Algebraic and Strong Splittings of Extensions of Banach Algebras

123 Pages·1999·4.73 MB·English

by  W. G. Bade

MEMOIRS
of the
American Mathematical Society

Number 656

Algebraic and Strong.
Splittings of Extensions
of Banach Algebras
W. G. Bade
H. G. Dales
Z. A. Lykova

January 1999 0 Volume 137 0 Number 656 (fifth of6 numbers) 0 ISSN 0065-9266

American Mathematical Society
Providence, Rhode Island
1991 Mathematics Subject Classification.
Primary 46H10, 461-125, 46H40; Secondary 46.115, 46.145, 46L35, 16E40.

Library of Congress Cataloging-in-Publication Data
Bade, W. G. (William G.), 1924-
Algebraic and strong splittings ofextensions ofBanach algebras / W. G. Bade, H. G. Dales,
Z. A. Lykova.
p. cm. — (Memoirs ofthe American Mathematical Society, ISSN 0065—9266 ; no. 656)
“January 1999, volume 137, number 656 (fifth of6 numbers)."
Includes bibliographical references.
ISBN 0-8218-1058-8 (alk. paper)
1. Banach algebras. 2. Ideals (Algebra) 3. Modules (Algebra) 4. Continuity. I. Dales,
H. G. (Harold G.), 1964— . II. Lykova, Z. A. (Zinaida Alexandrovna), 1954» . III. Title.
IV. Series.
QA3.A57 no. 656
[QA326]
5IOS—dc21
[512’.55] 98-46541
CIP

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Contents
. INTRODUCTION
. THE ROLE OF SECOND COHOMOLOGY GROUPS 21
. FROM ALGEBRAIC SPLITTINGS T0 STRONG SPLITTINGS 39
. FINITE-DIMENSIONAL EXTENSIONS 55
. ALGEBRAIC AND STRONG SPLITTINGS OF FINITE-DIMENSIONAL EXTENSIONS 79
. SUMMARY 105
REFERENCES 107
vii
ABSTRACT
Let A be a Banach algebra, and let 2 : 0 —v I —» ‘21 3+ A —) 0 be an extension
of A, where Qt is a Banach algebra and I is a closed ideal in 21. The extension splits
algebraically (respectively, splits strongly) ifthere is a homomorphism (respectively, con-
tinuous homomorphism) 0 : A -—> at such that 7r00 is the identity on A.
We consider first for which Banach algebras A it is true that every extension of A
in a particular class of extensions splits, either algebraically or strongly, and second for
which Banach algebras it is true that every extension of A in a particular class which
splits algebraically also splits strongly.
These questions are closely related to the question when the algebra 2t has a (strong)
Wedderburndecomposition. Themaintechnique forresolvingthesequestionsinvolves the
Banach cohomology group H2(A,E) for a Banach A—bimodule E, and related cohomol—
ogy groups.
Later chapters are particularly concerned with the case where the ideal I is finite-
dimensional.
We obtain results for many ofthe standard Banach algebras A.
Keywords: Banach algebra, extensions, Wedderburn decomposition, Hochschild co-
homology, finite—dimensional extensions, tensor algebra, derivation, point derivation, in-
tertwining map, automatic continuityI strong Ditkin algebra, C"—algebra, group algebra,
convolution algebra, continuouslydifierentiablefunctions, Beurlingalgebras, formalpower
series.
viii
1. INTRODUCTION
Let 91 be afinite-dimensional algebra (over C). Then the Wedderbumprincipal theo-
rem asserts that there is asubalgebra 93 of 91 such that 91 is the direct sum
91=EB® rad9l,
where rad91 is the radical of 9(. Motivated by this theorem, many authors have studied
when an infinite-dimensional Banach algebra 91 has an analogous decomposition. Imme-
diately we see that we must distinguish the cases of a Wedderbum decomposition of 91,
where there is a subalgebra B of 91 with 91 = ‘3 (D rad9l, and of a strong Wedderbum
decomposition of 91, where there is a closed subalgebra % of 91 with this property.
The first context in which this question was studied was that in which 91 is aspecified
non—semisimple Banach algebra. However, our main interest in the present work is to
regard the quotient algebra
A = fl/rad91
asbeingspecified, andtodiscussquestionsoftheexistenceofdecompositionsoftheBanach
algebras 91 when 91 is an arbitrary member of a particular class of Banach algebras. In
fact we shall work in a more general situation; we shall consider extensions of a Banach
algebra A, where an extension of A is defined as a certain short exact sequence
Z:O—>I—L>91—7r—)A-—»0
(see Definition 1.2); the most important case is that where I = rad91. The extension
splits algebraically (respectively, splits strongly) ifthere is ahomomorphism (respectively,
acontinuous homomorphism) 0 : A _. 91 suchthat 1r 0 0 =iA, theidentity on A. Weare
seeking to determine when each such extension 2 of A splits algebraically, when each
suchextensionsplitsstrongly, and whentheexistenceofan algebraicsplittingimpliesthat
there is a strong splitting; the latter question is an ‘automatic continuity’ question.
Ourmost extensiveresultsapplytoalgebraicsplittingsoffinite-dimensionalextensions
of (especially commutative) Banach algebras; this question has not previously received
much attention. A prominent feature ofthis study is that certain reductions to the one-
dimensional case that apply in the caseofstrong splittings are no longer available, and so
we must engage directly with the general finite-dimensional case.
The main established technique that is used to consider splittings of extensions of a
Banach algebra A is to calculate H2(A,E), the second Banach cohomology group of A
withcoefficientsinaBanach A-bimodule E. (Thistechniquebuildsonanearlieralgebraic
method ofHochschild.) We also consider a related cohomology group fl2(A,E) (see §2).
In certain situations, this technique is very effective, but it does have limitations. First,
the algebraic theory only applies in the special case where the extension 2 is singular,
Manuscript received by the editor January 6, 1997
2 W. G. BADE, H. G. DALES, Z. A. LYKOVA
i.e., I2 = 0. (However, in certain cases, results obtained in the singular case lead to
results in the more general, nilpotent, casein which I" = 0 for some n E N.) Second, the
cohomology theory as applied to Banach algebras requires the a priori assumption that
the short exact sequence 2 be admissible; i.e., that the closed ideal I be complemented
in the Banach algebra 91 as aBanachspace. Ingeneral, there is no reasonforsuch ashort
exactsequencetobeadmissible; wedescribeacounter—examplein §1. However, inthecase
where I is a finite-dimensional ideal in 21, the sequence X: is automatically admissible.
Further, we can reduce the case ofthe strong splitting of an arbitrary finite-dimensional
extension to that ofsingular, finite-dimensional extensions (see Theorem 1.8), and so the
cohomology theory gives the full story in the case ofstrongsplittings offinite-dimensional
extensions; if A is commutative, we can even reduce to one-dimensional extensions (see
Theorem 4.4). In general, the question of the algebraic splitting of finite-dimensional
extensions ofa Banach algebrais more difficult because there appears to be no reduction
to the singular case, but we can obtain such a reduction in the case offinite—dimensional
extensions of commutative Banach algebras. However, in this context, there is certainly
no reductionofthefinite-dimensional, singular casetothat ofone-dimensionalextensions.
The seminal results connecting the theory of Wedderburn decompositions of Banach
algebraswiththesecondBanachcohomologygroupswere givenbyProfessorH. Kamowitz
in 1962. Later the theory ofWedderburn decompositions ofBanach algebras was further
developed somewhat independently by the Moscowschool led by Professor A. Ya. Helem-
skii and by certain Western authors, particularly Professor B. E. Johnson of Newcastle,
England. (More details ofthe history ofour subject are given later in this introduction.)
In fact, the existing results in the literature are rather scattered; to give a full picture of
the subject, we have tried to collect these scattered results for ease of future reference,
and occasionally we have given a proofofexisting theorems.
The second part ofthis chapter introducesour notation and formulates moreprecisely
the questions which we wish to consider; it also gives some elementary reductions, and
summarizes some earlier work.
In Chapter 2, we shall define the cohomology groups H2(A,E), H2(A,E), and
ERA,E), and explain their role in our theory. The question whether or not the fact
that fi2(A,E) = {0} implies that H2(A,E) = {0} is related to the question of when
the existence of an algebraic splitting implies that there is a strong splitting; we define
intertwining maps, and draw attention to their significant role in the theory. It is a very
relevant question for us to determine when all intertwining maps from a Banach algebra
A into a Banach A-bimodule E are automatically continuous.
Chapter 3 is devoted to adiscussion ofthe question when we can deducethe existence
of a strong splitting of an extension from that of an algebraic splitting; we shall give a
variety ofexamples. The results obtained allow us to exhibit extensions ofmany standard
Banach algebras such that the extension does not split even algebraically.
In Chapter 4, we shall concentrate on finite-dimensional extensions, and obtain some
theorems giving sufficient conditions for all finite-dimensional extensions of a given Ba-
SPLITTINGS OF EXTENSIONS OF BANACH ALGEBRAS 3
nach algebra to split, either algebraically or strongly. For many standard examples of
Banach algebras, all finite—dimensional extensions split strongly, but we shall also give
various examples of finite-dimensional extensions which do not split even algebraically.
Two key theorems are Theorems 4.9 and 4.13, dealing with strong splittings and alge—
braic splittings, respectively. Special cases of these theorems assert the following. Let
A be a commutative, unital Banach algebra, and suppose that each maximal ideal of A
has a bounded approximate identity (respectively, an approximate identity). Then each
finite-dimensionalextensionof A splits strongly (respectively, splits algebraically). Chap-
ter 4 concludes with a further investigation of when all intertwining maps from Banach
algebras—now into finite-dimensional modules—are automatically continuous.
In Chapter 5, weshallapplythegeneralresultsofChapter4tosomespecificexamples,
concentrating on the case of extensions of commutative Banach algebras. In the case of
strong splittings, the case of finite-dimensional extensions can often be reduced to that
of one-dimensional extensions, but this is not possible for algebraic extensions, and the
results may depend on the dimension of the extension. For example, we consider the
algebra C(")(ll) of n-times continuously differentiable functions on ll; it will be shown in
Theorem5.7thatextensionsof C(")(ll) ofdimensionat most 71 split algebraically (butnot
necessarily strongly), but that there is such an extension ofdimension n+ 1 which does
not split algebraically. We shall also investigate some Banach function algebras related to
C(")(I[). In Theorem 5.12, we shall prove that all finite-dimensional extensions ofcertain
local Banach algebras of power series split algebraically; the algebraic calculations for
this result are rather complicated because it again seems that there is no straightforward
reduction from the n-dimensional to the 1—dimensional case.
Finally, in Chapter 6, weshall summarizetheresults thatwe haveobtained forvarious
classes ofBanach algebras, and raise some open questions.
We are very grateful to Dr. Olaf Ermert for some valuable comments on an earlier
version ofthis memoir and to Dr. H. Steiniger and Dr. Y. Selivanov for some corrections.
This work was supported by three agencies. First, Z. A. Lykova was awarded aRoyal
Society Fellowship to enable her to visit the University ofLeeds to work with H. G. Dales
in the period March—June, 1993, and an RFFI grant 93-011—156; she thanks the School
ofMathematics at Leeds and the Department ofMathematics at Berkeley for hospitality
while this work was carried out. Second, W. G. Bade and H. G. Dales were awarded
Collaborative Research Grant No. 940050 by NATO, enabling them to visit each other.
They acknowledge with thanks this financial support.
We now give a fuller description of our main results, establish some preliminary no-
tations and conventions, give precise definitions for the questions that concern us, prove
somegeneralresultsthatwill be usedthroughoutthememoir, andsummarizesomeearlier
results in this area.
Throughout this memoir we shall be concerned with extensions ofa fixed Banach al-
gebra A; the following is a full definition of the context in which we shall work. For
4 W. G. BADE, H. G. DALES, Z. A. LYKOVA
general background in Banach algebra theory, see Bonsall and Duncan ([BoDu]) and
Palmer ([Pa2]), for example.
An arbitrarily specified ‘algebra’ is a linear, associative algebraover the complex field
(C.
An ideal in an algebrais always ‘two—sided'. The (Jacobson) radicalofan algebra A is
denoted by radA; by definition, the algebra A is semisz'mple if radA = {0} and radical
if radA = A. Let S be a subset of A, and let n E N. Then
sin]: {al-v-anzal,...,an€S},
and
s" = 1mSin],
the linear span of SW ; in the case where I is an ideal of A, I" is also an ideal.
Let A be an algebra. An ideal I in A is nilpotent if I" = {0} for some n E N;
clearly, if radA is finite-dimensional, then radA is nilpotent. A character on A is an
epimorphism np : A -—> C. The set ofcharacters on A is the character space, denoted by
in, and we always suppose that <I>A has the relative weak—* topology from the algebraic
dual space of A; in this topology, a)” —' (p in (PA if and only if 90,,(a) _—> 90(a) for each
a E A. The kernel ofacharacter go is denoted by MW 50 that M.p is amaximal modular
ideal ofcodimension one in A.
The identity ofa unital algebra A is denoted by 6,4, or sometimes by e. The algebra
formed by adjoining an identity to a non—unital algebra A is denoted by A# , so that
A# = Ce(9A; in the case where A is a Banach algebra, A# is also a Banach algebra.
For an element a in an algebra A, we write
A(e—a)={b—ba:b€A};
this notation does not imply that A has an identity.
An ideal I in A is algebraically finitely generated if there exist a1,...,a,L e I such
that
I=a1A# +...+ anA# ={a1b1 +-.-+ anbnzb1,...,bneA#}.
Let A be a unital algebra. The set of invertible elements in A is denoted by InvA,
and the spectrumof a E A is
a(a)={CEC:(e—a¢ InvA};
in the case where A is not unital, the spectrum of a is
0(a) ={CeC:(e—a¢ InvA#}U{O}.
In each case, an element a is quasi—nilpatent if 0(a) C {O}; the set ofquasi-nilpotents of
A is denoted by D(A). We have redA C D(A). In the case where A is a commutative
Banach algebra,
radA = {a e A : ”11330”mum = 0} = mm“, : (p 6 am} = 1104).
SPLITTINGS OF EXTENSIONS OF BANACH ALGEBRAS 5
Let A bean algebra. The algebrawith thesamestructure as A, but withthe opposite
multiplication, is denoted by A°p .
Let E and F be linear spaces (respectively, Banach spaces). Then we write £(E,F)
(respectively, B(E,F)) for the linear space ofall linear (respectively, all bounded linear)
maps from E into F; we write £(E) and B(E) for £(E,E) and [3(E,E), respectively.
The identity map on E is 2'3. The range and kernel of T E £(E,F) are denoted by
imT and ker T,
respectively. The dual space ofa Banach space E is denoted by E’.
Recall that a sequence
--—>Xn£> n+1 T"—+>1Xn+2 _,...
oflinear spaces Xn and maps Tn e £(Xn,Xn+1) is exact if
imTn = ker Tn+1 (n E Z).
Let A beanalgebra, andlet E bealinearspace. Then E is aleft A-modale (respectively,
a right A-module) ifthere is a bilinear map (a,m) H a- m (respectively, (a,:1:) H :1: ~ a),
A x E ——> E, such that a ~ (b - 2:) = ab - m (respectively, (:17 ' a) - b = z - (ab)) for
a,b e A and :c E E. The space E is an A—blmodule if it is both a left A-module and a
right A—module and if
a-(m-b)=(a-m)'b (a,b€A,x€E).
For example, an ideal I in A is an A—birnodule with respect to the product in A. Let E
and F be A—bimodules. Then we set
A£A(E,F) ={TE£(E,F):T(a - x) =a - Tm, T(:c - a) =Tz - a (aEA, :1: EE)}.
In the case where A is a Banach algebra and E and F are Banach A—birnodules, we set
ABA(E,F) = A£A(E,F) DB(E,F).
Let E be a left A—module. Then
A‘E={a-:c:a€A, zEE} and AE2 linA-E.
The left module E is left annihilator if A - E = {0}. Similarly, a right A-module E is
right annihilator if E-A = {O}, and an A—bimodule is annihilator if
A-E=E-A={0}.
An A—bimodule E is symmetric (or commutative) if
a~z=$-a (aEA, zEE).
A symmetric A-bimodule over a commutative algebra A is termed an A-modale.
Let A be a unital algebra, with identity e. Then an A-bimodule E is unital if
e-:r=:1:-e=:z: (mEE).
6 W. G. BADE, H. G. DALES, Z. A. LYKOVA
Supposethat A is anon-unitalalgebraandthat E is an A-bimodule. Then E is aunital
A#-bimodule for the operations
(ae+a)-:c=a:r.+a-x, x-(ae+a)=ar+zva (06C, aeA, xEE).
An A-bimodule E over a Banach algebra A is a Banach A-bimodale ifit is aBanach
space and ifthere is aconstant C > 0 such that
|la' ac|| S Cllall llzll, llz - all S Old” Hill (a E A, I 6 E)-
By transferring to an equivalent norm on E, we may suppose that C = 1, and we
shall do this throughout. For example, a closed ideal in a Banach algebra A is a Banach
A-bimodule. Again, let E and F be Banach left A-modules over a Banach algebra A.
Then B(E,F) iseasilycheckedtobeaBanach A-bimodulewith respecttotheoperations
defined by
(axT)(:c) = a-Tm, (T x a)($) = T(a-:r) (:c E E) (1.1)
for a E A and T E B(E,F).
Let A and B be algebras. The (algebraic) tensor product of A and B is denoted by
A(8B; in this algebra,
(a1®b1)(a2®b2) = a1a2®b1b2 (04,012 EA, b1,b2 E B).
Now suppose that A and B are Banach algebras. Then A()9B is a normed algebrawith
respect to the projective norm || - II", where
”le1r = inf{leajlllle-ll:z = Zaj®bj}3
i=1 i=1
the completion of A 69 B with respect to ||‘||1r is the projective tensor product
(A®B, || - II") of A and B. We also write || - ”7, for the projective norm on A2: set
llall,r = inf{leajllllbjll =0 = Zajbj} (GE/12%
j=1 j=1
Clearly we have
Hall S llall,r (0 6 A2).
Let A be a Banach algebra. Then A®A is a Banach A—bimodule with respect to the
module operations defined by the conditions
a-(b®c) = ab®c, (b®c)-a = b®ca (a,b,c€A).
Let A be a Banach algebra. A net (e; : A E A) in A is a left (respectively, right)
approximate identity if e,\a —> a (respectively, ac), —v a) for each a E A; a net which
is both a left and a right approximate identity is an approximate identity. Let A be a
Banach algebra with a bounded left approximate identity. Then Cohen's factorization
theorem ([BoDu, 11.11]) asserts that Alz] = A.
We now give our formal definition of an algebraic extension; the word ‘algebraic’ is
inserted to stress the distinction from an ‘extensionofa Banach algebra’, tobe defined in
Definition 1.2.