CONNES-AMENABILITY OF BIDUAL AND WEIGHTED SEMIGROUP ALGEBRAS
MATTHEW DAWS
Abstract
We investigate the notion of Connes-amenability, introduced by Runde in [10], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of aσWC-virtual diagonal, as introduced in [13], especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing that these behave in the same way as C∗-algebras with regards Connes-amenability of the bidual algebra. We also show that for each one of these cancellative semigroup algebras l1(S, ω), we have thatl1(S, ω)is Connes-amenable (with respect to the canonical predualc0(S)) if and only ifl1(S, ω)is amenable, which is in turn equivalent toSbeing an amenable group, and the weight satisfying a certain restrictive condition. This latter point was first shown by Grønbæk in [6], but we provide a unified proof. Finally, we consider the homological notion of injectivity, and show that here, weighted semigroup algebras do not behave like C∗-algebras.
1. Introduction
We first fix some notation, following [2]. For a Banach spaceE, we letEbe its dual space, and forµ∈Eandx∈E, we writeµ, x =µ(x)for notational convenience. We then have the canonical map κE : E → E defined by κE(x), µ = µ, xforµ ∈ E, x ∈ E. For Banach spacesE andF, we writeB(E, F )for the Banach space of bounded linear maps betweenEand F; we writeB(E, E)=B(E); we writeTfor the adjoint ofT ∈B(E, F ). We use the notion of Banach leftA-modules, right modules and bimodules as in [2].
A linear mapd : A → E between a Banach algebra A and a Banach A-bimoduleEis aderivationifd(ab) = a·d(b)+d(a)·bfora, b ∈A. Forx ∈ E, we defineδx : A →E byδx(a) = a·x−x·a. Thenδx is a derivation, called aninner derivation.
A Banach algebraA is amenable if every derivationd : A → E to a dual bimodule is inner. For example, a C∗-algebraA is amenable if and only ifA is nuclear; a group algebraL1(G)is amenable if and only if the locally compact groupGis amenable (which is the motivating example). See [14] for further discussions of amenability and related notions.
Received September 5, 2005.
LetEbe a Banach space andF a closed subspace ofE. Then we naturally, isometrically, identifyFwithE/F◦, where
F◦= {µ∈E:µ, x =0 (x∈F )}.
Definition1.1. LetEbe a Banach space andE∗be a closed subspace of E. LetπE∗ : E →E/E∗◦be the quotient map, and suppose thatπE∗ ◦κE
is an isomorphism fromEtoE∗. Then we say thatEis adual Banach space withpredualE∗.
WhenAis a dual Banach space with predualA∗which is also a submodule ofAwe say thatA is adual Banach algebra.
For a dual Banach algebraA with predualA∗, we henceforth identifyA withA∗. Thus we get a weak∗-topology onA, which we denote byσ(A,A∗). As noticed by Runde (see [10]), there are very few Banach algebras which are both dual and amenable. For von Neumann algebras, which are the motiv- ating example of dual Banach algebras, there is a weaker notion of amenablity, called Connes-amenability, which has a natural generalisation to the case of dual Banach algebras.
Definition1.2. LetA be a dual Banach algebra with predualA∗. LetE be a BanachA-bimodule. ThenE is aw∗-BanachA-bimoduleif, for each µ∈E, the maps
A →E, a→
a·µ, µ·a areσ (A,A∗)−σ(E, E)continuous.
Then(A,A∗) is Connes-amenable if, for each w∗-BanachA-bimodule E, each derivationd:A →E, which isσ(A,A∗)−σ(E, E)continuous, is inner.
Given a Banach algebraA, we define bilinear mapsA×A →Aand A×A→Aby
·µ, a = , µ·a µ·, a = , a·µ (∈A, µ∈A, a∈A).
We then define two bilinear maps,♦:A×A→Aby
, µ = , ·µ ♦, µ = , µ· (, ∈A, µ∈A).
We can check thatand♦are actually algebra products, called thefirstand second Arens productsrespectively. ThenκA :A →Ais a homomorphism with respect to either Arens product. When = ♦, we say thatA isArens regular. In particular, whenA is Arens regular, we may check thatAis a dual Banach algebra with predualA.
Theorem 1.3. Let A be an Arens regular Banach algebra. When A is amenable,A is Connes-amenable. IfκA(A)is an ideal in A andA is Connes-amenable, thenAis amenable.
Let A be a C∗-algebra. Then A is Arens regular, and A is Connes- amenable if and only ifA is amenable.
Proof. The first statements are [10, Corollary 4.3] and [10, Theorem 4.4].
The statement about C∗-algebras is detailed in [14, Chapter 6].
Another class of Connes-amenable dual Banach algebras is given by Runde in [11], where it is shown thatM(G), the measure algebra of a locally compact groupG, is amenable if and only ifGis amenable.
The organisation of this paper is as follows. Firstly, we study intrinsic characterisations of amenability, recalling a result of Runde from [13]. We then simplify these conditions in the case of Arens regular Banach algebras.
We recall the notion of an injectivemodule, and quickly note how Connes- amenability can be phrased in this language. The final section of the paper then applies these ideas to weighted semigroup algebras. We finish with some open questions.
2. Characterisations of amenability
LetEandF be Banach spaces, and form the algebraic tensor productE⊗F. We can normE⊗F with theprojective tensor norm, defined as
uπ =inf n
k=1
xkyk:u= n k=1
xk⊗yk
(u∈E⊗F ).
Then the completion of(E⊗F, · π)isE⊗F, theprojective tensor product ofEandF.
LetAbe a Banach algebra. ThenA⊗Ais a BanachA-bimodule for the module actions given by
a·(b⊗c)=ab⊗c, (b⊗c)·a=b⊗ca (a∈A, b⊗c∈A⊗A).
Define"A :A⊗A →A by"A(a⊗b)=ab. Then"Ais anA-bimodule homomorphism. LetM ∈(A⊗A)be such that
a·M =M·a, "A(M)·a=κA(a) (a ∈A).
TheM is avirtual diagonal forA. It is well-known thatA is an amenable Banach algebra if and only ifA has a virtual diagonal.
Definition 2.1. LetA be a dual Banach algebra with predualA∗, and letEbe a BanachA-bimodule. Thenx ∈ σWC(E)if and only if the maps A →E,
a→a·x, x·a areσ (A,A∗)−σ(E, E)continuous.
It is clear that σWC(E) is a closed submodule of E. TheA-bimodule homomorphism"Ahas adjoint"A :A→(A⊗A ). In [13, Corollary 4.6]
it is shown that"A(A∗)⊆σWC((A⊗A)). Consequently, we can view"A as a mapA∗→σWC((A⊗A)), and hence view"Aas a mapσWC((A⊗ A)) →A∗ =A, denoted by"˜A. LetM ∈σWC((A⊗A))be such that
a·M =M·a, a"˜A(M)=a (a∈A).
TheM is aσWC-virtual diagonal forA.
Theorem2.2. LetAbe a dual Banach algebra with predualA∗. Then the following are equivalent:
(1) A is Connes-amenable;
(2) A has aσWC-virtual diagonal.
Proof. This is [13, Theorem 4.8].
In particular, we see that a Connes-amenable Banach algebra is unital (which can of course be shown in an elementary fashion, as in [10, Proposi- tion 4.1]).
3. Connes-amenability for biduals of algebras
Recall Gantmacher’s theorem, which states that a bounded linear mapT : E →F between Banach spacesE andF isweakly compact if and only if T(E) ⊆ κF(F ). We writeW(E, F )for the collection of weakly compact operators inB(E, F ).
Lemma3.1. Let Ebe a dual Banach space with predual E∗, letF be a Banach space, and letT ∈B(E, F). Then the following are equivalent, and in particular each imply thatT is weakly compact:
(1) T isσ(E, E∗)−σ(F, F)continuous;
(2) T(F)⊆κE∗(E∗);
(3) there existsS∈W(F, E∗)such thatS=T. Proof. That (1) and (2) are equivalent is standard.
Suppose that (2) holds, so that we may defineS∈B(F, E∗)byκE∗◦S= T◦κF. Then, forx∈Eandy ∈F, we have
x, S(y) = T(κF(y)), x = T (x), y,
so that S = T. Then S(F) = T(F) ⊆ κE∗(E∗), so that S is weakly compact, by Gantmacher’s Theorem, so that (3) holds.
Conversely, if (3) holds, as S is weakly compact, we have κE∗(E∗) ⊇ S(F)=T(F), so that (2) holds.
It is standard that for Banach spacesEandF, we have(E⊗F ) =B(F, E) with duality defined by
T , x⊗y = T (y), x (T ∈B(F, E), x⊗y∈E⊗F ).
Then we see, fora, b, c ∈ A and T ∈ (A ⊗A) = B(A,A), thata · T , b⊗c = T (ca), band thatT ·a, b⊗c = T (c), ab = T (c)·a, b so that
(1)
(a·T )(c)=T (ca), (T ·a)(c)=T (c)·a (a, c∈A, T :A →A).
Notice that we could also have defined (E ⊗F ) to be B(E, F). This would induce a different bimodule structure onB(A,A), but we shall see in Section 4 that our chosen convention seems more natural for the task at hand.
Proposition3.2. LetA be a dual Banach algebra with predualA∗. For T ∈B(A,A)=(A⊗A), define mapsφr, φl :A ⊗A →Aby
φr(a⊗b)=TκA(a)·b, φl(a⊗b)=a·T (b) (a⊗b∈A ⊗A).
ThenT ∈σWC(B(A,A))if and only ifφr andφlare weakly compact and have ranges contained inκA∗(A∗).
Proof. ForT ∈B(A,A) = (A ⊗A), defineRT, LT : A →(A ⊗ A) by RT(a) = a·T and LT = T ·a, for a ∈ A. By definition, T ∈ σWC(B(A,A))if and only ifRT andLT areσ(A,A∗)−σ(B(A,A), (A⊗A))continuous. By Lemma 3.1, this is if and only if there existϕr, ϕl ∈ W(A ⊗A,A∗)such thatϕr =RT andϕl =LT.
Fora⊗b∈A⊗Aandc∈A, we see that
c, ϕr(a⊗b) = RT(c), a⊗b = c·T , a⊗b = T (bc), a
= TκA(a), bc = TκA(a)·b, c = φr(a⊗b), c, c, ϕl(a⊗b) = LT(c), a⊗b = T ·c, a⊗b = T (b), ca
= a·T (b), c = φl(a⊗b), c.
Thus κA∗ ◦ϕr = φr and κA∗ ◦ϕl = φl. Consequently, we see that T ∈ σWC(B(A,A))if and only ifφrandφlare weakly compact and take values inκA∗(A∗).
The following definition is [13, Definition 4.1].
Definition 3.3. LetA be a Banach algebra and letE be a BanachA- bimodule. An elementx ∈Eisweakly almost periodicif the maps
A →E, a→a·x, x·a
are weakly compact. The collection of weakly almost periodic elements inE is denoted by WAP(E).
Lemma 3.4. Let A be a Banach algebra, and let T ∈ B(A,A) = (A⊗A). Letφr, φl :A⊗A →Abe as above. ThenT ∈WAP(B(A,A)) if and only ifφr andφlare weakly compact.
Proof. LetRT, LT :A →B(A,A)be as in the above proof. By defin- ition,T ∈WAP(B(A,A))if and only ifLT andRT are weakly compact.
We can verify that
φr ◦κA =RT, φl◦κA =LT, RT ◦κA⊗A =φr, LT ◦κA⊗A =φl, which completes the proof.
Corollary3.5. LetAbe a unital, dual Banach algebra with predualA∗, and letT ∈B(A,A) = (A ⊗A). The following are equivalent, and, in particular, each imply thatT is weakly compact:
(1) T ∈σWC(B(A,A));
(2) T (A)⊆κA∗(A∗),T(κA(A))⊆κA∗(A∗), andT∈σWC(B(A,A)); (3) T (A)⊆κA∗(A∗),T(κA(A))⊆κA∗(A∗), andT∈WAP(B(A,A)).
Proof. Let eA be the unit of A, so that for a ∈ A, we have T (a) = φl(eA ⊗a)andTκA(a) = φr(a⊗eA), which shows that (1) implies (2);
clearly (2) implies (1).
AsA∗is anA-bimodule, (2) and (3) are equivalent by an application of Lemma 3.4 and Proposition 3.2.
Theorem3.6. LetAbe a dual Banach algebra with predualA∗. ThenA is Connes-amenable if and only ifAis unital and there existsM ∈(A⊗A) such that:
(1) M, a·T −T ·a =0fora∈AandT ∈σWC(W(A,A)); (2) κA ∗"A(M)=eA, whereeA is the unit ofA.
Proof. AsσWC((A ⊗A)) is a quotient of (A ⊗A), this is just a re-statement of Theorem 2.2.
When A is an Arens regular Banach algebra, A is a dual Banach al- gebra with canonical predualA. In this case, we can make some significant simplifications in the characterisation of whenAis Connes-amenable.
For a Banach algebraA, we define the mapκA⊗κA :A⊗A →A⊗A by
(κA ⊗κA)(a⊗b)=κA(a)⊗κA(b) (a⊗b∈A ⊗A).
We turnA⊗Ainto a BanachA-bimodule in the canonical way. ThenκA⊗ κAis anA-bimodule homomorphism. The following is a simple verification.
Lemma3.7. LetA be a Banach algebra. The map ιA :B(A,A)→B(A,A); T →T,
is anA-bimodule homomorphism which is an isometry onto its range. Fur- thermore, we have that(κA⊗κA)◦ιA =IB(A,A). DefineρA :A⊗A→ (A⊗A)by
ρA(τ), T = T, τ (τ ∈A⊗A, T ∈B(A,A)=(A⊗A)).
ThenρA is a norm-decreasingA-bimodule homomorphism which satisfies ρA◦(κA ⊗κA)=κA⊗A.
For a Banach algebraA, it is clear thatW(A,A)is a sub-A-bimodule ofB(A,A)=(A ⊗A).
Theorem3.8. Let A be an Arens regular Banach algebra such thatA is unital, and letT ∈ B(A,A) = (A⊗A). Then the following are equivalent:
(1) T ∈ σWC(B(A,A)), where we treat B(A,A) as an A- bimodule;
(2) T=Sfor someS∈WAP(W(A,A)), where now we treatW(A,A) as anA-bimodule.
Proof. We apply Corollary 3.5 toA, so that (1) is equivalent toT being weakly compact, T (A) ⊆ κA(A), T(κA(A)) ⊆ κA(A), and T ∈ WAP(B(A,A)). Thus, if (1) holds, then there existsT0 ∈ W(A,A) such thatT =κA◦T0, and there existsT1∈W(A,A)such thatT◦κA = κA◦T1. LetS=T0◦κA ∈W(A,A). As before, we can check thatS=T1
and thatS = T. We know that the maps LT, RT : A → B(A,A),
defined by LT() = T ·and RT() = ·T for ∈ A, are weakly compact. DefineLS, RS : A → B(A,A)is an analogous manner, using S∈W(A,A). Fora ∈A,S·a∈W(A,A), so for∈Aandb∈A,
(S·a)(), b = , (S·a)(b) = , S(b)·a
= a·, S(b) = S(a·), b.
Thus, fora∈A and, ∈A, we have that
ιA(LS(a))(), = (S·a)(), = , S(a·) = S()·a, , so thatιA(LS(a))()=S()·a, and hence thatιA(LS(a))=S·a=T·a = T ·κA(a)=LT(κA(a)). Thus we have thatLS =(κA ⊗κA)◦RT ◦κA, so thatLSis weakly compact. A similar calculation shows thatRSis also weakly compact, so thatS∈WAP(W(A,A)). This shows that (1) implies (2).
Conversely, if (2) holds, thenLSandRSare weakly compact. AsSis weakly compact,T (A)= S(A)⊆ κA(A)andT(κA(A))= S(κA(A))
=κA(S(A))⊆κA(A), andT is weakly compact. Thus, to show (1), we are required to show thatLT andRT are weakly compact.
Fora, b∈A and∈A, we have
(a·S)(), b = , S(ba) = a·S(), b.
Then, for, ∈Aanda ∈A, we thus have
RS(ρA(⊗)), a = (a·S), ⊗ = (a·S)(),
= , a·S() = ·a, S()
= κA(a), S() = S()·, a.
Hence we see thatRS(ρA(⊗))=S()·. LetU =RS ◦ρA :A⊗ A→A, so that asRSis weakly compact, so isU. Then, for, , -∈A, we have that
U(-), ⊗ = -, S()· = ♦-, S()
= S(-), = (-·S)(), ,
so that U(-) = - ·T, that is, U = RT, so that RT is weakly compact.
Similarly, we can show thatLT is weakly compact, completing the proof.
Theorem3.9. Let A be an Arens regular Banach algebra. ThenA is Connes-amenable if and only ifAis unital and there existsM ∈(A⊗A) such that:
(1) "A(M)=eA, the unit ofA;
(2) M, a·T−T·a =0for eacha∈Aand eachT ∈WAP(W(A,A)). Proof. By Theorem 3.6, we wish to show that the existence of such anM is equivalent to the existence ofN ∈(A⊗A)such that:
(N1) κA "A(N)=eA;
(N2) N, ·S−S· =0 for each∈Aand each S∈σWC(B(A,A)).
We can verify thatιA ◦"A = "A ◦κA, so that (N1) is equivalent to
"AιA(N) = eA. ForS ∈ σWC(B(A,A)), we know thatS = Tfor someT ∈ WAP(W(A,A)), by Theorem 3.8. That is, the mapsφr andφl, formed usingT as in Proposition 3.2, are weakly compact. Then, for∈A, φr(), φl()∈B(A,A), and we can check that
φr()(a)=κA T(a·), φl()(a)=T (a)· (a∈A).
Thenφr(), φl()∈B(A,A)are the maps
φr()()=·T(), φl()()=T() ( ∈A), where we remember thatT(A)⊆κA(A). Consequentlyφr(), φl()
∈B(A,A)are given by
φr()()=T(), φl()()=T()· ( ∈A), whereAis anA-bimodule, asAis Arens regular. That is,φr()=·S andφl() =S·. Hence (N2) is equivalent to
0= N, φr()−φl() = N, ιA(φr()−φl())
= ιA(N), φr()−φl(),
for each∈AandS∈σWC(B(A,A)). That is, (N2) is equivalent to φrιA(N)−φlιA(N)=0 (S∈σWC(B(A,A))).
Asφr andφl are weakly compact,φrandφltake values inκA(A), and so (N2) is equivalent to
0= φrιA(N)−φlιA(N), κA(a) = ιA(N), φr(κA(a))−φl(κA(a)), for eacha ∈ A and eachS ∈ σWC(B(A,A)). However,φr(κA(a))− φl(κA(a))=a·T −T ·a, so that (N2) is equivalent to
0= ιA(N), a·T −T ·a (a ∈A),
for eachT ∈W(A,A)such thatφr andφl are weakly compact.
Thus we have established that (N1) holds forNif and only if (1) holds for M =ιA(N), and that (N2) holds forNif and only if (2) holds forM =ιA(N), completing the proof.
We immediately see thatAamenable implies thatAis Connes-amenable.
Furthermore, ifA is itself a dual Banach algebra, then Corollary 3.5 shows that ifAis Connes-amenable, thenAis Connes-amenable: notice that ifeA
is the unit ofA, then
κA ∗(eA)a, µ = eA ·a, κA∗(µ)
= κA(a), κA∗(µ) = a, µ (a ∈A, µ∈A∗), so thatκA ∗(eA)is the unit ofA.
4. Injectivity of the predual module
LetA be a Banach algebra, and letEandF be Banach leftA-modules. We writeAB(E, F )for the closed subspace of B(E, F )consisting of left A- module homomorphisms, and similarly writeBA(E, F )andABA(E, F )for rightA-module andA-bimodule homomorphisms, respectively. We say that T ∈ AB(E, F )isadmissible if both the kernel and image ofT are closed, complemented subspaces of, respectively, EandF. IfT is injective, this is equivalent to the existence ofS∈B(F, E)such thatST =IE.
Definition4.1. LetAbe a Banach algebra, and letEbe a Banach leftA- module. ThenEisinjectiveif, wheneverFandGare Banach leftA-modules, θ ∈AB(F, G)is injective and admissible, andσ ∈ AB(F, E), there exists ρ∈AB(G, E)withρ◦θ =σ.
We say thatEisleft-injectivewhen we wish to stress that we are treating Eas a left module. Similar definitions hold for right modules and bimodules (writtenright-injectiveandbi-injectivewhere necessary).
Let A be a Banach algebra, letE be a Banach leftA-module, and turn B(A, E)into a leftA-module by setting
(a·T )(b)=T (ba) (a, b∈A, T ∈B(A, E)).
Then there is a canonical leftA-module homomorphismι:E →B(A, E) given by
ι(x)(a)=a·x (a∈A, x∈E).
Notice that if E is a closed submodule of A, then B(A, E) is a closed submodule of(A⊗A) =B(A,A), andιis the restriction of"A :A→ B(A,A)toE.
Similarly, we turnB(A ⊗A, E)into a BanachA-bimodule by (a·T )(b⊗c)=T (ba⊗c), (T ·a)(b⊗c)=T (b⊗ac)
(a, b, c∈A, T ∈B(A⊗A, E)).
We then define (with an abuse of notation)ι:E→B(A ⊗A, E)by ι(x)(a⊗b)=a·x·b (x∈E, a⊗b∈A⊗A), so thatιis anA-bimodule homomorphism.
We can also turnB(A, E)into a rightA-module by reversing the above (in particular, we need to take the other possible choice in Section 3 leading to different module actions as compared to those in (1).)
Proposition4.2.LetAbe a Banach algebra, and letEbe a faithful Banach left A-module (that is, for each non-zero x ∈ E there exists a ∈ A with a·x=0). ThenEis injective if and only if there existsφ∈AB(B(A, E), E) such thatφ◦ι=IE.
Similarly, ifEis a left and right faithful BanachA-bimodule (that is, for each non-zerox ∈ Ethere exists a, b ∈ A witha·x = 0andx·b = 0).
ThenEis injective if and only if there existsφ ∈ABA(B(A ⊗A, E), E) such thatφ◦ι=IE.
Proof. The first claim is [4, Proposition 1.7], and the second claim is an obvious generalisation.
Again, there exists a similar characterisation for right modules. The above result is useful, as it allows us to work withA and not its unitisation (which is the usual approach).
LetA be a dual Banach algebra with predualA∗. It is simple to show (see [13]) that ifA∗is bi-injective, thenAis Connes-amenable. Helemskii showed in [8] that for a von Neumann algebraA, the converse is true. However, Runde (see [13]) and Tabaldyev (see [15]) have shown thatM(G), the measure algebra of a locally compact groupG, while being a dual Banach algebra with predual C0(G), has thatC0(G)is a left-injectiveM(G)-module only whenGis finite.
Recall that Runde (see [11]) has shown thatM(G)is Connes-amenable if and only ifGis amenable.
Similarly, it is simple to show (using a virtual diagonal) that ifAis a Banach algebra with a bounded approximate identity, thenA is amenable if and only ifAis bi-injective.
Let E andF be Banach left A-modules, and let φ : E → F be a left A-module homomorphism which is bounded below. Thenφ(E)is a closed
submodule ofF, so thatF/φ(E)is a Banach leftA-module. Hence we have ashort exact sequence:
−−−−→φ
0−−−−→E F −−−−→→ F/φ(E)−−−−→0.
P
If there exists a bounded linear mapP :F →Esuch thatP ◦φ =IE, then we say that the short exact sequence isadmissible. If, further, we may choose P to be a leftA-module homomorphism, then the short exact sequence is said tosplit. Similar definitions hold for right modules and bimodules.
Proposition4.3.LetA be a Banach algebra, letEbe a Banach leftA- module, and consider the following short exact sequence:
−−−−→ι
0−−−−→E B(A, E)−−−−→→ B(A, E)/ι(E)−−−−→0.
P
ThenEis injective if and only if this short exact sequence is admissible and splits.
Proof. See, for example, [14, Section 5.3]. Notice that whenA is unital, the short exact sequence is certainly admissible.
Proposition4.4.LetAbe a unital dual Banach algebra with predualA∗, and consider the following admissible short exact sequence ofA-bimodules:
(2)
"A
−−−−→
0−−−−→A∗ σWC((A ⊗A))
P −−−−→→ σWC((A ⊗A))/"A(A∗)−−−−→0.
ThenAis Connes-amenable if and only if this short exact sequence splits.
Proof. Notice that "A certainly maps A∗ into σWC((A ⊗ A)) = σWC(B(A,A)), and that Corollary 3.5 shows that we can define P : σWC(B(A,A))→A∗byP (T )=T (eA)forT ∈σWC(B(A,A)).
Suppose that we can choosePto be anA-bimodule homomorphism. Then letM =P(eA), so that fora∈AandT ∈σWC(B(A,A)),
a·M−M·a, T = eA, P (T ·a−a·T ) = a−a, P (T ) =0, so thata ·M −M ·a. Also "A(M) = (P ◦"A)(eA) = eA, so thatM is a σWC-virtual diagonal, and hence A is Connes-amenable by Runde’s theorem.
Conversely, letMbe aσWC-virtual diagonal and defineP :σWC(B(A, A))→Aby
P (T ), a = M, a·T (a∈A, T ∈σWC(B(A,A)).
Let (aα) be a bounded net in A which tends toa ∈ A in the σ(A,A∗)- topology. By definition,aα·T→a·Tweakly, for eachT∈σWC(B(A,A)), so that P (T ), aα → P (T ), a. This implies that P maps into A∗, as re- quired. Then, forµ∈A∗,
a, P "A(µ) = M, a·"A(µ) = M, "A(a·µ)
= eA, a·µ = a, µ (a∈A), so thatP "A =IA∗. Finally, we note that
P (a·T ·b), c = M, ca·T ·b = b·M, ca·T
= M·b, ca·T = P (T ), bca
= a·P (T )·b, c (a, b, c∈A, T ∈σWC(B(A,A))), so thatP is anA-bimodule homomorphism, as required.
Let A be an Arens regular Banach algebra. By reversing the argument Theorem 3.8, we can show that"A : A →B(A,A)actually maps into WAP(W(A,A)). Furthermore, if A is unital, then we may define P : WAP(W(A,A))→Aby
P (T ), a = eA, P (a) (a∈A, T ∈WAP(W(A,A))).
Then we have that
P "A(µ), a = eA, a·µ = µ, a (a∈A, µ∈A).
Proposition4.5. Let A be an Arens regular Banach algebra such that Ais unital, and consider the following admissible short exact sequence of A-bimodules:
(3)
"A
−−−−→
0−−−−→A WAP(W(A,A))
P −−−−→→ WAP(W(A,A))/"A(A)−−−−→0.
ThenAis Connes-amenable if and only if this short exact sequence splits.
Proof. This follows in the same manner as the above proof, using The- orem 3.9.
5. Beurling algebras
LetSbe a discrete semigroup (we can extend the following definitions to locally compact semigroups, but for the questions we are interested in, the results for non-discrete groups are trivial). AweightonSis a functionω:S→R+such that ω(st)≤ω(s)ω(t) (s, t ∈S).
Furthermore, ifS is unital with unituS, then we also insist thatω(uS) = 1.
This last condition is simply a normalisation condition, as we can always set ˆ
ω(s)= sup{ω(st)ω(t)−1: t ∈ S}for eachs ∈S. Fors, t ∈ S, we have that ω(st)≤ ˆω(s)ω(t), so that
ˆ
ω(st)=sup{ω(str)ω(r)−1:r ∈S}
≤sup{ ˆω(s)ω(tr)ω(r)−1:r ∈S} = ˆω(s)ω(t).ˆ Clearly ω(uˆ S) = 1 and ω(s)ˆ ≤ ω(s) for each s ∈ S, while ω(s)ˆ ≥ ω(s)ω(uS)−1, so thatωˆ is equivalent toω.
We form the Banach space l1(S, ω)=
(ag)g∈S ⊆C:(ag):=
g∈S
|ag|ω(g) <∞
.
Thenl1(S, ω), with the convolution product, is a Banach algebra, called a Beurling algebra. See [1] and [3] for further information on Beurling algebras and, in particular, their second duals.
It will be more convenient for us to think ofl1(S, ω)as the Banach space l1(S) together with a weighted algebra product. Indeed, forg ∈ S, let δg ∈ l1(S)be the standard unit vector basis element which is thought of as a point- mass atg. Then eachx∈l1(S)can be written uniquely asx =
g∈Sxgδgfor some family(xg)⊆Csuch thatx =
g∈S|xg|<∞. We then define 6(g, h)= ω(gh)
ω(g)ω(h), δg8ωδh =δg8 δh =δgh6(g, h) (g, h∈S), and extend8tol1(S)by linearity and continuity.
For example, if ω and ωˆ are equivalent weights on S, then define ψ : l1(S, ω)→l1(S,ω)ˆ byψ(δs)= ˆω(s)ω(s)−1δs. Asωandωˆ are equivalent,ψ is an isomorphism of Banach spaces. Then ψ(δs 8 δt) = ω(st)ω(s)−1ω(t)−1ω(st)ω(st)ˆ −1δst = ψ(δs) 8 ψ(δt), so thatψ is a homo- morphism.
For a setI, we define the spacesc0(I)andl∞(I)in the standard way. We write(ei)i∈I for the standard unit vector basis ofc0(I)(or its image inl∞(I)), so thatδj, ei =δi,j, the Kronecker delta, forj ∈I.
For a semigroupSands ∈S, we define mapsLs, Rs :S→Sby Ls(t)=st, Rs(t)=ts (t ∈S).
If, for eachs ∈ S, Ls andRs are finite-to-one maps, then we say thatS is weakly cancellative. When Ls and Rs are injective for each s ∈ S, we say thatSiscancellative. WhenSis abelian and cancellative, a construction going back to Grothendieck shows thatSis a sub-semigroup of some abelian group.
However, this can fail to hold for non-abelian semigroups.
Proposition5.1. Let S be a weakly cancellative semigroup, let ω be a weight onS, and letA = l1(S, ω). Thenc0(S) ⊆ l∞(S) = Ais a sub-A- module ofA, so thatl1(S, ω)is a dual Banach algebra with predualc0(S).
Proof. Forg, h∈Sanda=(as)s∈S ∈l1(S, ω), we have eg·δh, a = eg, δh8 a = eg,
s∈S
asδhs6(h, s) =
{s∈S:hs=g}
as6(h, s).
AsS is weakly cancellative, there exists at most finitely manys ∈ S such thaths = g, so thateg·δhis a member ofc0(S). Thus we see thatc0(S)is a right sub-A-module ofA. The argument on the left follows in an analogous manner.
Notice that the above result will hold for some semigroups S which are not weakly cancellative, provided that the weight behaves in a certain way.
However, it would appear that the later results in this section do not easily generalise to the non-weakly cancellative case.
Following [3, Definition 2.2], we have the following definition.
Definition5.2. LetIandJbe non-empty infinite sets, and letf :I×J → Cbe a function. Thenf clusters onI ×J if
n→∞lim lim
m→∞f (xm, yn)= lim
m→∞ lim
n→∞f (xm, yn),
whenever(xm)⊆Iand(yn)⊆J are sequences of distinct elements, and both iterated limits exist.
Furthermore,f 0-clusters onI×J iff clusters onI×J, and the iterated limits are always 0, when they exist.
From now on we shall exclude the trivial case when our (semi-)group is finite.
Theorem5.3. LetSbe a discrete, weakly cancellative semigroup, and let ωbe a weight onS. Then the following are equivalent:
(1) l1(S, ω)is Arens regular;
(2) for sequences of distinct elements(gj)and(hk)inS, we have
j→∞lim lim
k→∞6(gj, hk)=0, whenever the iterated limit exists;
(1) 60-clusters onS×S.
Proof. That (1) and (2) are equivalent for cancellative semigroups is [1, Theorem 1]. Close examination of the proof shows that this holds for weakly cancellative semigroups as well. That (1) and (3) are equivalent follows by generalising the proof of [3, Theorem 7.11], which is essentially an application of Grothendieck’s criterion for an operator to be weakly compact. Alternatively, it follows easily that (2) and (3) are equivalent by considering theopposite semigrouptoSwhere we reverse the product.
In [1] it is also shown that ifGis a discrete, uncountable group, thenl1(G, ω) is not Arens regular for any weightω. Furthermore, by [1, Theorem 2], ifGis a non-discrete locally compact group, thenL1(G, ω)is never Arens regular.
We shall consider both the Connes-amenability ofl1(S, ω) andl1(S, ω) (with respect to the canonical predualc0(S)) as, with reference to Corollary 3.5 and Theorem 3.8, the calculations should be similar.
Proposition5.4. LetIbe a non-empty set, and letX⊆l∞(I)be a subset.
Then the following are equivalent:
(1) Xis relatively weakly compact;
(2) Xis relatively sequentially weakly compact;
(3) the absolutely convex hull ofXis relatively weakly compact;
(4) if we definef : I×X→Cbyf (i, x) = x, δifori ∈I andx ∈X, thenf clusters onI×X;
Proof. That (1) and (2) are equivalent is the Eberlien-Smulian theorem;
that (1) and (3) are equivalent is the Krein-Smulian theorem. That (1) and (4) are equivalent is a result of Grothendieck, detailed in, for example, [3, Theorem 2.3].
It is standard that for non-empty setsIandJ, we have thatl1(I)⊗l1(J )= l1(I ×J ), where, fori ∈ I andj ∈J,δi⊗δj ∈ l1(I)⊗l1(J )is identified withδ(i,j) ∈l1(I×J ). Thus we have(l1(I)⊗l1(J )) =B(l1(I), l∞(J )) =
l1(I ×J ) = l∞(I ×J ), where T ∈ B(l1(I), l∞(J )) is identified with (T(i,j))∈l∞(I×J ), whereT(i,j)= T (δi), δj.
For a non-empty setI, the unit ball ofl1(I)is the closure of the absolutely- convex hull of the set{δi : i ∈ I}, so that for a Banach space E, by the Krein-Smulian theorem, a mapT :l1(I)→Eis weakly compact if and only if the set{T (δi):i∈I}is relatively weakly compact inE.
Proposition5.5. Let S be a weakly cancellative semigroup, letω be a weight onS, and letA =l1(S, ω). LetT ∈B(A,A)be such thatT (A)⊆ κc0(S)(c0(S))andT(κA(A))⊆κc0(S)(c0(S)). ThenT ∈W(A,A), andT ∈ WAP(W(A,A))if and only if, for each sequence(kn)of distinct elements ofS, and each sequence(gm, hm)of distinct elements ofS×Ssuch that the repeated limits
limn lim
m T (δhm), δkngm, lim
n lim
m 6(kn, gm) (4)
limn lim
m T (δhmkn), δgm, lim
n lim
m 6(hm, kn) (5)
all exist, we have that at least one repeated limit in each row is zero.
Proof. ThatT is weakly compact follows from Gantmacher’s Theorem (compare with Corollary 3.5). To show thatT ∈WAP, by Lemma 3.4, we are required to show that the mapsφr andφl are weakly compact. We shall show thatφl is weakly compact if and only if one of the repeated limits in the first line (4) is zero; the proof thatφr is related to (5) follows in a similar way. We have that
φl(δ(g,h))=φl(δg⊗δh)=δg·T (δh) (g, h∈S).
By Proposition 5.4,φl is weakly compact if and only if the function S×(S×S)→C; (k, (g, h))→ δg·T (δh), δk
= T (δh), δkg6(k, g) (g, h, k∈S) clusters onS×(S×S). AsT is weakly compact, the function
S×S→C; (g, h)→ T (δg), δh (g, h∈S) does cluster onS×S.
Let (kn)be a sequence of distinct elements of S, and let (gm, hm) be a sequence of distinct elements ofS×Ssuch that the iterated limits
(6) lim
n lim
m T (δhm), δkngm6(kn, gm), lim
m lim
n T (δhm), δkngm6(kn, gm)
