag, New York, 1971.
Ginn & Company, Boston.
10.
, On one-parameter subgroups of *-automorphisms on operator algebras and the corresponding unbounded derivations, Amer. J. Math. 98 (1976), 427-440.
11.
, On co~autative normal *-derivations II, J. Functional Analysis 21 (1976), 203-208.
12.
, Operator algebras in dynamical systems, to appear in the series of Encyclopedia of Mathematics.
13.
T. Kato. Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
14.
P. J~rgensen, Trace states and KMS states for approximately inner dynamical one-parameter groups of *-automorphisms, Corom. Math. Phys. 53 (1977), 135-142.
DEPARTMENT OF MATHE¥ATICS FACULTY OF HUMANITIES p~ SCIENCES NIHON UNIVERSITY TOKYO, JAPAN
Contemporary Mathematics Volume 32, 1984
REMOTELY ALMOST PERIODIC FUNCTIONS Donald Sarason This paper concerns a generalization of the notion of almost periodicity which, to my knowledge, has not appeared previously in the literature.
I call
functions which are almost periodic in this generalized sense remotely almost periodic functions.
The term "asymptotically almost periodic" would perhaps be
preferable had it not already been used by M. Frechet [4] to refer to a related but rather more restricted generalization of almost periodicity. almost periodic functions form a closed subalgebra,
RAP.
of
of bounded, uniformly continuous, complex valued functions on The main result to be established here is that algebra, by
AP,
RAP
The remotely
BUC, R,
the algebra the real line.
is generated, as a Banach
the algebra of Bohr almost periodic functions, and another
algebra, called
SO, consisting of functions which oscillate slowly at
One can define
RAP
by slightly modifying the definition of
~.
AP.
The
discussion here will be limited to the real line, although it will be clear that a similar development is possible in a more general context. number, we let T t
t+a.
T
If
f
whose value at
t
a
BUC.
=
stand for the transformation on
a
is a function on is
f(T t). a
R,
then
* T f a
a
is a real
R of translation by
a:
will stand for the function
The functions we shall deal with all belong to
We shall measure the distance between two functions
by means of the supremum norm:
If
dist(f,g)
dist~(f,g)
= lim
=
IIf-gli. ~
f
and
g
in
BUC
We also define
suplf(t)-g(t)I.
It I""""
e is a positive number, the real number a is called an e-translation number of the function f provided dist(f,T *f) ~ e. We shall call a a a * remote e-translation number of f provided dist (f,Ta f) ~ e. The function f If
~
e > 0, its e-translation numbers form a relatively dense set. (A subset of R is said to be relatively dense if there is a bounded interval each of whose translates contains at least belongs to
AP
if it is in
one point of the set.) for every Like
e > 0, AP,
BUC
and, for every
We shall say that
f
is in
RAP
if it is in
BUC
and,
its remote e-translation numbers form a relatively dense set. the class
RAP
is a closed suba1gebra of
BUC.
The proof of
this statement is for the most part immediate, the only difficulty being the verification that
RAP
is closed under the formation of sums.
The same
© 1984 American Mathematical Society 0271-4132/84 51.00 + 5.25 per page
237
SARASON
238
di.fficulty arises with one does for
AP, and one can overcome it for
in the same way
[1, p. 36].
AP
The algebra
RAP
obviously contains
AP.
It also obviously contains
R that
the algebra of continuous functions on AP + GO
RAP
vanish at
GO'
The linear span
00.
is easily seen to be a closed algebra; it consists of the functions
,
that Frechet termed asymptotically almost periodic in the paper cited above. Another sUbclass of f
in
BUG
such that
RAP
is
* f-Taf
SO,
which by definition consists of all functions
is in
obviously a closed subalgebra of
Co
for every
BUC;
a.
The class
SO
is
AP,
it is nearly disjoint from
the
only functions common to both algebras being the constant functions. The considerations that led me to the paper.
RAP
will be mentioned at the end of
The bulk of the paper will be devoted to the proof of the following
assertion. THEOREM.
RAP
is the closed subalgebra of
BUC
generated by
AP
The proof will be indirect and will involve an analysis of Gelfand space (space of multiplicative linear functionals) of sis will reveal how
M(RAP)
can be built from
identify the functions in the algebras
AP, SO
transforms on the appropriate Gelfand spaces.
M(AP) and
and RAP
and
SO.
M(RAP) ,
RAP.
M(SO).
the
The analyWe shall
with their Gelfand
Each of these algebras is a
C*-algebra, so each i.s identified with the algebra of all continuous functions on its Gelfand space. We shall regard the real line, each point of
R,
as a subset of
R becomes a dense open subset of
*
M(RAP).
M(RAP)
the transformation T a
T extends to a homeomorphism of a acts as an isomorphism of the algebra RAP
and that homeomorphism is the desired extension of T
RAP. For
Under a
R,
in
onto itself.
In
onto itself; that isomor-
phism is induced by a homeomorphism of the Gelfand space by
by identifying
R with the corresponding evaluation functional on
this identification, fact,
M(RAP)
T . a
M(RAP)
onto itself,
We denote the extension
also.
a
The space M~(RAP).
M(RAP) - R
(the "fringe" of
M(RAP»
will be denoted by
It consists of two connected components, which can be thought of as
the fibers of
M(RAP)
denote the closure of
above
{T x:x E R}; a
the transfonnation group closed subset of
and
M(RAP)
ing transformation group.
-""
.
For
x
in
Mco (RAP)
we let
the latter set is the orbit of
x
is the smalles t {T:a E R}. The orbit closure S~ x a which contains x and is invariant under the precedClearly, a function in
RAP
belongs to
SO
if and
only if it is constant on
Qx for each x in M00 (RAP). The first main step in the proof of the theorem will be to show that each
of the orbit closures
Q x
is a replica of
M(AP).
Next, we shall show that
two orbit closures which are not identical are actually disjoint.
Finally, we
239
REMOTELY AUtOST PERIODIC FUNCTIONS
shall show that two distinct orbit closures can be separated by a function in Once that has been done, the theorem will be almost immediate.
SO.
M (RAP) and, for f "" whose value at t is f(TtX).
Let us fix an function on
R
in
x
because any remote e-translation number of T.* f.
number of
The function
x
g.
ishes on
T* f
RAP,
denote by
T*f
T*f is in AP, x is an ordinary e-translation
vanishes identically if and only if
x
the
x
The function
f
T* as a map of
Hence, we may as well regard
x
in
C(Q)
x
van-
f
into
x
AP,
and when so regarded it obviously preserves norms and is an algebraic isomorphism. For
s
R,
in
let
*
T (e ) = e (x)e , x s *8 S
Then
T
the range of
e
denote the exponential function
s
so the range of
T
is therefore der.c;e in
x
*
contains
x
AP.
mum norm), so it equals
AP.
Hence,
x
s
= e ist
By Bohr's theorem,
s
From the observation at the end of T*
the preceding paragraph we know that the range of
* T
e.
e (t)
is closed (in the supre-
x
gives an isomorphism of
C(Q) x
onto
AP.
*
The preceding discussion shows the
is dense in
T AP
x
AP,
so
AP IS2 x
is
C(Q ). Actually, as we shall now see, APIQ = C(Q). To establish x x x this we need only to show that APIS2 x is closed, which we can do by showing
dense in
that the restrict ion map from proving
IIfll
00
=
liTx*fll
last equality when
for all
00
f
to
AP
in
f
preserves norms.
We actually need only to prove the
AP.
is an exponential polynomial in
As is well known, the space
M(AP)
as addition.
phically) in in
M(AP).
M(AP)
w.
Now suppose that
(Here,
a finite set.)
R with its natural image
as a dense subgroup; we identify
The restriction of the functional AP.
We write the group operation on
The real line is embedded (continuously but not homeomor-
which we denote by nomial in
AP.
can be identified with the Bohr group,
the dual of the discrete real line [5, p.33l]. M(AP)
That amounts to
x
to
f = Z C(s)e s
C stands for a function on
AP s R
is an element of
M(AP)
is an exponential polywhich is
0
except on
From the equality T* f (t) x
'Ie
= Zs C(s)T xs e (t) = l C(s)e (x)e (t) s s s = Zs C(s)e s (w)e s (t) = Zs C(s)e s (w+t)
= f(w+t) , ,,:
we see that the values taken by by
f
on the coset
equality
=
IIflloo
w + R of
T f on R are the same as the values taken x M(AP). As that coset is dense in M(AP), the
* IITxfll""
follows, and so the equality
and
are in
AP1Qx
= C(Qx)
is
established.
LEMMA 1.
If
x
tical or disjoint.
y
Moo (RAP) ,
then
Q x
and
Q
y
are either iden-
240
SARASON To prove this we need only to show that, if
in
~.
on
C(~)
y
~x'
is in
then
x
is
Suppose y is in ~. Because API~ = C(~) and the restriction y x y y map is an isometry, the functional f ~ f(x) on AP can be regarded as acting point
in
Since
~.
y
f(x) for all f Because f (z) x it must be that z = x, so x is in
is in
z
= C(Q
Apls~
z
and so is represented by a point
y
in
~.
),
(x j )
Let
(Yj)
and
the same directed set)
as desired.
be two convergent nets in M".,
such that
~
~
QXj
~~~~~~~~~~~~~~~
Th.Em.
the x and because
for every
Yj
~,
c
y
AP,
x x lemma is established.
LEMMA 2.
~
(indexed by
x
Let
j.
lim x J'
=
= ~Y•
x
By virtue of Lemma 1, to prove this we need only to show that
.
The
y
is in
For that we need only to show that, for each f in RAP and each positive x number e, there is a real number c such that If(y) - f(T x)1 ~ e. ~
c
Fix
e.
and
f
real number
such that
every interval on f.
For each
interval for all
[-aj,-aj+L],
x.) - f(T
!f(Yj) - f(T
x.)
cj J
I
M(RAP) ,
e12.
<
so there is a
,
Choose
L > 0
such that
contains a remote el2-translation number of bj
c j = aj+b j •
of
f
in the
,f(z) - f(T b z)! ~ e/2
We have
j
x.)'
=
cj J
If(T
for all
< e
x.) - f(T b T x.)1 ~ e12. j aj J j. The net (c.) is bounded, so it has
aj J
J
~
Then
c.
f(T z) c
uniformly with respect to
so f (T
X ) = f (T x.) + [f (T x.) - f (T x.)] cj j c J cj J c J
f(Yj) ~ f(y),
Also
I
x. J
With no loss of generality we can assume the original net
converges, say to in
x.)
aj J
~
is in
YJ'
so, in particular,
aj J
a convergent subnet.
z
L
and let
M~(RAP),
in
z
If(YJ') - f(T
of length
R
the point
j
select a remote el2-translation number
j
If(T Hence
For each
~
f (T x). c
If(y) - f(TcX), ~ e,
so we can conclude that
as desired.
The proof of Lemma 2 is complete. LEMMA 3.
x'
If
and
there is a function
x" g
M (RAP)
are points in
in
SO
such that
g (x') '/:; g (x") •
Under the hypotheses of Lemma 3, the sets by Lemma 1. and
h
=1
Hence, there is a function on
APIQ
is an isomorphism of
point
i(x)
tion
g
on
h
in
Q ,
x
RAP
and ~" x such that
are disjoint, h
=0
on
~
x
,
~
". x We have seen that, for
x
then
such that
00
such that M00 (RAP)
x AP
f(i(x» by setting
in
M (RAP),
onto
=
f(O) g(x)
00
the restriction map from
AP
C(~).
to
Hence, ~ contains a unique x x for every f in AP. Define the func-
= h(i(x».
To complete the proof of
241
REMOTELY ALMOST PERIODIC FUNCTIONS Lemma 3 we have only to show that extend
g
g
is continuous.
to a continuous function on all of
function in
RAP,
M(RAP) ,
the continuity of
g
SO
because it
say with limit
x.
To prove
we need only to prove that
g(x) = lim g(x j ). Passing to (i (xj converges, say to y.
»
a subnet if necessary, toTe can assume the net
(This involves no loss of generality.) J
in other words, to a
and that function will actually belong to
will be constant . on every ~ x • Let (x j ) be a convergent net in Mm(RAP) ,
lim f (i(x.»
Once that is done we can
= f (0).
By Lemma 2,
If
Y is in
= h(i(x» = lim
gex)
f
is in
AP
so
52, x
then
f(y).
y" i (x).
Consequently
h(i(Xj » = lim g(xj ),
and the proof of Lemma 3 is complete. The proof of the theorem is now all but finished.
We have seen that any
two points in the same 52
are separated by a function in AP, and Lemma 3 x tells us that any two points not in the same 52 are separated by a function x in SO. The algebras AP and SO therefore together. separate the points of M(RAP).
To complete the proof of the theorem we need only to invoke the Stone-
Weierstrass theorem. The information obtained in the course of the proof above gives some insight into the structure of ral image in
M(RAP) ,
However
t
M(AP).
for
in
M(RAP).
\ve continue to identify
and we also identify it with its natural image in we let
R,
wt now for the sake of claR is not homeomorphically embedded in M(AP).) Let J be the
rity, because
M(RAP)
into the product space
t
and
M(AP) x M(SO)
pair whose first member is the restriction of ber is the restriction of in
R.
The map
the theorem.
Hence
orbit closures
52
restriction of
x
x
M(AP) x {v}.
isomorphism of
x
to
SO.
is a homeomorphism between
J
SO
with to
in M(RAP)
x
AP
to the
and whose second
J(t) = (wt,t)
SO,
then
v
M(RAP)
If
x
is in
M (SO),
lies in
me~
for
t
onto
J(Q )
the image
C (52 ), x
v
x
belongs to
Mco (RaP)
whose restriction to
SO
net in
R which converges to
v
point of that net in
M(RAP). M(RAP) ,
in
v.
AP
-
M (SO),
x,
and then take for
M(AP) x M (SO). ~
v
is the
is a subset
API~x
then there is an
To get such an
M(SO),
and
J(52) x
onto
and the
R
is an
must actually be all of
We see therefore that
and the product
we may identify with
is
and its image.
M_(RAP) and
OD
But because the restriction map of AP
x
are the singleton points of
M~(RAP).
in
On the other hand, if
Mm (RAP)
to
In particular,
M(AP) x {v}.
between
x
which sends
is clearly continuous, and it is one-to-one by virtue of
J
The level sets of
of
M(SO).
denote the corresponding functional in
(We make the distinction between
map from
R with its natu-
J
x
in
simply take a x
any cluster
gives a homeomorphism
The full range of
J,
which
is the union of the preceding product and the
242
SARASON
set
{(w ,t):t E R}. t
.. L
I was led to the space If
is
f
bol
a
function in
RAP
during some musings on Toeplitz operators •
of the real line, the Toeplitz operator with sym2
is, by definition, the operator on the Hardy space H of multiplication by f followed by projection onto H2. An interesting problem, and one f
which is only very imperfectly understood, is that of describing the spectrum of a Toeplitz operator in terms of its orily resolved when the symbol is in discussed in f)]), and an algebra [6].
~bol.
CO'
SAP
The problem has been satisfactAP + Co
AP,
(all these cases are
which lies between
AP + Co
My hope was that Toeplitz operators with symbols in RAP
and
RAP
might be sus-
to analysis. The results above suggest that this hope could probably be realized if Toeplitz operators with symbols in SO were understood. How-
c~ptible
ever, we are a long bols in
SO; M(SO),
fram being able to handle Toeplitz operators with sym-
new ideas are undoubtedly needed.
The space space,
~~y
SO
has arisen in other connections [2], [7].
seems mysterious; I am unable to announce any interesting pro-
perty of it which is not fairly obvious. mind:
Its Gelfand
Here is one question that comes to
Is there a nontrivial homeomorphism of
M.. (SO)
onto itself?
REFERENCES 1.
H. Bohr, Almost Periodic Functions, 1951.
2.
H.O. Cordes and J).A. Williams, An algebra of pseudo-differential operators with non-smooth symbol, Pacific J. Math 78 (1978), 279-290.
3.
R.G. Douglas, Banach algebra techniques in the theory of Toeplitz operators", CBMS Regional Conference Series in Mathematics, no. 15, Amer. Math. Soc., Providence, 1972.
4.
M. Frechet, Les fonctions asymptotiquement presque-periodiques, Rev. Sci. 79 (1941), 341-354.
5.
C.E. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, 1960.
6.
D. Sarason, Toeplitz operators with semi-almost periodic symbols, Duke Math. J. 44 (1977), 357-364.
7.
D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols, Indiana Univ. Math. J. 26 (1977), 817-838. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CA 94720
Chelsea Publishing Co., New York,
Contemporary Mathematics Volume 32, 1984
ANALYTIC MULTIFUNCTIONS, q-PLURISUBHARMONIC FUNCTIONS AND UNIFORM ALGEBRAS * Zbigniew Slodkowski
The purpose of this paper is to show how ideas connected with the notion of an analytic multifunction have led naturally to a solution of Basener's conjecture [4] on higher order Shilov boundaries of tensor products of uniform algebras.
The method presented here seems to be of interest in that it leads
far away from the initial problem:
we start from uniform algebras and we end
up studying convex functions, working in between with k-maximum sets, q-pseudoconvex domains and q-plurisubharmonic functions.
In the outline given below
the details are omitted; however, the basic arguments are described and all relevant propositions are formulated.
The complete presentation is to be given
in 118] and 119]. Shortly after the conference, in Summer 1983, the author has obtained considerable modifications of the proof of Basener's conjecture, to be included in I1i].
Here the initial approach, presented actually during the Rickart Con-
ference, is followed; however, a brief description of the new method is given in Sec. 7. 1.
AN~YTIC
MULTlFUNCTIONS IN DIMENSION ONE
Analytic multifunctions (a mu1tifunction=set valued function) were introduced by K. Oka Ill] and further studied by T. Nishino [10], H. Yamaguchi [25],
J. Wermer 123, 24],
z.
Slodkowski [15-17], B. Aupetit [1, 2], E. Vesentini [20,
21], and more recently by T. Ransford [12] and A. Zraibi [26]. recall only some basic results on analytic multifunctions:
Here we
they will serve as
background and motivation for our further discussion.
*Dedicated
to Professor Charles Rickart.
Acknowledgment. This research was carried out in Scuo1a Normale Superiore di Pisa, under financial support from Consiglio Nazionale delle Ricerche. The author~s participation in the Rickart Conference was made possible by financial support from Yale University. The author is grateful to those institutions, and personally to Professors Edoardo Vesentini and Philip Curtis for their hospitality and help.
243
244
SLODKOWSKI K:G ~ 2' ,
Let us recall that an upper semicontinuous multifunction
(G
c,
is open)
(cf. [11]. [15]) if the set
is said to be analytic
U={(z,w) E GXI:w ~ K(z)}
is pseudoconvex.
The main motivation for studying analytic multifunctions comes from the following fact. TIlEOREM 1.1 US].
(a)
Let
X be a
~_omplex
an analytic operator-valued function. z
-+
a(T(z»:G (b)
z
-+
Let
-+
2'
B:mach space a-::td
T:G
-+
B(X)
bl:.
Then the multifunction
is analytic.
A be a uniform algebra and
g(f-l(z»:f(MA)\f(oA)
2'
-+
f,g E A.
Then the multifunction
is analytic.
To avoid overlapping with other papers in these Proceedings (cf. those by B. Aupetit I2], D. Kumagai [9] and J. Wermer [24]). we will not discuss connections of this result with spectral theory and uniform algebras. The definition given above is often not easy to check in concrete instances.
The following characterization (Th. 1. 2) gives somt;' more convenient
conditions, e.g., (iii), (v)
and
(vii),
and perhaps also some insight into
the nature of analvtic multifunctions. To specify the terminology we will say that a function local maximum property on a locally compact set every compact set
K
C
X
nY
Let
G
c,
the inequality max flK ::: max fla~
be open and let
continuous compact-valued function. U-GxC~.
(i) (ii)
x,
For every function
For every polynomial
monic.
,
2
X~(z,w):z
if for
holds (cf.
be an upper semi-
E G, w E K(z)}
and
plurisubharmonic in a neighborhood of is subharmonic in
p(z,w),
..
G
its absolute value
Ip(z,w)
I
has the
X.
Plurisubharmonic functions have the local maximum property on If
G'
hood of the set
(vi)
u(z,w),
v(z)=max uj{z}XK(z)
local maximum property on
function
-+
Y)
has
U is pseudoconvex.
(iii)
(v)
K:G
[_w,~)
Then the following conditions are equivalent:
the function
(Iv)
Denote
-+
X (intersecting
J. Wermer I23]).
TIlEOREM 1.2 I15J.
f:Y
C
G and
u(z,zl, ••• ,zn)
is plurisubharmonic in a neighbor-
{(z,zl' ••• 'zn ):z E G',z.1 E K(z), i=1,2, ••• ,n},
v(z)=max{u(z,zl, ••• ,zn):zl, ••• ,zn E K(z)} The function
(z,w)
-+
X.
-log dist(w,K(z»:U
then the
is subharmonic in -+
[-=,+=)
G'.
is plurisubhar-
ANALYTIC MULTI FUNCTIONS
245
lit
(vii) For every z E G and for every a,b E t, the function z ~ log max{/z-wa-bl-l:w E K(z)} is subharmonic where it is defined. Part of this theorem can be generalized beyond the context of compact valued multifunctions, so that the relation between local maximum property and pseudoconvexity becomes more conspicuous: 2
THEOREM 1. 3 [15]. Let V be open in t , X C V and u=V\X. Assume that is closed in V. Then the following conditions are equivalent: (i) (ii) (iii)
is locally pseudoconvex at each point of
U
V
X
n au.
Absolute values of polynomials have the local maximum property on Plurisubharmonic functions have the local maximum property on
X.
X.
This theorem does not extend directly to higher dimensional situations; to this end conditions (i)-(iii) have to be suitably modified (cf. Sec. 2). The next result, a partial converse to Theorem 1.1, suggests that, at least from the point of view of applications, the definition of analytic multifunctions in dimension ODe is optimal.
THEOREM 1.4 [15]. Let G be an open and bounded subset of C and let K:G ~ 2 C be an analytic multifunction, such that sup{ Iwl:z E G, w E K(z)} < -. Then: (a)
there exists an analytic operator-valued function
T:G
~
H is a complex separable Hilbert space, such that a(T(z» (b)
.. K(z)
for every
z E G;
there exist a uniform algebra
f(~A)\f(aA) ~
G,
g(f-l(z»
and for every • K(z).
A and
z E G we have
f,g E A such that
B(H) ,
where
246
2.
SLODKOWSKI ANALYTIC MULTIFUNCTIONS IN SEVERAL DIMENSIONS AND q-PSEUDOCONVEXITY In view of numerous interesting applications of analytic multifunctions Aupetit, [1]), Already in the
(in dimension one) to spectral theory and uniform algebras (cf. it seems desirable to extend this notion to several dimensions.
proof of Theorem 1.4 the author found it necessary to introduce (in [15]) the following generalization of the analyticity condition (namely of (ii) of Th. 1.2): (WA)
An upper semicontinuous multifunction
to have property
(WA)
K:G
-+
If' ,G
2
if for every plurisubharmonic function
in a neighborhood of the graph
(K),
the function
v(z)
=
..,k
C
..
,
is said
u(z,w), defined
max ul{z}xK(z)
is
plurisubharmonic in G. The practical usefulness of the class
(WA)
is due mainly to the follow-
ing properties: Ki: G -+ 2eni , i-l, ••• ,s have property s 1 2 K(z) .. K (z) x ••• x K (z), mapping G into 2.1.
If
perty
(WA) ,
en ,m=m l
then
+••. + m5'
has pro-
(WA) •
2.2.
K:G
If
map, where
-+
em 2 , G c en,
W contains the graph of
multifunction of class
(WA)
has property
then
K,
(WA) •
These and other properties of the class 4, 5])
1
and F:W -+ c: is anlanalytic z -+ F({z}XK(z»:G -+ 2~ is a
(WA)
(cf. Slodkowski [15, Secs.
had motivated the author to define analytic multifunctions in several
dimensions by condition
(WA).
It appeared, however, that the class
(WA) ,
though useful, is too large, and for this class extensions of Theorem 1.2, 1.3 and 1.4 cannot be proven. In this paper we present higher dimensional versions of Theorems 1.2 and 1.3, using a condition stronger than
(WA).
(The problem of generalizing
Theorem 1.1 and 1.4 will be studied elsewhere.) It 1s rather clear that if one wants to extend Theorem 1.2 to multifunctions K:G -+ 2tm , G c c: k , then the conditions (i), (ii), (iii) and (iv) have to be changed.
For example, as
be strengthened; in contrast, as weaker.
k
increases then condition
(iii) has to
m increases, condition (i) should become
More specifically we consider the following definitions.
A locally closed subset maximum property of order L of codimension
k
in
k, en
X of 0
~
en k
~
is called a k-maximum. set (or has n-l)
intersecting
have the local maximum property on
X
n L.
if for every complex hyperplane X,
absolute values of polynomials
ANALYTIC MULTI FUNCTIONS
a
For
~
k
~
n-2
most natural examples of k-maximum sets are complex
analytic varieties of pure dimension ...,n
247
k+l.
The only (n-l)-maximum sets in
are open ones. We define a weaker notion of pseudoconvexity by a modified kontinui-
tatsatz. Let
U eVe
~
n
be open.
We say that
U is q-pseudoconvex
V if
in
there does not exist a (q+2)-dimensional complex hyperplane L and a sequence of (q+l)-dimensional analytic discs M, n ~ 0, such that the following condin
M c U n L, n ~ 1; M U aM ~ M U aMo; aMn ~ aMo; aMn C U, n ~ 0; n n n 0 Mo c L n V, while Mo n aU n V = ¢. Of course all open subsets of t n are (n-l)-pseudoconvex, and O-pseudotions hold:
convexity is just the usual pseudoconvexity. In order to generalize conditions (ii) and (iv) of Theorem 1.2 we have to replace the class of plurisubharmonic functions by the larger one of q-plurisubharmonic functions studied by Hunt and Murray [6]. A smooth function in t n is q-plurisubharmonic if its complex Hessian has at each point at most
q
negative eigenvalues.
18] that an upper semi-continuous function only if for every smooth
u
It can be proven [slodkowski, is q-plurisubharmonic if and
(n-q-l)-plurisubharmonic function
f
the sum
u+f
has the local maximum property. G c ~k
Let
THEOREM 2. 3.
continuous multifunction. U = Gxt~.
be open and let Denote
X
=
(ii)
~
a:m
2
be an upper semi-
{(z,w):z E G, w E K(z)}
and
Then the following conditions are equivalent:
U is (m-l)-pseudoconvex in
(i)
K:G
m
Gxa: •
For every smooth (k-l)-plurisubharmonic function
neighborhood of
X,
u
defined in a
the function v(z)=max ul{z}XK(z)
is (k-l)-plurisubharmonic. (iii) (iv) on
X
is a (k-l)-maximum set.
All (k-l)-plurisubharmonic functions have the local maximum property
X. Later on we will give similar generalization of Theorem 1.3 (cf. Theorems
4.1 and 8.1 which, considered together, generalize both Theorems 1.3 and 2.3). It is tempting to consider multifunctions satisfying (any of the equivalent) conditions of Theorem 2.3 as the class of analytic·multifunctions (in sever~l
dimensions).
The notion so obtained is stronger than
are natural examples of multifunctions analytic in this sense:
(WA).
There
248
SLODKOWSKI (a)
em
F:U ~ ~, U c is analytic if and only if em {F(z)}:U ~ 2 is analytic.
The single-valued map
the multifunction (b)
Let
Assume that
z
~
Ck
B be an open ball in
Z c Bx~m
and let
Z is the polynomially convex hull of
Z
n
be a compact subset. (aBXCm). Then Z is
(the closure of) the graph of an analytic multifunction, unless (c)
Z
3B x em.
C
If
A is a uniform algebra, and fl, ••• ,fk,gl, ••• ,gm E A, then -1 K(z) = (gl' ••• '&m)(F (z», where F = (fl, •.• ,f k ) defines an analytic multik-l k-l function in F(MA) \F (a A ), where a A denotes the (k-l)-th boundary of A (cf. Sec. 3 for definition). Note that (c) generalizes Th. 1.1, Part (b). moment if part (a) of Th. 1.1
It seems unknown at this
can be generalized to several dimensions (with
the usual spectrum replaced by the joint spectrum of J. L. Taylor). 3.
BASENERtS CONJECTURE AND k-MAXIMUM SETS Higher order boundaries in the spectrum of a uniform algebra were intro-
duced by R. Basener [3] and N. Sibony [14].
As it was shown by these authors
and D. Kumagai [8, 9], these boundaries are useful in studying several dimensional analytic structures in the spectrum of a uniform algebra. o Let a A denote the usual Shilov boundary. The k-th order boundary of is defined as the closure (in MA) of the set
U
k (fl,···,fk)EA where
A
aO AIZ(fl,···,f k ) ,
Z(fl, ••• ,fk)-{x E MA:fi(x)=O, i=l, ••• ,k}. R. Basener has asked in [4] if 0.1) 'V
where
A ~ B denotes the smallest closed subalgebra of
the monomials
f(x)g(y), f E A, g E B.
C(MAx~)
containing
Cf. D. Kumagai [8] for some applications
of this conjecture. In this paper we will sketch the proof of this conjecture. sider first the set every
k
~=MA\aA.
It follows from the definition of
f l , ••• ,f k E A the intersection
local maximum property. variety of codimension context.
Since k,
~
n
Z,
where
Let us conk
a A that for
Z=Z(fl' ° ° • ,f k ). has the
Z is an abstract analogue of an analytic
one is bound to think about k-maximum sets in this
The similarity of the two situations is still more striking due to
the following fact.
ANALYTIC MULTI FUNCTIONS
PROPOSITION 3.1.
Le~
n Xc C
be locally closed.
249
Then X is a k-maxiJDUJIt set
if and only if for every k-tuple of holo.morphic functions fl, ••• ,fk , defined in a neighborhood of X, the intersection Xn Z(f1, ••• ,fk ) has the local maxiJDum property with respect to absolute values of analytic functions. In the equation (3.1) the inclusion
,~It
is rather easy to check.
The
reverse inclusion aJDounts to proving that the set
has the local maxiJDum property of order
n.
Since this set can be represented
as (3.2)
where ~ are as above, and Yr = MB"a~, we have to deterJDine the order of the maxiJDum property of the summands (which are locally closed subsets of (3.2». We consider a slightly JDOre general setting. Recall after Rickart [13] that a pair (X,A) , where X is a topological space and A is a subalgebra of C(X), is called a natural system if I E A, A separates points of X and the weak topology induced by A on X agrees with the given one. We say that (X,A) with X locally compact, has the k-maximum property (or the maxiJDum property of order k) if for every k-tuple fl, ••• ,fk E A, the functions
If I , f E A have local maximum property on
Z(f1, ••• ,fk)-{x E X:fi(x)-O,i=l, •.• ,k}. follows from the following theorem.
By (3.2) the Basener's conjecture
THEOREM 3.2 •. Let
X and Y be locally compact and let (X,A) and (Y,B) be natural systems. If (X,A) has the maximum property of order k ~ 0, then ... (XXY, A ® B) has the k-maximum property as well. If, in addition, (Y,B) has the t-maximum property, then order k+t+l.
¥
(XxY, A ® B)
has the local maximum property of
A natural strategy to prove this theorem would be to obtain first its finite dimensional version: THEOREM 3. 3 •
respectively.
Let If
X and
Y be locally closed subsets of
X is a k-maximum set,
set as well. If, in addition, (k+t+l)-maximum set.
k
~
0,
then
XXY
a:n and is a k-maximum
Y is a t-maximum set, then XXI
is a
250
SLODKOWSKI The proof of the finite dimensional case is based on the close relation-
ship between k-maximum sets, q-pseudoconvex sets and q-plurisubharmonic functions, hinted at in Theorem 2.3.
It will be discussed in Secs. 4-6.
indicate how the special case yields Theorem 3.2.
Now we
The link between the finite
dimensional situation and the general one is provided by the next three facts. PROPOSITION 3.4. Let (X,A) (k ~ 0, X locally compact).
be a natural system with the k-maximum property Let
U C X and
fl, •.. ,f
--
m
E A.
Assume that the
map
is proper.
Then
F(U}
is a k-maximum set.
LEMMA 3.5. Let (X,A) be a natural system with X-locally compact. Let a compact set K C X, an open covering {U t } of X, and an r-tuple r (gl, ••• ,gr) E A be given. Then there is an open subset U of X, and functions (i)
fl, ••• ,fm E A such that: U:> K;
{fl,···,fm} ~ {gl •••• 'gr};
(ii) (iii) fiber of
the map F
=
F
is contained in some
LEMMA 3.6.
Let
X there exist
containing in some
IIgi - Pi
X, gl, ••. ,gr E A,
F
=
(f l Iu, ••• ,fmlu):u ~ F(U)
m is proper and each
a natural system. B
> 0
and covering
U, K cUe X,
Assume {U t } of
such that
is proper and each fiber is
U •
t'
there are polynomials
PO,Pl, ••• ,Pr
such that
(f l ,··· ,fm) 11K < B;
(iii) Then
C
(X,A)
fl, ••• ,fm E A and an open subset
the map
(ii)
K
C t
Ute
X be locally compact and
that for every compact set
(i)
(fllu, ••• ,fmlu}:u ~ F(U)
(X,A)
is a k-maximum set.
F(U)
has the k-maximum property.
PROOF OF THEOREM 3.2 (Sketch).
According to the
3.6 we have (roughly .,v m (f l ' ... , fm) E (A ~ B) , such
speaking) to find sufficiently many m-tuples that the map
F
=
(fll u, ••• ,fmlu)
is proper, has small fibers, and. F(U)
a (k+$.H)-maximum set for suitably chosen functions
Lemm~
U
C
X x Y.
We approximate given
by polynomials in elementary functions
and using Lemma 3.5 we choose tuples
is
g1 x gi, g1 E A, g'1 E B), , " (fi, ... ,f~,)E Am, (f ,f;,,) E Bm
1, ...
ANALYTIC MULTIFUNCTIONS
containing gI and g1 respectively. If we choose take F' = (filu, ••• ,f~,lu) and F" alike, and set
1, ...
251
U as the product (fl, ••• ,fm) -
U' xU",
(fi ® l, .•• ,f~ ® 1, 1 ® f ,1 ~ f;), F· (fllu, ••• ,fmlu), then F(U) .. F'(U') x F"(U") is a (k+t·H)-maximum set by Proposition 3.4 and Theorem 3.3.
We omit further details.
REMARK. If X and Yare complex varieties of pure dimension, then Theorem 3.3 is equivalent to the well-known assertion that diml(Xxy) = dimCX+dimeY. PROBLEM.
Is Theorem 3.2 still true without the assumption that
X is locally
compact? 4.
DUALITY BETWEEN k-MAXIMUM SETS AND
q-PSEUDOCONVEX DOMAINS
The following two results are crucial to our method of proving Theorem 3.3. The Duality Theorem 4.1, which generalizes Th. 1.3, makes it possible to translate statements on k-maximum sets into assertions on q-pseudoconvex domains, while Proposition 4.2 allows us to reduce problems concerning q-pseudoconvex domains to questions on q-plurisubharmonic functions -- more amenable to analytic techniques. DUALITY THEOREM 4.1. is q-pseudoconvex in set.
Let U, V be open in en, U c V and X = v\p. Then U V (0 ~ q ~ n-2) if and only if X is an (n-k-2)-maximum
PROPOSITION 4.2. Let U,V be open in following conditions are equivalent: (i)
U is q-pseudoconvex in
en, U c V, 0 ~ q ~ n-2.
Then the
V;
(ii) the canonical exhaustion function ·.z -+ -log dist(z ,aU) subharmonic near V n au;
is q-pluri-
(iii) there is a neighborhood W of V n au and a q-plurisubharmonic function u:W n U -+ [--,~), such that lim u(z) = m, for every z E V n au. z '-+z Let us see how these results work.
If
X,Y
are as in Th. 3.2, take open
sets VI c t n , V2 c ~ ~, such that X C VI' Y c V2 and X,Y are closed in VI' V2 respectively. Set Xl = XxV 2 , YI = VlxY. Then Xl and YI are kl and t1-maximum sets respectively, where kl .. k+m, tl .. n+t. (We check this for Xl' If to show that
U .. VlXX, U x V2 is
then VI x V2\X x V2 = U x V2 • By Th. 4.1 it suffices «n+m)-k l -2) = (n-k-2)-pseudoconvex in VI x V2 • It
is indeed so because the canonical exhaustion function of
U,
composed with the
252
SLODKOWSKI
projection of
U x V 2 onto U, gives an (n-k-2)-plurisubharmonic function satisfying Proposition 4.2 (iii». By these observations Theorem 3.3 is implied by the following assertion. INTERSECTION THEOREM 4.3.
V.
sobsets of
Let
Assume that
perty of orders
kl
and
Xl
Vc (N
n
Yl
be open and
;~,
Xl' YI relatively closed Xl' Yl have the maximum pro-
and
t 1 respectively
(0::: kl ,t 1 ::: N-I).
Then
Xl
n YI
is an (N+l-kl-tl)-maximum set. This result can be viewed as a generalization of the classical estimate of the dimension of the intersection of complex submanifolds. By the Duality Theorem 4.1, the Intersection Theorem 4.3 is equivalent to the following statement about the relative complements 4.4. in
Ul
= V\XI'
U2 = V\X2 •
UI , U2 eVe (n be open. If Ul and U2 are q- and r-pseudoconvex then Ul U U2 is (q+r+l)-pseudoconvex (in V) •
Let
V, Let
be continuous exhaustion functions for
u l ' u2
r-plurisubharmonic near
V
n aU l
and
V
n au 2 ,
UI
and U2 ' respectively. Set
q- and
u(z) ..
Then
u
is a continuous exhaustion function; by Proposition 4.2 it is enough
to prove that
u
is (q+r+l)-plurisubharmonic near
V
n a(Ul
U U2 ).
The next
theorem suffices to yield this. THEOREM 4.5. ulB
and
vlB
Let
B be an open ball in
(n
and
u,v E C(B).
are respectively q- and r-plurisubharmonic in
is (q+r+l)-plurisubharmonic in
Assume that B.
Then
min(u,v)
B.
Of course, because of its local nature, the theorem is true for an arbitrary open set
B.
Also the continuity assumption can be omitted.
Our proof of Theorem 4.5 is closely connected with the generalized Dirichlet problem studied by Hunt and Murray [6]. 5.
OPERATIONS ON q-PLURISUBHARMONIC FUNCTIONS AND THE GENERALIZED DIRICHLET PROBLEM It is easy to prove Th. 4.5 in case one of the functions is smooth:
253
ANALYTIC MULTIFUNCTIONS
5.1. A smooth q-plurisubharmonic function has the property (P q,r ): For every r-plurisubharmonic function v, the function min(u,v) is (q+r+l)-plurisubharmonic. If we knew that continuous q-plurisubharmonic functions could be approximated (locally) by smooth ones of this class (cf. Hunt and Murray [6]), the last observation would imply Th. 4.5.
Since we do not, we take a longer way:
we
prove that a continuous q-plurisubharmonic function can be obtained (locally) from smooth q-plurisubharmonic functions by some simple operations, repeated (infinitely) many times; moreover these operations preserve property More specifically, we let
AP
(P
) •
q,r denote the smallest class of upper semi-
q
which contains the class
continuous functions defined on open subsets of
of all smooth q-plurisubharmonic functions and is closed with respect to the operations:
(a)
functions; (d)
upper semi-continuous envelope of the supremum of a family of
(b)
restriction to a subset;
local correction:
such that
having given
lim sup ul (z')
~
u(z)
u
(c)
translation by a vector;
D and
in
for every
ul
in
n ~Dl'
zED
z '-+z
v(z.) z
to be
u(z)
for
z E D\Dl ~
max(u(z), u l (z»
and
for
E1\. It is easy to see that the class
(a) -(d)
) is preserved by operations q,r AP q • Therefore Theorem 4.5 is a consequence
by 5.1 - contains
and
(P
of the following result.
nmOREM 5.2.
If
o
where
~
q !: n-l,
!, and q-plurisubharmonic in B is an open ball in Cn , then ulB E AP (B). is continuous on
u
--
By properties of the class continuous function
v
in
AP
B,
q
open in
B.
such that
,n,
We claim that
v
==
u.
vlB E AP (B). q
vlaB
If not, then
Using methods of
= ulaB
we have to show that
u-v ~ 0
in
H.
-vIH,
it follows that
u+V
Let
u
and
v
is con-
(u-v) laH ulH
(u-v) IH ~ 0
=0
and
(u-v) IH
is
and (n-q-l)-plurisubharmonic by the next theorem (and the ,n).
be q- and r-plurisubharmonic respectively.
is (q+r)-plurisubharmonic.
is
is both q-pluri-
local maximum property of (n-l)-plurisubharmonic functions in
THEOREM 5.3.
v
To complete the proof of Theorem 5.2
Since
the sum of the q-plurisubharmonic function
and
H= {z E B:v(z) < u(z)}
and by arguments of Hunt and Murray [6], ulH
subharmonic and (n-q-l)-plurisubharmonic.
function
q
there exists a greatest upper semi-
Bremermann [5] and Walsh [22], one checks that tinuous on
B,
Then
254
SLODKOWSKI This result, as was hinted
the uniqueness of the
sol~tion
at by Hunt and Murray [6], is equivalent to to the generalized Dirichlet problem studied by
these authors. Cf. [18, Secs. 5 and 6] for more details. in our opinion incorrect, of Th. 5.3 in case 6.
(M. Kalka [7] gives a proof, q=r=(n/2)-1.)
REGULARIZATION OF q-PLURISUBHARMONIC FUNCTIONS BY MEANS OF CONVEX FUNCTIONS To prove Theorem 5.3 we approximate q-plurisubharmonic functions by func-
tions which, although not smooth, exhibit some regularity, and then prove the theorem for the approximations. The standard way of smoothing up a function
u
is to consider the con-
B(O,e), and /g=l. Since the class of q-plurisubharmonic functions is not closed with respect to the summation, this method is useless in this context. The fact that the supremum of a family of q-plurisubharmonic functions is q-plurisubharmonic suggested to us to introduce a new type of convolution. volution
u*g,
DEFINITION.
where
Let
convolution of
g
u,g u
is smooth,
and
Let
u
g
(i)
Set
(n.
The supremum-
Cn •
Let
is the function
=sup{ u(y) g(z-y):y
E
tn} •
be a bounded nonnegative function in
g (0) = 1 n un • u* s g. Then
~
~
0
smooth functions such that n·l,2, ••••
C
be bounded nonnegative functions on
u* g(z) s THEOREM 6.1.
supp g
there are constants
L(n)
g n
~
0
and supp g -n
g
be
n
B(O,l/n),
C
such that the functions
are convex on (ii)
if
u
is continuous and q-plurisubharmonic near
q-plurisubharmonic near A function
v
K and converge to
such that
v(z) +
u
uniformly on
K,
then
u
n
are
K.
~lzl2 is convex for some L ~ 0, will
be said to have lower bounded real Hessian.
In other words, it is a function
whose real Hessian in the sense of distribution theory is a vector measure with values in the convex set of symmetric matrices with lowest value bounded from below by -L. By some results in convex analysis, such a function has at almost every point a second order differential in the local (namely Peano) sense.
In such points real and complex Hessians can be defined.
possible the following characterization.
This makes
255
ANALYTIC MULTIFUNCTIONS THEOREM 6.2. Then
u
Let
u
be a function with lower bounded real Hessian in (0 ~
is q-plurisubharmonic
x E U the complex Hessian of
point
q ~ n-l) u
at
n
Uc C •
if and only if at almost each x
has at most
q
negative eigen-
values. The necessity of this condition is rather easy; as for the sufficiency, it can be reduced to the next theorem, whose proof, based on ideas from geometric measure theory, is omitted. THEOREM 6. 3. let
L
~
Let
be convex in
u
N
B(O,r) c R , r > 0, u
Ixl
Assume that for almost every
0.
the real Hessian of
u
at
is greater than
x
<
r L.
~
the largest eigenvalue of Then
sup u(x)
Ixl
With Theorem 6.2 at hand one proves Theorem 5.3, assuming that and
v
have lower bounded real Hessian
complex Hessians.
and
0, u(O) - 0, ~
2
Lr /2. u
just by counting eigenvalues of the
Using the approximation from Theorem 6.1 we obtain Theorem
5.3 for
u,v
continuous and nonnegative.
These two restrictions can be easily
lifted.
By this we completed the long line of arguments we needed to establish
Basener·s conjecture. 7.
AL~ERNATIVE
AN
APPROACH TO BASENER'S CONJECTURE
After the Rickart Conference the author has found a new method of proving the finite dimensional version of Basener's conjecture Theorem 3.3.
It is
based on a new connection between k-maximum sets and q-plurisubharmonic functions:
the mediation of q-pseudoconvex domains is not required.
THEOREM 7 .1. XX(z) a~k
be ~
a
n-l,
Let if
xcVcCn , x E X and
if and only i f
-00
V °Een and X locallx closed in v. Let if x E V\X. Then X is a k-maximum set,
Xx
is an (n-k-l)-plurisubharmonic function in
Helped by this characterization and the identity Xxny the Intersection Theorem 4.3 directly from Theorem 5.3.
= Xx + Xy '
V.
we get
Then Theorem 3.3 is
obtained from Theorem 4.3 in a similar way as it was done in Section 4. (Except for the proof that XXCm is (k+m)-maximum set if X is a k-maximum set, which also follows from Theorem 7.1.)
The derivation of Theorem 3.2 from the finite
dimensional Theorem 3.3 remains unchanged. In this approach Theorem 4.5 is not used; conversely, it can be obtained from the Intersection Theorem 4.3 via the Duality Theorem 4.1. to be found in (19].
The details are
256
B.
SLODKOWSKI FURTHER PROPERTIES OF k-MAXIMUM SETS In this section we present only some of the properties of k-maximum sets.
THEOREM B.l.
Let
X be a locally closed subset of
,n
and let
0 ~ k ~ n-l.
Then the following conditions are equivalent. (i)
X is a k-maximum set.
(ii)
All k-plurisubharmonic functions have local maximal property on
(iii)
For every tuple
min(lpo(z)I, ••• ,lpk(z)
I)
Po,PI, ••• ,Pk
X.
of complex polynomials, the function
has local maximum properties on
X.
Note that this fact and Theorem 4.1 taken together implv Theorem 2.3.
that
X
=
n
Xl' ••• 'Xs c ~ -be k-maximum sets, o ~ k ~ n-l. Assume Xl U X2 ••• UXs is locally closed in ~n. Then X is a k-maximum
PROPOSITION B.2.
Let
set as well. PROPOSITION 8.3. k-maximum sets,
Let 0
~
V c ,n k
~
n-l.
be open and let Assume that every
Xt
be a monotone family of Xt
is a closed subset of
V.
Then
9.
(a)
the closure (in
(b)
the intersection
V)
of
nXt
U Xt
is a k-maximum set;
is a k-maximum set, provided it is non-empty.
CONCLUDING REMARK We hope we were able to convince the reader that the notion of k-maximum
sets and q-plurisubharmonic functions might have further applications in functional analysis.
It seems, in particular, that these tools can be applied in
the theory of analytic phenomena in general function algebras, developed by Professor Charles Rickart (cf. [13]).
ANALYTIC MULTI FUNCTIONS
257
REFERENCES 1. 2.
B. Aupetit, Analytic multivalued functions in Banach algebras and uniform algebras, Advances in Math. 44 (1982), 18-60. B. Aupetit, Geometry of pseudoconvex open sets and distribution of values of analytic multivalued functions, these Proceedings, 15-34.
3.
R. F. Basener, A generalized Shilov boundary and analytic structure, Proc. Amer. Math. Soc. 47 (1975), 98-104.
4. 5.
R. F. Basener, Boundaries for product algebras, preprint. H. Bremermann, On a generalized Dirichlet problem for plurisubharmonic functions and pseudoconvex domains. Characterization of Shilov boundaries, Trans. Amer. Math. Soc. 91 (1976), 246-276. L. R. Hunt and J. J. Murray, q~Plurisubharmonic functions and a generalized Dirichlet problem, Mich. Math. J. 25 (1979), 299-316.
6. 7. 8. 9.
M. Kalka, On a conjecture of Hunt and Murray concerning q-plurisubharmonic functions, Proc. Amer. Math. Soc. 73 (1979), 30-34. D. Kumagai, Subharmonic functions and uniform algebras, Proc. Amer. Math. Soc. 78 (1980), 23-29. D. Kumagai, Maximum modulus algebras and multidimensional analytic structure, these Proceedings, 163-168.
10.
T. Nishino, Sur les ensembles pseudoconcaves, J. Math. Kyoto Univ. 1-2 (1962), 225-245.
11.
K. Oka, Note sur les familIes des fonctions ana1ytiques multiformes etc., J. Sci. Hiroshima Univ. 4 (1934), 93-98. T. J. Ransford, Analytic mu1tivalued functions, Essay, Cambridge University.
12. 13. 14.
C. E. Rickart, Natural function algebras, Universitext, Springer-Verlag, 1979. N. Sibony, Multi-dimensional analytic structure in the spectrum of a uniform algebra, in Conference on Spaces of analytic functions (Kristiansand, Norway 1975), Lect. Notes in Math. No. 512, Springer-Verlag, (1976),139165.
15.
Z. Slodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), 363-386.
16.
Z. Slodkowski, A criterion for subharmonicity of a function of the spectrum, Studia Math. 75 (1982), 37-49.
17.
Z. Slodkowski, Uniform a~gebras and analytic multifunctions, to appear in Rendiconti Acad. Naz. dei Lincei. Z. Slodkowski, The Bremerman-Dirich1et problem for q-p1urisubharmonic functions. An application of convex analysis to complex analysis, Annal! della Scuolla Normale Superiore di Pisa (to appear).
18.
19. 20. 21. 22.
Z. Slodkowski t Local maximum properties and q-plurisubharmonic functions in uniform algebras, submitted for publication. E. Vesentini, Caratheodory distances and Banach algebras, Advances in Math. 47 (1983), 50-73. E. Vesentini, Non-subharmonicity of the Hausdorff distance, Rendiconti Ace. Naz. Linei (8) vol. 74 (1983), to appear. J. B. Walsh, Continuity of envelopes of p1urisubharmonic functions, J. Math. Mech. 18 (1968), 143-148.
258
SLODKOWSKI
23.
J. Wermer, Maximum modulus algebras and singularity sets, Proc. of the
24.
Royal Soc. of Edinburgh, 86 A (1980), 327-331. J. Wermer, Green's functions and polynomial hulls, these Proceedings, 273-278.
25.
26.
H. Yamaguchi, Sur une uniformite des surfaces constantes d'une fonction enti~re de deux variables complexes, J. Math. Kyoto Univ. 13 (1973), 417-433. A. Zralbi, Sur 1es fonctions analytiques multiformes, Universite Laval, Ph.D. Thesis, 1983.
SCUOLA NORMALE SUPERIOR PISA ITALY
INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSAW, POLAND
Contemporary Mathematic8 Volume 32, 1984
ALGEBRAIC DOMAINS IN STEIN MANIFOLDS Edgar Lee Stout A connected noncompact Riemann surface Rj , j
= 1,2, ••• ,
c~mpact,
of
Schottky double, say
Rj ,
•• R can be exhausted by a sequence
bordered Riemann surfaces, and each
which is a compact
R!eman~
and admitting an antiholomorphic involution a:Rj boundary ~f
Rj ,
•
fixed pointwise and carries
~
Rj
admits a
surface containing
Rj
int Rj
Rj
that !eaves Cl~Rj' onto
Rj/Rj •
the
The
surface
Rj , as a compact Riemann surface, admits the structure of a projective algebraic curve. This construction can be used as the basis for the study of function theory of
R and was used by Behnke and Stein [2] to construct non-
constant ho10morphic functions on
R.
It is natural to wonder whether there is an analogous construction in the context of Stein manifolds and whether, should such a construction exist, it has applications.
We shall show below that the answer to the first question is
in the affirmative: manifold.
There is rather similar construction on an arbitrary Stein
In addition, we shall apply the result to obtain some information
about certain covering spaces. Our main result is the following. THEOREM.
M and
Given a Stein manifold
a neighborhood
U of
K in
M'
M
a compact set
K,
there exists
M that is biholomorphically equivalent to a holo-
morphically convex Runge domain We emphasize that
in
U'
in an affine algebraic manifold,
is a manifold,
!.~.,
M'.
has no singularities.
The reader will recognize that the argument we give to prove the theorem is, in its essentials, an adaptation of an argument used by Nash [8] to approximate differentiable manifolds by real algebraic manifolds. (2) PROOF OF THEOREM.
Let
r
= dim
M.
We
ding theorem for Stein manifolds, that Denote by
TM
assume, as
we
may by virtue of the embed-
M is a submanifo1d of ¢n for some n.
the holomorphic tangent bundle of
M and by
T n
the
C
•Dedicated to Charles Rickart on the occasion of his retirement • ••Research supported in part by Grant lACS81-00768 from the National Foundation.
Science
co 1984 American Mathematical Society 0271·4132/84 S1.oo + S.2S per page 259
STOUT
260
holomorphic tangent bundle of ,no Also, let N denote the normal bundle of the embedding M~ Cn • If T n lM denotes the restriction of T n to M, then
,
t
we have an exact sequence of vector bundles
°~ T" ~ T,n 1M l
N ~ 0,
M
h:N ~
and this sequence splits so that there is a bundle map
T
1M with
goh
(In
the identity map on hog
(For this splitting, see [5, p.256].)
The endomorphism
T 1M is effected by an n x n matrix
of
Cn
with entries
Xj,k
.!..,!!..,
x2 ... X,
X(z)
N.
functions holomorphic on
X is idempotent.
has, for each
z E M,
M.
As
goh
is the identity,
N is a bundle of rank n-r, the matrix For every z E M, the kernel of the
As
rank n-r.
matrix X is the subspace
where
M.
Also, for every
X(z)
°
af -(z) = for all f E rCM)} z j=lj·a z j reM) is the ideal in O(ln ) consisting of the functions that vanish on
= Tz (M) =
T
n n {a E C : Za
z E
M,
we let
= Nz (M)
Nz
be the range of the matrix
so that
eM}, though, in general, z these spaces are not orthogonal in the sense of the standard Hermitian inner
This is a subspace of product on
that is complementary to
Cn.
We define a map
p:M x en ~ cn p(z,a)
The map p(z,a)
pC· ,O) =
T
p(z,O)
Mx {O}
carries
= z,
and if
by
=
z + X(zla.
biholomorphically onto
a EN, z
then
p(z,a)
If
z + a
=
a E T , then z since X(z) acts
as the identity on N. Finally, if we denote by N(M) the submanifold z {(z,a) E M x ,n: a EN} so that, in a natural way, N(M) may be identified z with the bundle N, then there is a neighborhood U of Mx {O} in N(M) that is carried biholomorphically by The map
n:~ ~
M given by
p
onto a neighborhood
=
n(p(z,a})
z
carries
~
~
of
M in
holomorphicallyonto
it is a holomorphic retraction. Consider now a compact subset convex compact subset of relative to
M.
We take
M
with
K of K
C
M, and let
int E,
with
int
E be a polynomially denoting interior
E to be the closure of a Stein Runge domain in
M.
be a polynomial convex compact set of the form n -1 (E) n W, where W is an open neighborhood of M in Cn such that Wc~. We require int E to be
Let
E
261
ALGEBRAIC DOMAINS
Cn •
Runge in
Let X = (Xj,k)l~,ksn be a matrix with polynomial entries that approximates X together with its first-order derivatives very closely on a neighborhood of n
M,
in
E
'V
and let
n:t
n
~
be a polynomial map that approximates
together with its first-order derivatives very well on a neighborhood of z E E, XGrr(z)}
Then for every
very well.
x(1T(z»
will have rank at least
For every
In particular, if our approximation is good enough, E~,
z
n-r
the matrix
so its characteristic polynomial is imation to
X(n(z»,
for each X(n(z}}
z E
E.
is idempotent of rank n-r, and
±~r(~_l)n-r.
As
x6T(z})
is a good approx-
the characteristic polynomial P(z}(~)
r
roots near zero and
n-r
count roots with multiplicity.)
= det(XGrr(z»
- ~I) z E '" E.
roots near one when For
z E E,
PI(Z)(~}
let
P(z) (~,
nomial the roots of which are the small roots of
(Of course, we be the monic poly-
Thus, for
z E E,
'V
M(z), z E E,
(~)
= ± p' (z)
(~)plI
(z)
(~)
•
the matrix M(z)
and define a set
P(z)(~).
we have the factorization P (z)
Denote by
P"(z) (~)
and let
be the monic polynomial the roots of which are the large roots of 'V
r.
is a polynomial matrix that approximates
X(n(z»
has
n
C
= p' (z)(X<1T(z»),
by
~
~ - {z E E:O - M~l(z - nCz})}.
We shall verify that
~o,
the part of
~
in
'V
int E,
possesses two properties:
If our approximations are close enough, then
1)
submanifold of 2)
int
The·set
E ~
that is carried one-to-One by
If
map of
V
is an r-dimensional
n Onto
int E.
is contained in an r-dimensional algebraic subvariety of
We first deal with 1). LEMMA.
~o
This will depend on the following fact. (3)
V is a reduced complex space and if onto a complex manifold, S,
.£h!!!.
~.
~
is a one-to-one holomorphic
is biholomorphic, and
V
!!.
nonsingular. PROOF. and
As
~
is one-to-one,holomorphic and onto,
dim V - dim S necessarily,
~ is an open map [7, p.132]. Thus, ~-l is continuous: ~ is a homeo-
morphism and so proper.
Let
V i denote the set of singularities of V, a s ng proper, possibly empty subvariety of V. By properness, ~(V i ) is a sub-1 s ng variety of S, and ~ :S\~(V i ) ~ V\V i is a biholomorphic map. The map 1 ·sng sng -1 ~ - I(S\~(V i » therefore continues holomorphically to S, and thus, ~ s ng
262
STOUT
is holomorphic.
~
The holomorphic map
has a holomorphic inverse, and so is
biholomorphic, and
V i is empty, as we wished to show. s ng Toward the proof of 1), we begin by performing a similar construction for
the matrix
X(n (z» •
We write
= Q'(z} ().,)QII(Z}().,)
det(X(n(z» - ).,I} with Q' (z) C).,) - ±).,r
and
Q" (z) (A.) ... (A._l}n-r.
The equation
=0
Q' (z)(X (n (z» ) (z - n (z»
M,
is satisfied on z
E~,
then as
for there
X
z - n (z).
Also, if
is idempotent and
Q'(z)(A.) = ±A. ,
o = X(n (z» (z this, write
z - p(z' ,a')
=
p (z ' ,a') Thus,
z=z'+a'
we have
with
z E
whence
and
Since
Z -
en
n Nn(z)'
= O.
a' E N
n (z)'
we have
rv
X(n(z»M(z)(z - n(z»
notice that the range of the matrix M(z)
and since
M(z) (z - n (z»
rv
valent to the equation
We have then
n(z) E T . n(z)
is defined by the equation
1:
To see
n (z) • n (p (z ' ,a'» = z'.
M.
The set
we have
z - n(z} E Nn(z}.
z -n(z} = a',
is equivalent to
z - n(z) E Nn(z).
But also,
(z' ,a') E NCM).
z + a',
satisfies it and if
- n (z»
z - n(z) E Tn(z) - ker X(n(z».
so that
z r
z - n(z),
= 0, which is equi-
To see this equivalence
is an (n-r)-dimensional subspace of
that is very near the range of the matrix
X (n (z»
since
M(z)
is a good
approximation to
X(n(z». Thus the matrix X(n(z» effects an isomorphism of n n ,..., n M(z)(e) onto X(n(z»(C), so the vector M(z)(z - n{z» E M(z){1 )
the space
is the zero vector i f and only i f the vector
n(z»
X (1T (z) )M(z) (z -
is the zero
vector. fiber
~ EE
Now fix -1 1T
(~).
This equation is equivalent to
~~(z):o X(~)M(z) (z
set
rank
n-r,
~~ (z)
and on
n
= X(n (z» (z
solution on
n
-1
(~)
un i que so 1ut i on on Thus,
EO
- n(z», then
n X(~) (C:)
dimensional subspace
by
and consider the equation
-1
(~)
rv
n E,
of
~~
n C,
n E, w-l(y) ~ II
~.,
~
EO
Notice that because
carries
- n(z}) = O.
n-l(~)
If we
into the fixed (n-r)-
it is a regular map, 1.e., a map of
1T
(z» •
As
~~ (z) = 0
itself, we conclude that
~~
. given
has a unique ~~(z)
= 0
has a
n "'E.
is a variety carried by
and so by the lemma,
X(~}M(z}(z
on the
rv
it is a small perturbation of the map
- n (z» = X(~)(z rv
M(z) (z - n(z» - 0
n
one-to-one onto the domain
int E,
is a complex manifold.
E was taken to be polynomially convex,
following convexity property:
XO
has the
ALGEBRAIC DOMAINS 3)
If
S
is a compact subset of
contained in
and thus,
LO ,
LO ,
263
the po1ynomia11y convex hull of
M~)
to verify that·the coefficients of the matrix ,....,
Zo E
is
is ho10morphically convex.
LO
This completes our discussion of I}; we now turn to 2). We fix a point
S
The first step is
satisfy poJynomial equations. p(z)~)
int E at which the polynomial rv
has distinct
,....,
X and n slightly, retaining the approximation properties we are interested in, so as to produce a new P with, generically, distinct roots. In a neighborhood, G, of Zo, P(z)(~) roots.
If no such point exists, we can alter
will have distinct roots. P~
are well-defined holomorphic functions in
fies the polynomial equation small roots of
PI(z)""'Pn~l.
Denote these roots by
p(z)(~), P
n-
r+l(z), ••. ,pn (z)
and, of course, each satis-
G,
= O.
P(z) (pj (z»
where the
Let
PI(z)"" ,p (z) be the n-r the large roots. By construction,
n-r P'(z)(~)
= IT (~-Pj(z»
j=l
= ~n-r _ p's
Since the subring
(Pl(z)+"+ P (z»~n-r-l+ •• '±P1(z) ••• p (Z). n-r n-r
t[z] consisting
of the polynomials in 1
matrix
t[z].
The matrix X(n(z»
= M(z)
p' (z) <X1T(z»
2
s
,....,
G,
are integral over the
zl"",zn
of
1
O(G),
it fol-
are also integral
s
has polynomial entries, so it follows that the
has coefficients integral over
theory of integral dependence,see [15].) is correct only in
..
l~j <j <';'<j ~-rPj (z)"'Pj (z)
lows that the coefficients over
!.~
satisfy monic polynomial equations,
C:[z].
(For the
Note that initially, the conclusion
but then, by analytic continuation, it is correct
E. The set L O is defined by the equation M(;:) (z - (;T(z» = o. If ,...., th ,...., (z - (n(z»). denotes the j coordinate of z - (rr(z», this equation is the J system of n equations given by throughout
n
L ~j (z)(z - (rr(z»j j""l k
= l, ••• ,n.
= 0,
Denote the left side of this equation by
is holomorphic on
,...., E
and is integral over
lIz].
nomial of minimal degree with cop.fficients from Then qk (~) with
qk,j(z) E C[z],
:: 0
=
L qk j
j=O
'
on the set where
C[z]
satisfied by
9k (z).
(z)~j
and, by minimality, qk,O
have
whence
dk
Let
9k (z). The function 9k qk be the monic po1y-
is not the zero polynomial.
We
264
STOUT
~o.
we have rid ourselves of singularities along
Denote by
N'sing
lar locus of N', by I(N'sing) the ideal in the coordinate ring, R, of N' consisting of the functions that vanish on N' sing The ideal TeN' sing } is finitely generated. Let gl, ••• ,gt E R generate it. We now choose the U of the statement of the theorem: Let U be a
M,
domain in
K cue E,U
be the domain in zero in
El,
E and
that corresponds to
U polynomially convex. U.
there are holomorphic functions
flg l +···+ ftg t approximate
N'
Runge in
=I
on
fl, ••• ,f t
EI.
As
gl, ••• ,gt
fl, .•• ,f t
Let
have no common
on
~l such that
By the polynomial convexity property of
uniformly on
fi'
by elements
R.
The function
g
U'
and contains
M'
N' sing·
is an affine algebraic manifold that contains
= hlg l +···+
htg t
belongs to
Thus, if we set
R
U'
hl, ••• ,h t
U,
we can
of the ring
and its zero variety misses
M' = N' ~ = U',
{p E N':g(p} ¢ O},
then
and the theorem is
proved. As an application of the theorem proved above, we can obtain a fact about covering spaces.
A simply connected noncompact Riemann surface is known to be
conformally equivalent to either the plane or the unit disc. responding result for Stein manifolds: dimension
n
There is no cor-
A simply connected Stein manifold of
need not be holomorphically equivalent to a domain in
in general, there will be topological obstructions to such
en;
for
an equivalence, and
even if there are no topological obstructions, there may be holomorphic obstructions,as was shown in [13]. However, Griffiths [4] obtained a uniformization theorem in the algebraic context. if
(In connection with Griffith's theorem, see also [3].)
V is an affine algebraic variety, then each nonsingular point
a Zariski neighborhood of en,
V'
V'
in
He shows that
..
p E V has
V such that the universal covering space,
.
V' ,
is biholomorphically equivalent to a bounded domain of holomorphy in
and, indeed, some information is available about the structure of
example, it is topologically a cell.
V'.
For
265
ALGEBRAIC DOMAINS We can apply Griffith's result in our context. Stein manifold, let
p E
U be a neighborhood of
M, let
holomorphically convex Runge domain under the biholomorphic map
Zariski neighborhood of
in
domain of
holomorp~
~j
~j'
n
in
~1'~2"."
Denote by one
N'
M be compact and, as in the theorem, let
K c
M that is biholomorphically equivalent to a
K in
p'
M be an n-dimensional
Let
U'
in an affine algebraic manifold Set
p'
=
E U',
and let
N'
be a
N that is uniformized by the simply connected
C.
Let
..
~
n:N'
the components of
N' n
be a covering map. -1
(U'
n N').
There may be only
there may be finitely many, and there may be infinitely many.
is a covering space of
N
U' (under the projection n).
Each
Moreover, each
~j
is necessarily a domain of holomorphy, as follows from Stein's theorem [12] to the effect that a covering space of a Stein space is a Stein space. In summary, our situation is this: Eo
such that the domain anyone of the
~j'
U\Eo
=
There is a variety C
u,
p ~ Eo,
admits a bounded domain of holomorphy,
'as a covering space'(with'coverinS projection
At this point, some further questions arise naturally. First of all, it is not clear that any of the domains simply connected. will be a domain in
~j
It is natural to ask whether the universal cover of (n.
D in
en,
is the universal covering space of
again (biholomorphically equivalent to) a domain in
in
en
is there a subvariety
E of
M
D
en?
Secondly, is it possible to globalize this construction:
M,
U\Eo
This is a special case of a more general question:
Given a domain of holomorphy
manifold
are necessarily
such that
M\E
Given a Stein admits a domain
as a covering space?
Finally, notice that the space version of the theorem proved above is false:
According to Whitney [14], not all local singularities are algebraic. REFERENCES Sario~
1.
L.V. Ahlfors and L. Princeton, 1960.
Riemann Surfaces, Princeton University Press,
2•
H. Behnke and K. Stein, Entwicklung analytischer Funktionen auf Riemannschen FlKche, Math. Ann. 120 (1949), 430-461.
3.
L. Bers, On Hilbert's 22nd problem; Mathemat~cal Develo~ments Arisin& from Hilbert Problems (Pro~ Sympos. l'ure Math., Northern Illinois Univ. DeKaib, Ill. (1974), pp.559-609.) Proc. Sympos. Pure Math. Vol XXVIII, Amer. Math. So~., Providence,1976.
4"
P. Griffiths, Complex analytic properties of certain Zariski open sets on algebraic varietie&} Ann. Math. (2) 94 (197l), 21-51.
5..
R.C. Gunning and H. Rossi, Theory of Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965.
266
STOUT ."
6•
N.V. Ivanov, A?proksimatisija gladkikh mnogobrazii veschestvenuo algebraischekimi mnozhectvami, Uspekhi Mat. Nauk 37 no. 1 (1982), 3-52. [English translation: Approximation of smooth manifolds by algebraic sets, Russian Math. Surveys 37(1982), 1-59.]
7•
R. Narasimhan,Introduction to Ana1vtic Spaces, Springer Lecture Notes, vol. 25, Springer-Verlag, Berlin, Heidelberg, New York, 1966.
8•
J. Nash, Real
9•
R. Remmert, Ho10morphe und meromorphe Abbi1dungen komplexer RKume, Math. Ann. 133 (1957), 328-370.
a1~ebraic
manifolds, Ann. Math. (2) 56 (1956), 405-421.
10.
loR. Shafarevich, Rllsic Al~ebraic Geometry., Springer-Verlag, Berlin, Heidelberg, New York, 1972.
11.
K. Stein, Analytische Zerlegungen komplexer Raume, Math. Ann. 132 (1956), 63-93.
12 •
K. Stein, Uber1agerungen Holomorph-vollstandiger komplexer Riume, Arch. Math. VII (1956), 354-361.
13 •
E.L. Stout and W.R. Zame, A Stein manifold topologically but not ho10morphically equivalent to a domain in Cn , Adv. Math., to appear.
14 •
H. Whitney, Local prpperties of analytic varieties, Differential and Combinatorial To~ology (A Symposium in Honor of Marston Morse., pp. 205-244), Princeton University Press, Princetun, 1965. O. Zariski and P. Samuel, Commutative Algebra, vol. I, Van Nostrand, Princeton, 1958.
15 •
FOOTNOTES (1)
For the theory of the Schottky double, see [1].
(2)
For a survey of the recent developments related to Nash's work, see [6].
(3)
Professor Remmert has kindly drawn my attention to ~ 32, p. 362 of his paper [9] for the result of this lemma. See also Satz 2, p. 74 of the paper [11] of Stein. As doing so requires little space, we include a proof of the lemma for the sake of completeness. DEPARTMENT OF MATHEMATICS UNIVERSITY OF WASHINGTON SEATTLE, WASHINGTON 98105
Contemporary Mathematics Volume 32, 1984
SETS OF BEST APPROXIMATIONS TO ELEMENTS IN CERTAIN FUNCTION SPACES Junzo Wada Let
A be a function space or a function algebra on a compact Hausdorff
space X and M = ~ = {f E A: f(x) = 0 for every x E F} for a closed subset F in X. Let P(f) be the set of best approximations to f in kr' for f E A. When A is a function algebra on X, it is well-known that if F is a p-set for
A,
kr
then
is an M-ideal in A and
F has the following
kr
property S: P(f) spans for any f E A\ky. We here investigate properties of P(f) in a function space or a function algebra when F is not neces-
...
sarily a p-set for A. In particular, in §2 it is shown that if FO ~ F, where FO = F n aA, then F has the property S above in certain function algebras, and the converse holds in function spaces with some conditions. 1.
PRELIMINARIES Throughout this paper
C(X)
X will denote a compact Hausdorff space.
By
we denote the Banach algebra of complex-valued continuous functions on
X with the supremum norm. A is said to be a function space (resp. a function algebra) on X if A is a uniformly closed linear subspace (resp. subalgebra) of C(X) containing constant functions and separating points in X. A function algebra on X is said to be logmodular if log IA-l I = {logtfl: f E A, -1
f E A} is uniformly dense in CR(X) , the set of real-valued continuous functions on X. We say that a function algebra A on X is a maximum modulus algebra if f is constant whenever f E A and If(x) I - IIfll maxlf(y)I for an x E X\a A, where aA denotes the Shilov boundary for A. yEX Let A be a function algebra. A is said to be an integral domain if f
=0
if If set
or
g - 0 whenever
f, g E A, fg
= O.
A function space A is analytic
f - 0 whenever f E A, feU) • 0 for a non-void open subset U in X. A is a function algebra and is analytic, then it is an integral domain. Let A be a function space or a function algebra on X. A closed subF
in X is said to be a peak set for
A if there is _n
f E A such
*
that f(x) = 1 for every x E F and If(x) I < 1 for every x F. A p-set is the intersection of peak sets for A. A closed subset F in X is a closed restriction set A if the restriction AIF of A to F is closed in C> 1984 American Mathematical Society 0271·4132/84 $1.00 + $.2S per page
267
268
WADA
C(F).
AIF
is said to have the norm preserving extension property if for any
F, IIgll '"' IIfllF = maxlf{x) I. xEF We call the set F an NPEP-set for A. AIF is said to have the bounded extension property i f for any f E A and each closed set G c X with G n F f E A there is agE A such that
and for each
= C/>,
g = f
on
there exists a
e > 0,
g
g E A such that
f
=
on
F,
IIfllF and IIgli G < e. We call such a set F a BEP-set for A. Let A be a function space and F be a closed subset in X. We consider the four conditions stated above: (1) F is a p-set for A, (2) F IIgll
=
is a BEP-set for
A,
F
(3)
is an NPEP-set for
F
(4)
is a closed
(3) and (3) ~ (4). Recall that if A is a logmodular algebra, then (1), (2), (3) and (4) are mutually equivalent (cf. [1]). In an arbitrary function space, in general, these are restriction set for
We see easily that (2)
A, and
A.
~
not necessarily equivalent. Let
E be a real Banach space.
A linear mapping
p2
closed linear subspace
E is an L-ideal if
L-projection. MO
= {f
'Ie
E
E.
of
We say a closed linear subspace
=0
E E*: f(x)
linear subspace in an M-idea1 in
E
A,
A,
(2)
F
(x E E).
M in
A
E is an M-ideal if
is an L-ideal in the conjugate space X and if
M is equal to the set
{f E A: f(x)
It
M is a closed
(I)
the following conditions are equivalent: for
is
N is the image of an
A is a function algebra on
for a p-set When
x E M}
to
E
M-ideals are also defined in a complex Banach space (cf. [3J).
is well-known that if
x E F}
for any
from
= P and IIxll = IIPxll + IIx - Pxll
said to be an L-projection if N in
P
=0
M is
for any
A ([3], [6]).
A is a function space on X,
the following is obtained in a way
similar to that in the proof of Theorem 7.6 of [6]. PROPOSITION 1.1. for any
x E F}
1!!
~ BEP-~
F
!2I
if
fi E M,
M
n n B(g +
3
"fi";;r; 1 fi , 1
i=l Let G - {x E X:
+
(i
= 1,2,3),
e) ~ C/>,
where
g E A, IIgil ::: 1, B(f,eS)
3
z
Ifi(x)J ~ e}. Then i=l From the hypothesis, 'there is an h E A, Ihex) I < e (x E G). We here see that follows that M is an M-idea1. Let
A be a Banach space and
here consider the set
that
an
M is
M-ideal in
=0
A.
As in the proof of Theorem 7.6 in [6], it is sufficient to show that
PROOF.
f E A.
~
A.
M = {f E A: f{x)
~
X.
A be a function space on
We call P(f)
PCf)
=
then
> 0,
t::
{h E A: "h - f" ::: eS}
eS >
G is a compact set disjoint from
h
=
g
on
g - hEM
F,
"h" - "gllF'
n( R B(g + fi'
1 +
e~.
It
i=l.
M a closed linear subspace in
p{£) = {g E M: IIg - fll ,. d{f ,M)
= inf"f
- hI!}
f
M.
the set of best approximations to
is a closed convex subset in
for
M.
In general,
A.
We
for
hEM
in P(f)
We see
is not
o. F.
SETS OF BEST APPROXIMATION
necessarily non-void. proximinal i f
A closed linear subspace
p(f) 1: cf>
ror any
subset
in
F
A is called
when
A is a function space or a function
= {g E A: g(x) = 0 for any x E F} for a closed F Even in this case, there is an example such that M is
X and
algebra on
M in
f E A.
P(f)
Thereafter we deal with
269
M= k
X.
not proximinal. Such an example is given as follows (llO]): on
D = {z E ¢: Izi
S
I}.
If
0, I, and if fez) = 1 - z, have IIf - gil > If(O) - g(O) shows that P(f) = cf>.
F
is the set
then
d(f,ky)
I =1
LEMMA 1.2.
~
Then
is non-void for any
P(f)
Let
{O,l}
= 1,
A be the disc algebra
consisting of two points g E kF'
and for any
by the maximum modulus principle.
A be a function space on f E A,
X and
that is,
F be an NPEP-set
we This
f2! A.
ky is proximinal.
fl E A such that fl = f on F and f l , then IIflll = IIfIl F , since F is an NPEP-set for A. If we put h = f h E k F • For any g E kF , IIf - hll = Ilflll = IIfllF S IIf - gil· Hence h E PCf)
PROOF.
and
For any
f E A,
there 1S an
P(f) t: cf>.
2.
THE SETS P (f) OF BEST APPROXIMATIONf Let
A be a function space on
and have the following
THEOREM 2.1.
~
~
spans
ky
A.
By
for any
We here consider closed subsets which are contained in a BEP·-set
FO (FO t: X),
subse~
F be a BEP-set for
ky is an M-iaeal, and hence PCf)
Proposition 1.1, f E A\kF.
X and
in P(f)
X.
A be an analytic function space on
If there exists a
BEP-~
is not a singleton tor any
FO
i2!
X and
A (FO t: X)
-----...,-
F be a closed containiTlS
F,
f E A\ky.
FO be a BEP-set for A such that F C FO ~ X. Let PCf) contain g for £ E A\kF • For any x E X\FO and for any e > 0, there is an and E A such that h h f - g on FO' IIh II = Ilf - gliF x,e = x, e x,e 0 E P(f) . For Ilf - (f - h = Ih (x) I < e. We here see that f - h x,e x,e )11 x,e E kyo Suppose that and f - h Ilhx,e II = Ilf - gil, 0 S IIf - gil = d(f,L) x,e -~
PROOF.
Let
g = f - h fOL' any x E X\FO and any e > O. Then If(x) - g(x) I = x,e Ih (x) I < e and it imp11es that (f - g)(x) - 0 for x E X\F O' and hence x,e f = g. This is a contradiction since f E A\~ and IIf - gil = d(f,kF ) > O. It completes the proof. Similarly, we have COROLLARY 2.2.
~
A be an integral domain and
F
a closed subset in
X.
270
WADA
If there exists a p-~ FO for A (FO ~ X) containing F, then P(f) !l not a singleton for any f E A\~. Let A be a function space or a function algebra on X. We say that a closed subset F in X has the property S if P(f) spans ~ for any f
E A\~.
We now consider a condition for F denotes t:he set THEOREM 2. 3. algebra. Let ~
ha~
F
~
F under which
{x E X: If (x) I !: IIfllF
for any
A be a function algebra on S,
...
wnere
=F
FO
S.
f E A}. a logmodu1ar
X and
F be a closed restriction set for the property
F has the property
-A.
!!
FO
~
...
~
q,
FO:::> F,
n aA•
PROOF. Let FO ~ q, and FO:::> F. Put B = ~laA. Then BtFO is closed in C(F O) since AIF is closed in C(F) and FO:::> F. Since B is logmodu1ar, FO is a p-set for B. If we put N = {g E B: g(y) = 0 for any B
then P (f 1 ) .. {g E N: IIg - f111a .. d(f 1 ,N)} spans N
for any
A
([4J).
From this we conclude that P(f) spans ky for f E A\ky. We next consider the converse. Let A be a function space on X• We say that a c10seJ subset F in X has the property C i f F n aA C FO' where
.
FO .. F n aA• If the Choquet boundary for closed subset F in X has the property Choquet boundaries, see [5], [8]).
.
coincides with aA, then any C (for Shi10v boundaries and A
Our main theorem is the following THEOREM 2.4. ~ A be a function space on X. Suppose that F is set for A having the property C. Then if F has the property S, x
FO : :> F.
an
NPEP-
then
.
PROOF: (.) Suppose that FO ~ F and FO ~~. Then there is an Xo E F with Xo t FO· It implies that there are an xl E F\aA and an f E A such that IIfliF < l!fllF - f(x 1 ). Since F is an NPEP-set for A, we can assume that
o
IIfli ..
1I£lI y
o
Let m be a representing measure on
aA
for
xl.
Then
= If dm!: tIf dml !: Iffl dm!: IIfll - f(x l ) • ... have that f = f(x 1 ) on ~upp m and FO n supp m .. q,. If f(x 1 )
From this we g E P(f), then
f(x1 ) .. f(x1 ) - g(x1 ) !: I If-a fdm !:
= I( f-g)dm!: II (f-g)dmJ
II f-gll
- f Ut1 ) •
This implies that now that if let
f ...- g .. f(x 1 ) on supp tn. H"nce g" 0 on supp m. Note x E aA\F O' then there is a
gex) - 0 whenever
g E~.
Then
gl(x}
= g2(x) if gl,g2 E A, and
SETS OF BEST APPROXIMATION gl
= g2
on
F.
Since F
is an NPEP-set for
A,
271
for any
h E A there is an
hI E A such that hI = h on F and I' h ~I ... II hll,. Hence for any h E A, I hex) I = I hI (x)1 5 II hIli = II ~I F. It implies that x E ~. and so x E FO by the property C. This is a contradiction. We take a point y in supp m.
..
+
Then there is a ~ E ~ with ~(y) O. Now, if P(f) spans ~, ,(x)'" 0 for x E supp m and , E~. This is a contradiction since ~ E ~ and ~(y) ~ O. It shows that PCf) does not span
~
for an f E A\~. (b) Suppose that FO
can prove that
P(f)
=~,
does not span
~
n aA =~.
1. Then we by the similar manner as in (a).
that is,
F
Put
f
5
Thus the theorem is proved. We easily have the following PROPOSITION 2.5.
~
A be a function algebra on X. Suppose that B - Ala A is a logmodular algebra and A is a maximum modulus algebra. If F is an NPEP-set for A, ~ P(f) is a singleton or spans ~ for any f E A. PROOF.
.
ky,
it is clear that PCf) is a singleton. Let f E A\~. If P(f) does not span ~, by Theorem 2.3 i t follows that FO :t F or ... FO = ~. If FO ~ ~ and FO l F, a contradiction arises since F is an NPEP-set for A and A is a maximum modulus algebra. If FO ... fIJ, then F is a singleton and P(f) = {f - f (a) } when F = {a}. If
f E
EXAMPLES
J.
We give some examples which have relations to theorems in §2. tl) FO(~
The following is an example for the case when there is no p-set
X) containing F. In this case, the conclusion of Corollary 2.2 fails. Let A be the disc algebra on D'" {z E a, Iz I 51} and F the set
{zl,z2} consisting of two points zl' z2 with IZil < 1 (i - 1,2). Then is a closed restriction set for A, but there is no p-set (~ X) for A
F
F. We show that ~ is proximinal and P(f) is a singleton for any f E A. We can assume without loss of generality that zl - 0, z2 - a, O
!
z:a
IIpllr .. d(p,Aln, p
where
as functions in
C(r),
r = f1
{z E 4:,
Izi
= 1}.
If we consider
f1'
hand
has an element of best approximation in Air,
272
WADA
and hence
f
has an element of best approximation in
~.
That
P(f)
is a
singleton is easily shown (cf. [9]). (2)
In Proposition 2.5, that
hypothesis is necessary. sisting of two points functions on
A.
not a singleton.
If
.. F na
f
=
In fact,
Ig(z) - 11 ~ 1
Let
A
A
U [1,2],
FO
then
=
and
=
=
F be the set
{0,2}
con-
A(K)
P(f)
does not span
{g E A: g(z)
1 ~ z ~ 2}.
for =
1, P(f)
a logmodular algebra), but we see that
= fi
be the algebra of continuous K which are analytic on the interior KO of K. Then F is
an NPEP-set for and
K
Let
0, 2.
A is a maximum modulus algebra in the
Here
. FO l
=0
for any
~
and also is z E fi, g(2)
Ala A is a Dirichlet algebra (so is
A is not a maximum modulus algebra. {2}
and
F
=0
In this case,
(cf. Theorem 2.4).
REFERENCES 1.
T.W. Gamelin, Uniform Algebras, Prentice Hall, 1969.
2·
T.W. Gamelin, Restrictions of subspaces of 112 (1964), 27H-286.
3.
B. Hirsberg, M-ideals in complex function spaces and algebras, Israel J. Math. 12 (1972), 133-146.
:4.
R. Holmes, B. Scranton and J. Ward, Approximation from the space of compact operators and other M-ideals, Duke Math. J. 42 (1975), 259-269.
5.
G.M. Leibowitz, Lectures on Complex Function Algebras, Scott Foresman, 1970.
6.
A. Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1-62.
7.
S.J. Poreda, A characterization of badly approximable functions, Trans. Amer. Math. Soc. 169 (1972), 249-256.
H..
C.E. Rickart, General Theory of Banach Algebras, Van Nostrand, 1960.
9.
J. Wada, On best approximation in function algebras, Tokyo J. Math. 1
C(X), Trans. Amer. Math. Soc.
o
(1978), 105-111. 10.
D.T. Yost, Best approximation and intersections of balls in Banach spaces, Bull. Austral. Math. Soc. 20 (1979), 285-300. WASEDA UNIVERSITY SHINJUKUKU, TOKYO JAPAN
Contemporary Mathematics Volume 32, 1984
GREEN'S FUNCTIONS AND POLYNOMIAL HULLS
if
J. Wermer
This is a report on recent joint work with Herbert Alexander on polynomially convex hulls. Let X be a polynomially convex compact set contained in the product set: {IXI 5
I} x {!wI < ... }
in Y
Put
(;2.
= X n {I XI -
I}.
Denoting the polynomially convex hull of a set (1)
X
We call (X.,w)
,. S,
IXI < 1.
Every point
(2)
o..o,wo )
IQo.. ,w o
for every polynomial
0
Q in X and
we assume
= Y.
X admissible provided that (1) holds, and that
with
If
S by
)1
in
X contains points
then satisfies
X
maxlQI
5
Y
w.
X is foliated by compact sets
X such that each X lies on a a a one-dimensional analytic variety V and has its boundary, relative to v , a a contained in Ixi = 1, then the classical maximum principle on each V gives a an explanation for the inequality (2). However, an example given in [1] shows
that varieties contained in A class of sets
X in general do not exist.
X which are foliated by analytic varieties is obtained
as follows.
We fix a polynomial
Assume that
Kp
is compact.
P
in X, w.
We put
Then the sets
Xc - { ~ ,w)1 P~ ,w) = c, Ixi 5 I}, foliate
and each
K
P
X
c
I cl
5 1,
has its boundary relative to the variety
{ixi = l}. Let now X be a compact set in
{p - c}
contained in the form (3)
*Dedicated
X
=
(2
... n
such that
K
n=l
to Charles Rickart.
X can be expressed in
,
Pn © 1984 American Mathematical Society 0271-4132/84 S1.00 + S.25 per page
273
WERMER
where
{p}
c K , n = 1, 2, • •• • Pn+l Pn For an arbitrary neighborhood
n,
K
).,W with
is a sequence of polynomials in
n
Pn
and K
X in
U of
q:2
compact for each
there then exists
n
such that X e K e U.
If
E X,
(~,w) o 0
then
Pn lies on the variety
(~,w) 0 0
avn
and that variety has its boundary U
n
=
{I).I
relative to
The maximum principle on
I}.
Vn = {Pn = Pn(Ao'Wo )} {I).I ~ I} contained in
now applies to each polynomial
Vn
Q to give IQ(A ,w ) I ~ maxlQI ~ o
Since
U
max
un {IA 1=1 }
avn
0
is an arbitrary neighborhood of
X,
(2)
IQI.
then follows.
Here, although
X itself may not contain any variety, it can be arbitrarily well approximated by varieties. QUESTION 1:
We now ask: Let
X be an admissible set.
When does
X admit a representa-
In particular, does every admissible set
X allow a representa-
tion (3)1 QUESTION 2: tion (3)1 For each
).
in
IAI ~ 1,
let us put
Regarding Question I, we have THEOREM 1:
We denote by
cap
(2
X be a compact subset of re~resentation
Let
is harmonic on
k
~
A,
Assume that
cap(XA) > O.
C.
Let
X has a
Then the function:
IAI < 1.
P be a polynomial in
some integer
IAI ~ 1.
contained in
(3) and also that for some
A~ log(cap XA)
PROOF:
the logarithmic capacity of subsets of
A and
w such that
K
p
is compact.
For
1,
P(A,W) where each with
aj
is a polynomial in
a (A) ¢ O.
o
~
and
a
o
is not identically
O.
Fix
Then po..,w)
= ao(A)(w-wl(~»···(w-wk().»· 0 and so
(A,Wj(~»
For each
j, P(A,Wj(A»
1).1 ~ I}
is compact, there is a constant
IWj(A)
I
~ C, j
=
= 1,2, ••• ,k.
E
{ipi
~
I}.
C independent of
Since
{ipi
~
A such that
1,
A
GREEN'S FUNCTIONS a 0 ..*)
Suppose now that a o (Av ) ¢ 0,
all
=0
a
v.
Then
P(~,w)
=
275
A* • We choose
for some
ao(Av)(w-wl(~»···(w-wk(~»'
with
so
Ip(Av'w)I ~ lao(Av) 1(lwl + C)···(lwl + C) for all wEt
and all
v.
Fix w in
Ip(\*,w)
so
0,
{(A,W) IIAI ~ 1, Ip(A,W) I ~ I}
Thus
trary to the compactness of all
I~
A in
IAI < I,
P
and let ~(A
*,w)
=
v
4~.
Then we get
O.
A = A* con-
contains the complex line
Hence
K .
[
ao
never vanishes on
IAI < 1.
For
then,
and so loglp(A,w)I
= 10glao(A)I
+ k loglwl + 10gl1 +···1.
QA the set {wllp(A,w)1 > I}. For fixed are all contained in e\QA and so 10glp(A,·)1
We denote by
A the zeros of
P(A,.)
is harmonic on QA.
Put O(A,W) = G(A,·)
is harmonic on QA'
~loglp(A,w)I, w E QA.
has a singularity at
w
=~
with development
there G(A,w)
= loglwl
+ h(A,w),
is harmonic on QA and at ~ • On aQA' I Po.., • ) I = I, so G - O. Thus GQ..,·) is Green's function for QA with pole at ~. Hence h(A'~) is Robin's constant for QA at GO, and, putting (Kp\ "" [\QA' we have
where
h
eap «Kp)A)
= e-h(A,'" ) ,
so
log cap«Kp \ ) =
-h(A,~).
By the above,
so Therefore Thus A4
log cap ( (Kp)A )
is a harmonic function on
IAI <
1.
00
n K , wi th each K compact and K C K • n=l Pn Pn PI\+1 Pn Then by the continuity properties of capacity. for each A, log cap(X )A Pn decreases to log cap(XA) as n 4 By hypothesis, log cap(XA) > for Suppose now that
X =
00.
some
A,
and by the preceding, for each
00
n,
log cap(X Pn
)A
is harmonic in
A.
276
WERMER
It follows that proved.
log cap(X>!
is harmonic in
A on IAI < 1,
as was to be
The answer to Question 2 is No, as is seen by the following two examples of admissible sets which admit no representation (3). EXAMPLE 1:
Let
x = X+
= {( A, w)
X
=
{( A, w)
and
X-
X+
We claim that
{(A,w) IIAI ::; 1,
I ::; 1},
jr.r2 - (A - 10)
11m w E X 11m w EX
> 0 }, < 0 }.
are connected components of
with
X,
Fix A.
Each point in
= "IIc +
w
with
Ic I :::
Since
1.
or
A - 10
v'c + A - 10
ASSERTION: PROOF:
X+
lies in
P
n
A - 10 ,
1m w > 0
It follows that
for each x+
x+
is
x+
n
{IAI
=
I}.
So
it
is a compact polyno-
X is
x+
A, c,
X
n {IAI =
I},
is admissible.
has no representation (3).
We pu t Q+
Let
= -Vc +
w
Also, since the Silov boundary of
the Silov boundary of
UX •
XA has the form
follows that our claim is correct. mially convex set.
= X+
Re ~ < 0 with
Let yr- denote the branch of the square root defined on ~ = 1.
X
=
[IAI < l]\X+, Q~
=
{w I(A'w) E Q+}.
g+ (A,·)
denote the Green' s function for Q+A with pole at 00. Suppose that X+ has a representation (3). Thus there are polynomials
as above with GO
n n=l For each
n,
we put
Green's function of
Q = {IAI < 1 }\K , n Pn (Qn) A with pole at
K . Pn
and we write 00.
Then, on
~ (A,·)
for the
~,
g (A,w) ... (d 1 P )log IF (A,w) I. n eg n n It follows tnat As
n
g
n
(\,w)
+
4
g (A,w)
{IAI < I} x {Iwl > R}
g+(A,w)
= loglwl
+ peA) +
Q n
as a function of
+
A, g (A,·) t g (A,·)
for each fixed
400,
that the function. large that
is pluriharmonic on
C
n
is pluriharmonic on Q+,
Iwl > R.
on
Q,.
~.
"-
X and w. It tollows
Choosing
R so
we have the expansion
al(A) altA) a 2 (A) a 2 (A) w + W + 2 + _2 + ••• ,
w valid for
+
w
Since g+ is pluriharmonic, we can choose a conjugate
function h+ such that
GREEN'S FUNCTIONS g+ (~.w) + ih+ (~,w) where
F
=
'2.77 F(~,w),
log w +
fI ~ I <
is a single-valued analytic function on
defining
exp[-(g+ + ih+)],
f =
{/ ~ I < I} x {lw I > R}, on
we see that
and vanishes for
f
w =
I}
x
Then,
is single-valued analytic on g+
Since
00.
is pluriharmonic 52+ U {w
can be analytically continued along each path in
f
{Iw I _~ R}.
=
oo}.
Since that domain is simply-connected, the resulting function, again written f,
is single-valued analytic on
is the ~reen's function of
g+(A,.)
Q~U
malone-one map of
~,aQ~
For fixed f(\,·)
If I
=1
+ aQ~
If(~,w)
and ~,
I=
c
with aQ+.
f (~, Vc
10)
A -
a constant
lei = 1. Hence
=
going to
00
O. f(~,·)
Hence
+ aQ\"
w in
to some neighborhood of
w
is a confor-
It follows
Since this is true for each
Then the variety
Vc +
w
If(~,Vc + ~ - 10) I
A-
1, IAI < 1.
is an analytic function of
aQ+.
Also
10,
< 1,
is
Also
I~I < 1.
on
IAI
Hence there is
with f(~,Vc + ~ - 10)
(4)
I~I < i.
= Yc'
~,
Differentiating (4) with respect to
we get
f\(~,Vc + A - 10) + fw(~,Ve + ~ - 10) .
(5)
w
for
1
+ aQ~.
extends analytically across
a subset of
+
00, f(A,·)
\,
+ aQ.
on
Fix
Q~ with pole at
to the unit disk, with
{oo}
extends analytically in
f
Also, since for fixed
= oo}.
is a real analytic simple closed curve.
extends continuously to that
Q+ U {w
Equation (5) holds for each (~,Vc + ~ - 10)
with
c
on
I~I ~ 1
Ic
and
+ A - 10
1\1
< 1.
The totality of points
1 = 1.
Icl
= 0,
1 2'1/c
=1
is precisely
+ aQ.
So we have
(6)
at each on
rl
~"2~.
(~,w)
+ E aQ.
For fixed
Since (6) holds on
+
aQ~,
f ~ CX., .)
(7)
at each point of
+
Q~.
~,
now,
and
f\(~,·)
f (\,.) w
are analytic
we have 1 + f w ( .\, • ) 2w
But for fixed + Hence at Q~.
~,
w
=
0
-+ f(~,w)
is a conformal map and so
never vanishes on w = 0, (7) gives a contradiction. We w conclude that x+ has no representation (3) • Since X+ is admissible, this f
gives a negative answer to Question 2. NOTE:
We did not appeal to Theorem 1 because we lacked enough information
about
cap(x~)
+
1n order to apply that result.
In the next example, we are
able to appeal to Theorem 1. EXAMPLE 2:
Put
x
{t~, w) II ~ lSI /4,
Iw(1 - ~w) I
<
I}.
278
WERMER
For each \ ~ 0, X\ splits into two components X~ unit disk. One can show by direct computation that nic in
\
for
1\1
< 1/4.
For a suitable
~= Then
r,
x:
and x~. +Here is the log cap X\ is not harmo-
0 < r < 1/4,
we put
u X+.
I\I~
\
x+ is po1ynomially convex, admissible, and in view of Theorem 1,
X+
admits no representation (3). We shall give the details for this example elsewhere. These examples leave open the following QUESTION 3:
Let
X be an admissible set.
section of connected components of sets NOTE:
For properties of
log(cap X\)
Can
X be expressed as the inter-
by an expression as in (3)? Pn in the more general context of analytic K
multiva1ued functions see [2], in particular Theorem 3.7.
REFERENCES 1.
J. Wermer, Po1ynomial1v convex hulls and analyticity, Arkiv for Matematik 129-135.
20 (1982),
2.
B. Aupetit, Analytic multivalued functions in Banach algebras and uniform algebras, Advances in Mathematics 44 (1982), 18-60. DEPARTMENT OF MATHEMATICS BROWN UNIVERSITY PROVIDENCE, RHODE ISLAND 02912
Contemporary Mathematics Volume 32, 1984
CONTINUOUS HOMOMORPHISMS AND DERIVATIONS ON BANACH ALGEBRAS
'Ie
Bertram Yood
1.
INTRODUCTION Let
T be a continuous homomorphism of a Banach algebra
subset of
A.
We investigate the properties of
every inner derivation of
A or (in case
iently many inner automorphisms of a non-commutative version of the
A.
T if
A onto a dense
T must permute with
A has an identity) with suffic-
The results are applied to help prove
Singer~ermer
theorem [5].
That theorem
asserts that if
D is a continuous derivation on a commutative Banach algebra
B,
R,
then
D(B)
commutative.
C
where
R is the radical of
We show that
D(B) c R if and only if
inner derivation modulo the radical. if the center of
If
Suppose that
D permutes with every
B is semi-simple or, more generally,
inner derivation.
PERMUTING PROPERTIES OF HOMOMORPHISMS Let
A be a Banach algebra with an identity
e.
Let
invertible elements and let
B is not
B is semi-simple, zero is the only continuous derivation on
B which permutes with every
2.
B.
G1 be the principal component of denote the inner automorphism, h(v)(x) = vxv-l
h(v)
continuous homomorphism of
v E G for which
the set of v E S(W)
if and only if
this one sees that A let
A onto a dense subset of
6a
SeW)
v
-1
W permutes with W(v) E Z,
where
h(v).
6 a (x)
G.
For each
Next let
W be a
SeW) denote It is readily shown that
Z is the center of =
v E G
A and let
is a multiplicative subgroup of
denote the inner derivation
G be the group of
xa - ax.
G.
A.
For each
Using a
in
These items are all
related. 2.1 THEOREM. of
A.
Let
W be a continuous homomorphism of
A onto a dense subset
The following statements are equivalent:
*Dedicated
to Charles E. Rickart on the occasion of his retirement. © 1984 American Mathematical Society 0271·4132/84 $1.00
279
+ $.25
per page
280
YOOD (a)
S(W)::> Gl ,
(b)
W permutes with every
(c)
(I-W)(A) c Z.
PROOF.
Assume
(a).
For each
B , yEA, Y
y
A and each scalar
in
A;O,
by [1, Prop. 8,
p. 88] we have exp(AB ) = h(exp(-Ay» • y
Now
exp(-Ay) E Gl ,
so that -1 A [exp( AO -1)]W. Y
Thus W(O + A(O )2/2! + ••• ) Y Y
=
(0 + A(O )2/2! + ••• ) W. Y Y
Inasmuch as the usual rules for power series are valid [2, Theorem 3.11.4], we let
to see that
WO
y
so that
=
implies
(a)
(b) •
Note that
(b)
holds if and only if W(xy - yx) = W(x)y - yW(x) for all
x, yEA.
By expanding the left side we find that this holds if and
only if Iy - W(y)]W(x) = W(x)Iy - W(y)] for all
x, yEA.
equivalent to
(c).
II, Prop. 8, p. 88] exp(y).
Hence
SeW)
Inasmuch as the range of Now suppose
contains every
Suppose that
(2)
W= I
Suppose that
h(v) ,
exp(y}.
W fulfills
W is an automorphism if
PROOF.
W permutes with every where
As
SeW}
v
0
y
(b)
(a)-(e)
has the form
is a group, it follows
in Theorem 2.1.
Then
Z is finite-dimensional,
Z is one-dimensional. Z is finite-dimensional.
Then
W - I - (I-W) , where Z.
I - W,
by Theorem 2.1, has its range in the finite-dimensional space
Then by the Riesz-Schauder theory, the range of
is
then by
S(W)::> Gl •
(1)
if
As
W permutes with each
from II, Prop. 7, p. 41] that 2.2 THEOREM.
(b).
W is dense, we see that
W is closed, so that
CONTINUOUS HOMOMORPHISMS AND DERIVATIONS
= A. That Z = p ..e:
W(A) that
v E Gl •
theory also now shows that )..
complex}.
W is one-to-one. v -~(v)
As observed above,
Then there is a scalar
)..(v)
28].
Next suppose for each
E Z
so that
W(v) - )..(v)v for each
v E Gl •
But by
of Theorem 2.1 there is also a scalar
(c)
~(v)
so
that ~(v)e.
v - W(v) This gives us (1 - )..(v»v
If
v
is not a scalar multiple of
W(v)
=v
Gl •
Now let
W(e~x)
if
v ~x.
2.3 COROLLARY. and
we must have
is a scalar multiple of
x E A.
=e +
e
= ~(v)e.
W(x)
Suppose that
S(W):l Gl •
~+o
We can choose
so that
e.
Let
W(P)
P,
C
where
If
A/K. of
A/K
W(K)
C
for all
~x E Gl
•
v
in
Then
P
is a primitive ideal of
A
P.
I" - W"
A and rr
be the canonical
K then W defines as usual a conA/X
by
- rr (W(x» •
be the identity operator on A/K.
2.1, the range of A/K.
v
onto a dense subset of
W" (rr (x»
S(wi')
C
K be a closed two-sided ideal in
A onto II tinuous homomorphism W
In
e +
=
Then
homomorphism of
Let
W(v)
Clearly
= x.
(I - W)(A)
PROOF.
Thus
so that
= 1.
)..(v)
If
~ Gl
SeW)
is contained in the center of
then, by Theorem A/K,
so that also
contains the principal component of the set of invertible elements of We apply this to the case
dimensional by 13, Cor. 2.4.5]. If
P
K· P.
A/P is oneIl II Then Theorem 2.2 shows that W - I •
is a 'primitive ideal of
A,
(see the proof of [3, Theorem 2.7.5]).
Here the center of then
Let
P
nZ
R(Rz)
is a maximal ideal of
be the radical of
RZ eRn Z. On the other hand, R n Z is an ideal of tained in its radical. Thus RZ = R n Z.
Then
2.4 COROLLARY.
Suppose that
inner automorphism of
Z is semi-simple.
Z
A(Z).
Z clearly con-
Then the identity is the only
A which permutes with every
h(v) , v E Gl •
YOOD
282
PROOF.
Suppose
h(y)(P)
=P
h(y), Y E G permutes with every
for all primitive ideals
Corollary 2.3 and the remark above on
h(v) , v E Gl • Then, as it follows from Theorem 2.1,
P,
Rz that
[I - h(y)](A)
RZ'
C
We have at hand an example of a case where the inner automorphisms form a non-trivial commutative group.
Z is two-dimensional.
in that example,
3.
Z is not semi-simple even though,
Of course
A NON-COMMUTATIVE SINGER-WERMER THEOREM Let
B be a commutative Banach algebra.
Singer and Wermer have shown
[5] that any bounded derivation on B maps B into its radical. We give a result about bounded derivations on a non-commutative Banach algebra A with
R which reduces to the Singer-Wermer theorem if
radical
:.
L. '.
~
3.1 THEOREM.
,
A is commutative.
I
The following statements concerning a continuous derivation
D
on A are equivalent: (a)
(D5 -5 D) (A)
(b)
D(A)
C
{x E A:
(c)
D(A)
C
R.
PROOF.
a
a
1sasmuch as
C
R for every
a E A,
xy-yx E R for all
D(R)
C
R,
in view of Sinclair's Theorem [4, Theorem 2.2]
there is no loss of generality in assuming generality in assuming identity
e
and extend
D = O.
and
(b)
a
are equivalent.
Now for each scalar
~
).,
= xD(a)
also
P
be a primitive ideal of exp(AD)(P)
tained in
P.
As
C
P.
D(e) cry.
Note that
- D(a)x.
Then we have
D(A)
C
Z and wish to show
0,
(exp(AD)-I)(A) Let
There is also no loss of
For otherwise we may adjoin an
D linearly by setting a
(a)
R=(O).
A has an identity.
(D5 -5 D) (x) Thus
x E A},
A.
Z.
By Sinclair's theorem,
Then by Corollary
A is semi-simple,
C
~3
the range of
l=exp(AD).
D(P)
C
exp(AD)-I
Therefore
P,
so that is con-
CONTINUOUS ROMOMORPHISMS AND DERIVATIONS
Hence we may let
A
-+
0
continuous derivation
to see that D
D=O.
283
The same arguments show that if a
permutes with every
E Gl ,
h(v), v
then
D(A) cR.;'
The proof of Theorem 3.1 gives the following. 3.2 COROLLARY.
If
Z
is semi-simple, zero is the only continuous derivation
which permutes with every inner derivation. In particular the conclusion holds if
4.
A
is semi-simple.
INVARIANT MANIFOLDS FOR DERIVATIONS Let
D
be a continuous derivation on a Banach algebra
Theorem 2.2] has shown that if A/K
is semi-simple, then
K
K
A.
Sinclair [4,
is a closed two-sided ideal of
is an invariant space for
D,
A
i.e.,
such that
D(K) C K.
We give other closed invariant subspaces not necessarily ideals. 4.1 PROPOSITION.
Let
L
be a closed linear subspace of
for all continuous automorphisms of derivation PROOF.
D
on
AFO,
exp( D)
Then
exp(AD)(L) = 1.
A-l(exp(AD)(x)-x) E L.
This shows that
is in
L.
We let
Then
D(L)
C
L
such that
T(L)=L
for each continuous
A.
For each scalar
Prop. 7, p. 87].
A.
A
A~O
As an example, let Then for each idempotent
is a continuous automorphism [I, Let
x E L.
Then
and use [2, Theorem 3.11.4] to see that L
D(x) E L.
be the closed linear span of the idempotents of p,
D(p)
is the limit of finite linear combinations
of idempotents. Next, let uD(v)
K
+ D(u)v E K.
be a two-sided ideal. In particular, if
in a semi-simple Banach algebra
A,
A.
For
u,v
in
K, D(uv) =
M is a minimal closed two-sided ideal then
D(1)
is contained in
M.
284
YOOD REFERENCES
1.
F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, 1973.
2.
E. Hille and R. S. Phillips, Functional Analvsis and Semi-groups, Amer. Math. Soc. Coll. Publ. 31, 19~7.
3.
C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, 1960.
4.
A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166-170.
5.
I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264.
DEPARTMENT OF MATHEMATICS PENNSYLVANIA STATE UNIVERSITY UNIVERSITY PARK, PA 16802
Contemporary Mathematics Volume 32, 1984
THE MAXIMAL IDEAL SPACE OF A COMMUTATIVE BANACH ALGEBRA William R. Zame 1.
I
INTRODUCTION I would like to survey some work which involves connections among algebra,
topology and analysis; specifically, the relationship between a commutative Banach algebra and its maximal ideal space.
Some of what I have to say is old,
some is more recent (although I think the point of view I wish to take is rather new) and some is quite new.
Let me begin at-the beginning.
It was discovered a long time ago that a compact Hausdorff space pletely determined by its ring
C(X)
of continuous (complex-valued) functions.
To be more precise, two compact Hausdorff spaces and only if the rings
C(X)
and
C(Y)
X is com-
X and
are isomorphic.
Yare homeomorphic if In an informal, but pre-
Cise, sense, this means that every topological property of construction on
X) is mirrored by an algebraic property of
construction on
C(X».
X (or topological C(X)
(or algebraic
The initial discovery of this fact led to a series of
attempts to "algebraize" topology; that is, to do topology in a purely algebraic way.
One of the most successful of these attempts was a purely algebraic form-
ulation of the dimension of a space (see [14]).
Although this branch of Mathe-
matics has largely (but not entirely) died out, its offshoots are still thriving. I 'd like to mention three of them which have been, and continue to be very fruitful. The first of these comes about if we replace the compact Hausforff space X by a non-compact space
Z.
Along this line I'd like to mention the invention
of real-compact spaces (by Hewitt and Nachbin) and the real-compactification, the systematic study of ultrafulters and the discovery of intimate connections between set-theory and topology (see [7]). The second (historically the most recent) arises if we replace which is a commutative C*-algebra, by a non-commutative C*-algebra.
C(X) , This has
given rise to a subject which Effros calls "non-commutative algebraic topology". Among the extremely rich and beautiful results of this subject, I should mention 1
Research supported in part by grants from the National Science Foundation. This represents a version of a lecture given at the Conference on Banach Algebras and Several Complex Variables (Yale University, June 1983) in honor of Professor Charles E. Rickart.
e 1984 American Mathematical Society 0271·4132/84 51.00
28!
+ $.25
per page
ZAME
the work of Brown, Douglas and Fillmore [6] on essentially normal operators and Ext,
of which generalizations and applications have been made by Effros,
Kaminker, Kasparov, Rosenberg, Schochet and others (see the references of [24] for example). The third generalization arises if we replace
C(X)
by an arbitrary com-
mutative (unital) Banach algebra; it is this generalization which I would like to discuss in some detail. PROGRAM.
The program may be. formulated in the following way:
To understand the algebraic structure of a commutative Banach algebra
A in terms of the topological structure of its maximal ideal space the space of continuous unital homomorphisms of weak-* topology).
A to
MA
(i.e.,
equipped with the
¢,
The prototypical result in this program is the Shilov idempotent Theorem [26, 3]:
is connected if and only if the algebra
the maximal ideal space
A has no idempotents other than cohomology group
o
H (MA,Z)
and
1.
More is true:
the
is isomorphic to the additive subgroup of
generated by its idempotents.
v
O-th
Cech A
(Here, and in all that follows, it should be
understood that the isomorphism is a natural one, induced by the Gelfand transform.)
The corresponding result for the first Cech cohomology group was found
by Arens [1] and Royden [25]:
the group
1
-1
H (MA,2) is isomorphic to A /exp A, the quotient of the group of invertible elements of A by the exponential
subgroup (which is the connected component of
A-I
containing the identity).
Arens [2] later discovered a matrix-valued generalization of this result, which Eidlin [12] and Novodvorskii [20] showed could be interpreted as providing an isomorphism between the groups algebraic K-theory. and of
KO(A).
K-l(MA) and Kl(A) of topological and Novodvorskii also established the isomorphism of KO(MA)
At about the same time, Forster [13] established the isomorphism
2
H (MA,2) with the Picard group Pic(A). Taylor [28] was the first to really understand the structure underlying all these results. Building on work
of Lin [8] he obtained far-reaching generalizations of the K-theoretic work as v 3 well as results on the third Cech group: the torsion subgroup of H (MA,J) is isomorphic to the Brauer group
Br(A).
(Taylor's work is sketched in [29];
see also [30] and [10].) In a rather different vein, I showed [32] that, for any finite group there is a bijection between the cohomology set Ext(A,G)
of Galois extensions of
A with group
1
H (MA,G) G.
If
G,
and the set G is abelian, both
these sets are abelian groups, and the bijection is a group isomorphism. I should say that the program of relating the algebraic structure of and the topological structure of
MA
is not merely a sterile exercise.
A For
example, the K-theory results were used by Sibony and Wermer [27] to solve a concrete problem in function algebras.
MAXIMAL IDEAL SPACE
287
To describe all this work in any detail would be a monumental task.
I
shall content myself with some additional explanation of the K-theoretic work, which reveals most clearly the underlying structure and connects with the new results I want to describe concerning complex cohomology and homotopy.
I will
first need to say some things about vector bundles (topological K-theory) and projective modules (algebraic K-theory) and their relationship (Swan's Theorem).
2.
VECTOR BUNDLES To make life a little simpler, let us assume here that
connected Hausdorff space; complex field.
k
will (temporarily) denote either the real or
By an n-dimensional vector bundle on
together with a continuous mapping each
x
in
X is a compact
rr
of
E onto
X we mean a space X.
E
We require that, for
E = rr-l(x) be equipped with the structure of an x n-dimensional vector space over k. In addition, we require a local triviality condition: 1
rr- (U)
X the fiber
there should be a covering of
is equivalent to
n
X by open sets
U such that
i.e. there is a homeomorphism ~: rr
UXk;
-1
n
(U) ~ Uxk
such that for each x in U, ~IE is a linear isomorphism of E with x x n {x}xk (regarded as a vector space in the obvious way). As usual, we are frequently interested only in equivalence classes of such bundles; we say (E,rr)
and
such that
tiE x
(E' ,rr') rr'
-t = rr
are equivalent if there is a homeomorphism (so that
t
maps the fiber
E x
t:
to the fiber
is a linear map. Vector bundles arise naturally in many contexts.
E
~
E') x
E' and
Familiar examples over
the reals are the Mobius band (a one-dimensional vector bundle over the circle), the tangent bundle of a manifold and the cotangent bundle and its exterior powers (which give rise to differential forms).
Over the complex numbers, the
most familiar examples are perhaps the holomorphic tangent bundle of a complex manifold, the bundles associated to divisors
(a
la the Riemann-Roch Theorem)
and the complexifications of real bundles. For our purposes we will restrict our attention to complex vector bundles; the set of equivalence classes of such bundles will be denoted by Vect(X).
The
formation of direct sums and tensor products of vector spaces extends naturally to vector bundles and descends to equivalence classes. Vect(X)
These operations make
into a commutative semi-ring; the zero element for addition (direct
sum) is the trivial bundle
Xx{O},
while the identity element for multiplica-
tion (tensor product) is the trivial bundle
Xx(.
Presented with a commutative semigroup, we naturally try to form a commutative group from it by taking formal differences.
In this case there is a
difficulty since formation of direct sums is not cancellative; equivalence of
288 El
ZAME
~
F with
E2
~
F
does not imply equivalence of
El
with
E2 • But, with look at formal dif-
a twist due to Grothendieck the construction still works: a,~
ferences
a -
~,
for
in
Vect(X) ,
same as
a' -
p'
if there is a class
y
and agree to regard in
Vect(X)
a -
~
as the
such that
a E9~' my = at E9 ~ E9 y.
The group resulting from this construction is called
KO(X);
it is easy to see
that tensor products also behave properly on formal differences, so becomes a commutative ring.
There is a natural homomorphism
KO(X)
Vect(X)
~ KO(X)
which is in general not one-to-one, reflecting the lack of cancellation in Vect(X).
° suggests that there are other K-groups and
The presence of the index this is indeed the case. KO(SX);
the suspension
collapsing the subsets
SX
of
° and
X
is the space formed from
and
xx{+l}.
is as
XX[-l,+l]
by
(The suspension of a circle is
The Bott Periodicity Theorem has a K-theoretic form-
KO(SSX) = KO(X).
K
of
xx{-l}
thus a two-sphere, etc.) ulation:
K-1 (X)
The most succinct way to define
Thus suggests defining all other K-groups in terms
K-1 : KP(X)
= KO(X)
KP(X)
= K-l(X)
for
p
for
even p
odd.
Topological K-theory was invented by Atiyah and Hirzebruch [5], generalizing a construction of Grothendieck (on sheaves over algebraic varieties). It has turned out to be widely useful. rich structure (for example,
K*(X) =
In part this is because it has a very ;
Kn(X)
carries a natural ring struc-
n=-
ture and is closely connected with the cohomology ring
*
H (X,Z»,
and in part
because, while passage from vector bundles to K-theory loses some information, it makes tractable calculations of a kind which are intractable at the level of vector bundles.
For more details and discussion, see the notes of Atiyah
[4] and the book by Karoubi [17]. Before leaving the subject of vector bundles, let us note one more thing. In the semi-ring
Vect(X),
we may look at the maximal multiplicative subgroup,
which consists of one-dimensional vector bundles; this subgroup is naturally isomorphic to
2
H (X,Z).
MAXIMAL IDEAL SPACE 3.
289
PROJECTIVE MODULES
R be a commutative ring with unit; for simplicity we assume
Let
no idempotents other than
0,1.
A (finitely-generated) module
projective if there is another (finitely-generated) module direct sum M $ M' and if
is a free module.
R has
Mover
M'
R is
such that the
Of course free modules are projective,
R is a local ring then all projective modules are free, but in general
there may be many projective modules which are 'not free. The direct sum and tensor product of projective modules are again projective modules; these operations make the set
Proj(R)
projective modules into a commutative semi-ring.
isomorphism classes of
Again, direct sum is not
cancellative, but again we may use the Grothendieck construction to form the ring
KO(R)
of algebraic K-theory.
As in the topological setting, there are other K-groups.
The group
Kl(R)
is formed from equivalence classes of invertible elements in matrix rings over R.
Failure of the Bott Periodicity Theorem in the algebraic case makes the
proper definition of the higher K-groups a difficult problem, which was solved by Quillen [21]. of
Proj(R).
max~al
Again we may isolate the
multiplicative subgroup
This subgroup consists of isomorphism classes of projective
modules of rank one, and is usually called the Picard group
4.
Pic(R).
SWAN'S THEOREM The constructions of
KO(X)
and
KO(R)
tween topological and alsebraic K-theory.
suggest certain similarities be-
These similarities are made precise
by SWAN'S THEOREM.
There is a natural isomorphism between
Vect(X)
and
Proj (C(X». PROOF.
Let us construct the correspondence from projective modules to vector
bundles. such that
Given a projective module M $ M'
endomorphism
P
= C(X)r, on
identified with an
for some
C(X)r rxr
complex matrices.
on the trivial bundle
C(X)
F
from
We can then use XXC
r
we choose a module
M'
This gives us an idempotent module M.
matrix of functions in F
Such an endomorphism may be C(X) ,
or what amounts to the
X into the space of idempotent to define a bundle endomorphism
by setting Q(x,)J
The range of
r.
whose range is
same thing, a continuous function rxr
Mover
= (x,F(x)·)J
•
Q is then the vector bundle we seek.
remain to be checked, but they are all routine.)
(Of course, many things
Q
290
ZAME
It follows immediately that universal constructions on
Vect(X)
Proj(C(X»
yield isomorphic objects; in particular, we obtain
COROLLARY.
KO(X) - KO(C(X».
COROLLARY.
H2 (X,Z)
and
= Pic(C(X».
K-THEORY AND BANACH ALGEBRAS
5.
It should be evident that, to obtain the K-theoretic results for commutative Banach algebras, we need to find a Banach algebra version of· Swan's Theorem.
Instead, we'll do something a bit different.
THEOREM.
Let
X be a polynomially convex compact subset of
be the algebra of (germs on Then
X of) holomorphic functions defined near
X.
In this case, let's describe the correspondence from vector bundles to
projective modules. to a vector bundle
Given a vector bundle (E' ,TT')
done easily by brute force. that
(E,n)
(That is,
Since
(EO,TT O)
rh(X,E O)
We "now appeal to a deep theorem of Grauert (EO,TT O)
on W which is equivalent
TT~l(U)
W, EO
is a complex
Ux~n can be chosen to be bi-
=
The projective module we seek is then the module
of holomorphic sections of
of a neighborhood of
we first extend it
W of
is a vector bundle on
manifold, and the local equivalences holomorphic mappings.)
X,
X·, this can be X is polynomially convex, we may assume
there is a holomorphic vector bundle
(E',TT').
on
on some open neighborhood
W is a polynomial polyhedron.
[15]:
f
O(X)
Vect(X) - Proj(O(X».
PROOF.
to
and let
(N
X into
EO
EO
near
for which
Xj
that is, holomorphic maps
TTO·f
= identity.
Verification
that this works is no longer routine, but depends on further results of Grauert: firstly, that the holomorphic equivalence class of that
EO
EO
is unique, and secondly,
has "enough" holomorphic sections.
At first glance, this Theorem might seem unrelated to the Banach algebra result we want.
In fact, together with a little additional information, it
turns out to be just what we need. The first piece of additional information we need is that the Theorem works perfectly well whether the index set
N is finite or infinite.
If
A
is a commutative Banach algebra, we may view MA as a compact, polynomially convex subset of the infinite-dimensional product space ~A [23]·. There is an obvious homomorphism from polynomials on vexity of
MA
A
C
onto
Aj
polynomial con-
allows us to extend this homomorphism to a homomorphism
9: O(MA) ~ A. This is just the familiar holomorphic functional calculus, recast in an infinite-dimensional setting, an invention of Taylor [28] based
MAXIMAL IDEAL SPACE on an idea of Craw [8].
~,
rather than in
291
(Taylor actually works in the linear dual space of
A
but that is merely a matter of convenience.)
The other piece of information we need concerns the behavior of modules under change of base rings. gives us a way to regard M,
If
R,S
are rings, a homomorphism
S as an R-module.
a: R
~
S
In that case, given an R-module
we can obtain an S-module by forming the tensor product
S @R M (as
a.: Proj(R) ~ Proj(S). In general, the behavior of a* is complicated, but if a is surjective and its kernel is contained in the Jacobson radical of R, then a. is an isomorphism. If we specialize to R = o(MA) , s = A, a = a, this is exactly our situation. Putting all these facts together yields Swan's Theorem for commutative Banach
R-modules).
This yields a homomorphism
algebras. THEOREM.
If
A is a commutative Banach algebra, then
Vect(MA)
= Proj(A).
We obtain the results of Novodvorskii and Forster as immediate corollaries. COROLLARY (Novodvorskii). COROLLARY (Forster).
2
H
o
= KO(A).
K (MA)
(MA'~)
= Pic(A).
I have mentioned that Taylor's approach to the infinite-dimensional functional calculus is slightly different from that used here.
In addition, he
uses an improvement of Grauert's results, due to Ramspott [22], which is more closely connected to classifying spaces.
Taylor's work in [29, 30] and mine in
[32] are based on rather different ideas.
6.
COMPLEX COHOMOLOGY The last section suggests that the ring
o(MA)
might be viewed as the
principal object of study, rather than the commutative Banach algebra self.
A it-
Although this point of view is unusual, it has several things to
recommend it.
For one thing, results for
O(MA)
tend to be more general and
can frequently be used to recapture results for some topological algebras which are not Banach algebras. ring than ted). while ring, while
A;
for example,
For another thing,
O(MA )
A usually is not, and
o(MA)
is a much "nicer"
is an integral domain (if
O(MA)
MA
is connec-
is "close" to being a Noetherian
A never is (unless it is finite-dimensional).
Finally, many
O(MA) yield much more familiar interpretations than The Picard group of O(MA) , for example, reduces
algebraic invariants of are possible for (assuming
MA
factorization.
A.
connected) to .the ideal class group, which is tied up to unique In particular,
2
H (KA,Z) = 0
if and only if every irreducible
292
ZAME
O(MA)
element of
is prime.
(This was first observed, in a different context,
by Dales [11].) Going one step further, we can adopt the viewpoint of algebraic geometry (schemes rather than rings):
the "correct" object of study is the sheaf
°
MA, viewed as a compact subset of ~ (we can as the algebra r(MA,O) of sections). If we adopt this view-
of holomorphic functions on
O(MA) point, we.., can obtain a great deal of additional information, including all the recover
complex Cech cohomology groups
~~.
The method has its genesis in some
work of Watts [31], although Watts' methods do not carryover to our context. So let
X be a compact polynomially convex subset of
(N,
(where
which we shall view as a sheaf on
X.
(I'll make some comments later
about translations back to the context of commutative Banach algebras.) each integer p ~ 1, write oP for the p-fold tensor product of itself, where we take tensor products as sheaves of C-algebras on each p, we define a coboundary operator dP : oP ~ Op+l: dl(a) 2
=a
d (a ®
and so forth. (*)
01
~)
N may
0 denote the sheaf of holomorphic functions
be finite or infinite) and let on
(N
°
For
with X. For
®1 - 1 ® a ,
=a ®
~
®1 - a ®1 ®
~
+ 1 ®a ®
~
This leads to the infinite complex 123
~ 02 ~ 03 ~
which is easily proved to be exact (and hence not very interesting from our point of view).
However, if we pass to sections we obtain a more interesting
complex (**)
The complex (**) is still a co chain complex (i.e.,
dP +l dP
= 0)
but is no .., longer exact, and its failure to be exact reflects exactly the complex Cech
cohomology of
X. v
The cohomology of (**) is the comElex Cech cohomoloS1 of a dimension shift. That is , TIlEOREM.
ker dl ker dP+~ /image dP
X,
with
= HO(X,(E) , = HP (X,a:)
for
p
~
1 •
In fact, the complex (**) comes with a natural multiplication, and from this multiplication we can recover the cohomology ring individual cohomology groups.
H* (X,C)
as well as the
The proof of this Theorem looks very much like
the proof of the DeRham Theorem for differentiable manifolds; crucial roles are of course played by vanishing theorems which come out of Cartan's Theorem B.
MAXIMAL IDEAL SPACE I have mentioned before that In fact, if we tensor with
C,
K* (X)
293
is closely tied up with
H*(X,Z).
we obtain (group) isomorphisms:
~
n
H (X,G:) •
n odd It might therefore seem that we have gone to a lot of trouble to obtain information about the cohomology groups more cheaply.
Hn(X,C)
that could have been purchased
That is not so, for a subtle but important reason.
The problem
n
~ H (X,C), for example, does not provide us with n even n knowledge of the individual groups H (X,C).
is that knowledge of
More importantly, the complex (**) itself contains quite a lot more information than just the cohomology ring.
What I have in mind is tied up with
ideas of Sullivan, Griffiths and Morgan [16] about differential forms and rational homotopy theory.
Roughly speaking, I believe that the complex (**)
determines, not just the complex cohomology ring of homotopy type".
X,
but also its "complex
(I hope to describe all this in detail elsewhere.)
In
particular, it should be possible to "see" some of the torsion in the groups HP(X,Z)
which is lost on passage to complex coefficients. X - MA, there are other natural viewpoints to take. For
If we begin with a commutative Banach algebra viewed as a subset of
~,
A and take
0 by the sheaf 0/1, where J is the subsheaf of 0 generated by the kernel of the functional calculus homomorphism 9: O(MA) ~ A. This leads to a complex with all the good properties of (**) and which is perhaps more easily interpretable directly in terms of A. example, we can replace the sheaf
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2.
R. Arens, To what extent does the space of maximal ideals determine the algebra?, in Function Algebras (F. T. Birtel, ed.), Scott Foresman, Chicago, 1966.
3.
R. Arens and A. P. Calderon, Analytic functions of several Banach algebra elements, Ann. Math. 62 (1955), 204-216.
4. 5.
M. F. At iyah , K-theory, Benjamin, New York, 1964.
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,
M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, Proc. Symposia Pure Math. 3, 7-38, Amer. Math. Soc., Providence, R.I., 1961. L. G. Brown, R. G. Douglas and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of.C*-a1gebras, Springer-Verlag LNM 345 (1973), 58-128.
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I. Craw, Galois extensions of a Banach algebra, J. Functional Anal. 27 (1978), 170-178.
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I. Craw and S. Ross, Separable algebras over a commutative Banach algebra, Pac. J. Math. 104 (1983), 317-336.
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H. G. Dales, The ring of holomorphic functions on a Stein compact set as a unique factorization domain, Proc. Amer. Math. Soc. 44 (1974), 88-92.
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V. L. Eidlin, The topological characteristics of the space of maximal ideals of a Banach algebra, Vestnik Leningrad Univ. 22 (1967), 173-174.
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O. Forster, Function theoretische Hilfsmittel im der theorie der kommutati kommutativen Banach algebren, Uber. Deutsch. Math. Verein. 76 (1974), 1-17.
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L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, New York, 1976.
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P. Griffiths and J. W. Morgan, Rational homotopy and differential forms, Birkhauser, Boston, 1982.
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M. Karoubi, K-theory:
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V. Ya. Lin, Holomorphic fiberings and mu1tivalued functions of elements of a Banach algebra, Functional. Anal. Appl. 7 (1973), 122-128.
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E. Novodvorskii, Certain homotopical invariants of spaces of maximal ideals, Mat. Z. 1 (1967), 487-494.
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D. Quillen, Higher algebraic K-theory, Springer-Verlag LNM 341 (1973), 85-147.
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K. Ramspott, Stetige und holomorphe Schitte in Bunde1n mit homogener Faser, Math. Z. 89 (1965), 234-246.
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C. E. Rickart, Analytic functions of an infinite number of complex variables, Duke Math. J. 36 (1969), 581-597.
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J. Rosenberg and C. Schochet, The classification of extensions of
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J. L. Taylor, Topological invariants of the'maximal ideal space of a Banach algebra, Adv. in Math. 19 (1976), 149-206.
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J. L. Taylor, Twisted products of Banach algebras and 3rd. Cech cohomology
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DEPARTMENT OF MATHEMATICS S.U.N.Y. AT BUFFALO BUFFALO, NY 14214
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