Banach algebras on semigroups and on their compactifications

178 Pages·2010·1.24 MB·English

by  H. G. Dales

MEMOIRS
of the
American Mathematical Society
Number 966
Banach Algebras on Semigroups
and on Their Compactifications
H. G. Dales
A. T.-M. Lau
D. Strauss
May 2010 • Volume 205 • Number 966 (end of volume) • ISSN 0065-9266
American Mathematical Society
Number 966
Banach Algebras on Semigroups
and on Their Compactifications
H. G. Dales
A. T.-M. Lau
D. Strauss
May2010 • Volume205 • Number966(endofvolume) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Dales,H.G.(HaroldG.),1944-
Banach algebras on semigroups and on their compactifications / H. G. Dales, A. T.-M. Lau,
D.Strauss.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 966)
“Volume205,number966(endofvolume).”
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4775-6(alk.paper)
1.Banachalgebras. 2.Measurealgebras. 3.Semigroups. 4.Functionalanalysis. I.Lau,
AnthonyTo-Ming. II.Strauss,Dona,1934-. III.Title.
QA403.D35 2010
512(cid:1).27—dc22 2010003552
Memoirs of the American Mathematical Society
Thisjournalisdevotedentirelytoresearchinpureandappliedmathematics.
Publisher Item Identifier. The Publisher Item Identifier (PII) appears as a footnote on
theAbstractpageofeacharticle. Thisalphanumericstringofcharactersuniquelyidentifieseach
articleandcanbeusedforfuturecataloguing,searching,andelectronicretrieval.
Subscription information. Beginning with the January 2010 issue, Memoirs is accessi-
ble from www.ams.org/journals. The 2010 subscription begins with volume 203 and consists of
sixmailings,eachcontaining oneormorenumbers. Subscription pricesareasfollows: forpaper
delivery,US$709list,US$567institutionalmember;forelectronicdelivery,US$638list,US$510in-
stitutionalmember. Uponrequest,subscriberstopaperdeliveryofthisjournalarealsoentitledto
receiveelectronicdelivery. Iforderingthepaperversion,subscribersoutsidetheUnitedStatesand
IndiamustpayapostagesurchargeofUS$65;subscribersinIndiamustpayapostagesurchargeof
US$95. ExpediteddeliverytodestinationsinNorthAmericaUS$57;elsewhereUS$160. Subscrip-
tion renewals are subject to late fees. See www.ams.org/customers/macs-faq.html#journalfor
moreinformation. Eachnumbermaybeorderedseparately;pleasespecifynumber whenordering
anindividualnumber.
Back number information. Forbackissuesseewww.ams.org/bookstore.
Subscriptions and orders should be addressed to the American Mathematical Society, P.O.
Box 845904, Boston, MA 02284-5904USA. All orders must be accompanied by payment. Other
correspondenceshouldbeaddressedto201CharlesStreet,Providence,RI02904-2294USA.
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Requests for such
permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety,
201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by
[email protected].
MemoirsoftheAmericanMathematicalSociety(ISSN0065-9266)ispublishedbimonthly(each
volume consisting usually of more than one number) by the American Mathematical Society at
201CharlesStreet,Providence,RI02904-2294USA.PeriodicalspostagepaidatProvidence,RI.
Postmaster: Send address changes to Memoirs, American Mathematical Society, 201 Charles
Street,Providence,RI02904-2294USA.
(cid:1)c 2010bytheAmericanMathematicalSociety. Allrightsreserved.
(cid:1) (cid:1) (cid:1)
ThispublicationisindexedinScienceCitation IndexR,SciSearchR,ResearchAlertR,
(cid:1) (cid:1)
CompuMath Citation IndexR,Current ContentsR/Physical,Chemical& Earth Sciences.
PrintedintheUnitedStatesofAmerica.
(cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 151413121110
Contents
Chapter 1. Introduction 1
Chapter 2. Banach algebras and their second duals 11
Chapter 3. Semigroups 27
Chapter 4. Semigroup algebras 43
Chapter 5. Stone–Cˇech compactifications 55
Chapter 6. The semigroup (βS,(cid:1)) 59
Chapter 7. Second duals of semigroup algebras 75
Chapter 8. Related spaces and compactifications 91
Chapter 9. Amenability for semigroups 99
Chapter 10. Amenability of semigroup algebras 109
Chapter 11. Amenability and weak amenability for certain Banach algebras 123
Chapter 12. Topological centres 129
Chapter 13. Open problems 149
Bibliography 151
Index of Terms 159
Index of Symbols 163
iii

Abstract
Let S be a (discrete) semigroup, and let (cid:2)1(S) be the Banach algebra which
is the semigroup algebra of S. We shall study the structure of this Banach algebra
and of its second dual.
We shall determine exactly when (cid:2)1(S) is amenable as a Banach algebra, and
shall discuss its amenability constant, showing that there are ‘forbidden values’ for
this constant.
The second dual of (cid:2)1(S) is the Banach algebra M(βS) of measures on the
Stone–Cˇech compactification βS of S, where M(βS) and βS are taken with the
firstArensproduct(cid:1). WeshallshowthatS isfinitewheneverM(βS)isamenable,
and we shall discuss when M(βS) is weakly amenable. We shall show that the
second dual of L1(G), for G a locally compact group, is weakly amenable if and
only if G is finite.
Weshallalsodiscussleft-invariantmeansonS aselementsofthespaceM(βS),
and determine their supports.
We shall show that, for each weakly cancellative and nearly right cancellative
semigroupS,thetopologicalcentreofM(βS)isjust(cid:2)1(S),andso(cid:2)1(S)isstrongly
Arens irregular; indeed, we shall considerably strengthen this result by showing
that, for such semigroups S, there are two-element subsets of βS \ S that are
determining for the topological centre; for more general semigroups S, there are
finite subsets of βS\S with this property.
We have partial results on the radical of the algebras (cid:2)1(βS) and M(βS).
Weshall alsodiscuss analogousresultsforrelatedspacessuch asWAP(S)and
LUC(G).
ReceivedbytheeditorSeptember8,2006.
ArticleelectronicallypublishedonJanuary25,2010;S0065-9266(10)00595-8.
2000 MathematicsSubjectClassification. Primary43A10,43A20;Secondary46J10.
Key words and phrases. Banachalgebras,secondduals, dual Banachalgebras,Arens prod-
ucts,Arensregular,topologicalcentre,stronglyArensirregular,introvertedsubspace,amenable,
weakly amenable, approximately amenable, amenability constant, ultrafilter, Stone–Cˇech com-
pactification,invariantmean,semigroup,Reessemigroup,semigroupalgebra,Munnalgebra,semi-
character,character,cancellativesemigroup,radical,minimalideal,idempotent,locallycompact
group,groupalgebra,measurealgebra,boundedapproximateidentity,diagonal.
(cid:1)c2010 American Mathematical Society
v

CHAPTER 1
Introduction
OuraiminthismemoiristostudythealgebraicstructureofsomeBanachalgebras
which are defined on semigroups and on their compactifications. In particular we
shall study the semigroup algebra (cid:2)1(S) of a semigroup S and its second dual
algebra; this includes the important special case in which S is a group. Here
(cid:2)1(S) is taken with the convolution product (cid:3) and the second dual (cid:2)1(S)(cid:1)(cid:1) is
takenwithrespecttothe first andsecond Arensproducts, (cid:1) and (cid:2); these second
dual algebras are identified with Banach algebras (M(βS),(cid:1)) and (M(βS),(cid:2)),
whichare,respectively,therightandlefttopologicalsemigroupsofmeasuresdefined
on βS, the Stone–Cˇech compactification of S. We shall also study the closed
subalgebras (cid:2)1(βS) of M(βS) and some related Banach algebras.
Much of our work depends on knowledge of the properties of the semigroup
(βS, (cid:1)), and, in particular, of (βN, (cid:1)). We wish to stress that (βN, (cid:1)) is a deep,
subtle,andsignificantmathematicalobject,withadistinguishedhistoryandabout
which there are challenging open questions; we hope tointroduce the power of this
semigroup to those primarily interested in Banach algebras. Indeed the questions
that we ask about Banach algebras are often resolved by inspecting the properties
of this semigroup, and sometimes require new results about it. So we also hope
that those primarily interested in topological semigroups will be stimulated by the
somewhat broader questions, arising from Banach algebra theory, that we raise
about (βS, (cid:1)). In brief, we aspire to interest specialists in both Banach algebra
theory and in topological semigroups in our work. For this reason we have tried to
incorporate general background from each of these theories in our exposition in an
attempt to make the work accessible to both communities.
This paper is partially a sequel tothe earlier memoir [21]. (For a correction to
[21], see p. 147 of the present work.)
Notation. Werecallsomenotationthatwillbeusedthroughout;forfurtherdetails
of all terms used, see [19] and [21].
Weshall use elementarypropertiesofordinal andcardinal numbers asgivenin
[19,Chapter1.1],forexample. Theminimuminfiniteordinalisωandtheminimum
uncountable ordinal is ω ; these ordinals are also cardinals, and are denoted by ℵ
1 0
andℵ , respectively, inthis case. The cofinality of anordinal α is cof α; a cardinal
1
α is regular if cof α=α.
TheContinuum Hypothesis (CH)istheassertionthatthecontinuumc=2ℵ0 is
equal to ℵ ; this hypothesis is independent of the usual axioms ZFC of set theory.
1
Results that are claimed only in the theory ZFC+CH are denoted by ‘(CH)’.
Let S be a set. The cardinality S is denoted by |S|, the family of all subsets of
S is P(S), and the family of all finite subsets of S is P (S). Let κ be a cardinal.
f
1
2 H.G.DALES,A.T.-M.LAU,D.STRAUSS
Then
[S]κ ={T ⊂S :|T|=κ} and [S]<κ ={T ⊂S :|T|<κ}.
The characteristic function of a subset T of S is denoted by χT; we set δs = χ{s}
for s∈S.
We set N = {1,2,...}, Z = {0,±1,±2,...}, and Z+ = {0,1,2,...}. The sets
{1,...,n} and {0,1,...,n} are denoted by N and Z+, respectively. The set of
n n
rational numbers is Q, I = [0,1], and the unit circle and open unit disc in the
complex plane C are denoted by T and D, respectively. The complex conjugate of
z ∈C is denoted by z.
Algebras. Let A be an algebra (always over the complex field, C). The product
map is
m :(a,b)(cid:4)→ab, A×A→A.
A
The opposite algebra toA isdenotedby Aop; this algebrahas the product givenby
(a,b)(cid:4)→ba, A×A→A. InthecasewhereAdoesnothaveanidentity,thealgebra
formed by adjoining an identity to A is A# (and A# =A if A has an identity); the
identity of A or A# is often denoted by e .
A
The centre of A is
Z(A)={a∈A:ab=ba (b∈A)}.
Anidempotent inAisanelementpsuchthatp2 =p;thefamilyofidempotents
inAisdenotedbyI(A). Forp,q ∈I(A),setp≤q ifpq =qp=p,sothat(I(A),≤)
is a partially ordered set; a minimal idempotent in A is a minimal element of the
set (I(A)\{0},≤).
Let I be an ideal in an algebra A, and let B be a subalgebra of A such that
A=B⊕I as a linear space. Then A is the semi-direct product of B and I, written
A=B(cid:1)I.
Let I bean ideal inan algebraA, andsuppose thatI has anidentity e . Then
I
we remark that e ∈ Z(A). Indeed, for each a ∈ A we have ae ,e a ∈ I, and so
I I I
e (ae )=ae and (e a)e =e a. Thus e ∈Z(A).
I I I I I I I
Let A be an algebra. We denote by R , N , and Q the (Jacobson) radical
A A A
of A, the set of nilpotent elements of A, and the set of quasi-nilpotent elements of
A,respectively; thesetsR , N , andQ aredefinedin[19], butR isdenotedby
A A A A
radA in [19]. We recall that R is defined to be the intersection of the maximal
A
modular left ideals of A, that R is an ideal in A, and that A is defined to be
A
semisimple if R ={0}. We always have the trivial inclusions:
A
R ⊂Q , N ⊂Q .
A A A A
In general, we have R (cid:8)⊂ N and N (cid:8)⊂ R ; further, neither Q nor N is
A A A A A A
necessarily closedunder either additionor multiplication inA. For anideal I inA,
we have R =I ∩R ; in the case where A/I is semisimple, we have R =R .
I A I A
A nilpotent element a ∈ A has index n if n = min{k ∈ N : ak = 0}; a subset
S of A is nil if each element of S is nilpotent; the radical R contains each left or
A
right ideal which is nil.
Let A be an algebra. For subsets S and T of A, we set
S · T ={st:s∈S, t∈T},
1. INTRODUCTION 3
and ST = lin S · T; we write S[2] for S · S; we define Sn inductively by setting
Sn+1 = SSn (n ∈ N). The set S is nilpotent if Sn = {0} for some n ∈ N. The
algebra A factors if A=A[2].
Let A be an algebra, and let a ∈ A. Then a is quasi-invertible if there exists
b∈A such that
a+b−ab=a+b−ba=0;
the set of quasi-invertible elements of A is denoted by q-InvA. A subalgebra B of
A is full if B ∩q-InvA = q-InvB. In the case where A has an identity, the set of
invertible elementsofAisdenotedby InvA, andthespectrum ofanelementa∈A
is denoted by σ (a) or σ(a), so that
A
σ (a)={z ∈C:ze −a(cid:8)∈InvA};
A A
if B is a full subalgebra of A and a∈B, then σ (a)=σ (a).
B A
A character on an algebra A is a non-zero homomorphism from A onto C; the
collection of characters on A is the character space of A and is denoted by Φ .
A
Suppose that Φ (cid:8)=∅, andtakea∈A. Thenthe Gel’fand transform of a is defined
A
to be (cid:1)a ∈ CΦA, where (cid:1)a(ϕ)= ϕ(a) (ϕ ∈ΦA); we define the Gel’fand transform of
A as
G :a(cid:4)→(cid:1)a, A→CΦA,
so that G is a homomorphism.
Let m,n∈N, and let S be a set. The collection of m×n matrices with entries
from S is denoted by M (S), with M (S) for M (S) and M for M (C).
m,n n n,n m,n m,n
In particular, M is a unital algebra; the matrix units in M are denoted by E ,
n n ij
so that
E E =δ E (i,j,k,(cid:2)∈N ),
ij k(cid:2) j,k i(cid:2) n
and the identity matrix in M is I = (δ ). Let A be an algebra. Then M (A)
n n i,j n
is also an(cid:2)algebra in the obvious way; the matrix (aij) is identified with the tensor
product {E ⊗a :i,j ∈N }, so that M (A) is isomorphic to M ⊗A. If A is
ij ij n n n
unital, weregardM asasubset ofM (A)byidentifyingE withE ⊗e . Inthe
n n ij ij A
case where A is commutative, the determinant, deta, of an element a ∈ M (A) is
n
defined in the usual way; the element a is invertible in M (A) if and only if deta
n
is invertible in A.
LetE bealinearspace. ThelinearspanofasubsetS ofE isdenotedbylin S.
Let S and T be subsets of E. Then
S+T ={s+t:s∈S, t∈T}.
The linear space of all linear operators from E to a linear space F is denoted by
L(E,F),andwewriteL(E)forthespaceL(E,E). Infact,L(E)isanalgebrawith
respect to composition of operators, with identity I , the identity operator on E.
E
Let A be an algebra, and let E be an A-bimodule with respect to operations
(a,x)(cid:4)→a · x and (a,x)(cid:4)→x · a from A×E to E. For subsets S of A and T of E,
set
S · T ={a · x:a∈S, x∈T},
and ST =lin S · T. A map D ∈L(A,E) is a derivation if
D(ab)=D(a) · b+a · D(b) (a,b∈A).
For x∈E, set
ad :a(cid:4)→a · x−x · a, A→E.
x