Central Beurling algebras: Weak amenability of the central Beurling algebras on [FC]$^-$ groups
Varvara Shepelska, Yong Zhang

TL;DR
This paper investigates the weak amenability of central Beurling algebras on [FC]$^-$ groups, extending known results from commutative cases and establishing conditions based on group properties and weights.
Contribution
It provides necessary and sufficient conditions for weak amenability of central Beurling algebras on specific groups, including polynomial weights on compactly generated [FC]$^-$ groups.
Findings
Weak amenability characterized for [FC]$^-$ groups.
Necessary conditions for [FD]$^-$ groups.
Weak amenability of $ZL^1(G, ext{poly})$ if and only if $ ext{degree} < 1/2$.
Abstract
We study weak amenability of central Beurling algebras . The investigation is a natural extension of the known work on the commutative Beurling algebra . For [FC] groups we establish a necessary condition and for [FD] groups we give sufficient conditions for the weak amenability of . For a compactly generated [FC] group with the polynomial weight , we prove that is weakly amenable if and only if .
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Taxonomy
TopicsAdvanced Operator Algebra Research · Geometric and Algebraic Topology · Advanced Topics in Algebra
Weak amenability of the central Beurling algebras on [FC]- groups
Varvara Shepelska
and
Yong Zhang
Department of Mathematics
University of Manitoba
Winnipeg, Manitoba
R3T 2N2 Canada
(Date: November 24, 2014)
Abstract.
We study weak amenability of central Beurling algebras . The investigation is a natural extension of the known work on the commutative Beurling algebra . For [FC]- groups we establish a necessary condition and for [FD]- groups we give sufficient conditions for the weak amenability of Z\hbox{L^{1}(G,\omega)}. For a compactly generated [FC]- group with the polynomial weight , we prove that is weakly amenable if and only if .
Key words and phrases:
weighted group algebras, center, [FC]- groups, [FD]- groups, compactly generated, tensor product
2010 Mathematics Subject Classification:
Primary 46H20, 43A20. Secondary 43A10.
‡ Supported by NSERC Grant 238949
1. Introduction
Let be a locally compact group. As it is customary, two functions equal to each other almost everywhere on with respect to the Haar measure will be regarded as the same. We denote the integral of a function on a (Borel) measurable subset of against a fixed left Haar measure by . The space of all complex valued Haar integrable functions on is denoted by . A weight on is a Borel measurable function satisfying
[TABLE]
Given a weight on , we consider the space of all complex valued Haar measurable functions on that satisfy
[TABLE]
With the convolution product and the norm , is a Banach algebra, called a Beurling algebra on . When , this is simply the group algebra . Let Z\hbox{L^{1}(G,\omega)} be the closed subalgebra of consisting of all f\in\hbox{L^{1}(G,\omega)} such that for all , where (). Then Z\hbox{L^{1}(G,\omega)} is a commutative Banach algebra, called a central Beurling algebra [19]. Indeed, Z\hbox{L^{1}(G,\omega)} is the center of . It is well known that Z\hbox{L^{1}(G,\omega)} is non-trivial if and only if is an [IN] group [22].
From [8, Remark 8.8], a measurable weight on is always equivalent to a continuous weight on , where the equivalence means that there are constants such that for almost all . The equivalence implies that the respective Beurling algebras and are isomorphic as Banach algebras. So are the central Beurling algebras Z\hbox{L^{1}(G,\omega)} and . For this reason, in our investigation we will most time assume the weight to be a continuous function.
The study of goes back to A. Beurling [4], where was considered. One may find a good account of elementary theory concerning the general weighted group algebra in [26]. Structure of the center of group algebras and the central Beurling algebras were substantially studied in [19, 20].
Amenable Beurling algebras are essentially isomorphic to amenable group algebras [11, 30]. This is no longer true for the weak amenability of them. Weak amenability of Beurling algebras for commutative groups has been extensively investigated and are well characterized [3, 12, 27, 31], while for non-commutative groups, one hardly sees a non-trivial example of a weakly amenable Beurling algebra. Recent investigations are in [5, 28, 29].
We are concerned with weak amenability of central Beurling algebras Z\hbox{L^{1}(G,\omega)}. When is commutative, Z\hbox{L^{1}(G,\omega)} coincides with . So investigation of weak amenability for Z\hbox{L^{1}(G,\omega)} is a natural extension of the study for the commutative groups. We notice that some investigations on amenability and weak amenability of , the case with the trivial weight , have been made recently in [2] and [1]. All theses studies even for amenability have answers only for particular cases of locally compact groups, in particular the ones in [2] and [1] are only on compact and some discrete groups.
We will mainly focus on [FC]- groups (groups with precompact conjugacy classes). We note that for the trivial weight it has been shown in [2] that is always weakly amenable for an [FC]- group . When is compact, the result has a simple direct proof. In fact, is generated by its minimal idempotents if is compact. So by a simple observation (see [17, Section 7]), is always weakly amenable for compact .
We recall here some standard notions concerning a locally compact group. More details regarding them may be found in [13, 14, 25].
For a locally compact group ,
- (1)
is an [IN] group if there is a compact neighbourhood of the identity that is invariant under all inner automorphisms; 2. (2)
is a [SIN] group if there is a compact neighbourhood basis of the identity such that each member of the basis is invariant under inner automorphisms; 3. (3)
is an [FC]- group if the conjugacy class for each has a compact closure in ; 4. (4)
is an [FD]- group if the closure of the commutator subgroup of is compact in (where the commutator subgroup of is the subgroup generated by all elements of the form , ).
Obviously, is an [FD]- group if and only if there is a compact normal subgroup such that is abelian. It is also obvious that [FD] [FC]-. The less obvious inclusion [FC] [IN] was shown in [21, Proposition 3.1].
In Section 2 we will be devoted to considering [FC]- groups. We will show that the projective tensor product ZL^{1}(G_{1},\omega_{1})\hbox{\widehat{\otimes}}ZL^{1}(G_{2},\omega_{2}) is weakly amenable if and only if both and are weakly amenable and, under some conditions, ZL^{1}(G_{1}\times G_{2},\omega_{1}\times\omega_{2})\simeq ZL^{1}(G_{1},\omega_{1})\hbox{\widehat{\otimes}}ZL^{1}(G_{2},\omega_{2}). We will also show, among others, that a condition characterizing the weak amenability of the Beurling algebra on a commutative group remains necessary for central Beurling algebras on [FC]- groups. In section 3 we will focus on the case for [FD]- groups, establishing some sufficient conditions for Z\hbox{L^{1}(G,\omega)} to be weakly amenable for this sort of groups . For a compactly generated [FC]- group , we will consider the polynomial weight . We will show that is weakly amenable if and only if . This last result notably generalizes [3, Theorem 2.4.(iii)-(iv)].
2. Central Beurling algebras on [FC]- groups
Let be a locally compact group, and let be the set of all topological automorphisms of onto itself. For any compact set of and any open neighbourhood of in we denote
[TABLE]
The family of all subsets of the form forms an open neighbourhood basis at the identity of . This defines a Hausdorff topology on , called the open compact topology on . It is well-known that this topology is completely regular [15, Theorems 4.8 and 8.4]. With this topology is a topological group (but it may not be locally compact) [15, Theorems 26.5 and 26.6]. All inner automorphisms of form a (completely regular) topological subgroup of , denoted by . For , let be the inner automorphism of implemented by , i.e.
[TABLE]
Clearly, the natural mapping is a continuous group anti-homomorphism from onto , so that is a continuous group homomorphism from onto .
Let be a semitopological semigroup, i.e. is a semigroup with a Hausdorff topology such that the product is separately continuous. Obviously, a topological group is a semitopological semigroup, in particular, belongs to this class. The space of all bounded complex-valued continuous functions on forms a Banach space with the supremum norm
[TABLE]
Indeed, with the pointwise product is a unital commutative C*-algebra whose identity is the constant function . Let and . The left translate of by is the function (). The right translate by is defined similarly. Clearly for each , and define bounded operators on . A positive linear functional is called a left invariant mean on if (i.e. is a mean) and (i.e. is left invariant) for all and all . Similarly, a right invariant mean on is a a mean satisfying for all and all . For a locally compact group , it is well-known that is amenable if and only if has an invariant mean, a mean on which is both left invariant and right invariant [10].
For two semitopological semigroups and , it is readily seen that if has a left invariant mean and if there is a continuous semigroup homomorphism from onto , then has a left invariant mean. It is also readily seen that if there is a continuous anti-homomorphism from onto , then the existence of a right invariant mean on implies the existence of a left invariant mean on .
Let be a locally compact group, and let . Then, for each we have , where (). Suppose further that is an [FC]- group. Then, as is well-known, is amenable (see [25] or [18]). The above discussion shows that has a left invariant mean, say . Note that , where is the Stone-Cěch compactification of [6, Corollary V.6.4]. Let and . Then
[TABLE]
is a compact subset of and the function is a continuous function on whose support sits in . As explained above, we may regard the left invariant mean as in . Restricting to , we obtain a positive finite Borel (probability) measure space . Note that is of finite Haar measure as a compact subset of . We then may apply Fubini’s Theorem to the function on and define by
[TABLE]
Clearly and . By the left invariance of it is readily seen that ( and ). Whence . Moreover, the Fubini Theorem ([15, Theorem 13.8]) asserts that
[TABLE]
Since is dense in , extends to a bounded linear operator from into , still denoted by . It is also easily seen that when . Therefore : is a Banach space contractive projection. Although is usually not a Banach algebra homomorphism, it is a -bimodule morphism if we view both and as natural -bimodules. But we will not use this feature.
If is a left invariant mean on (), then is a left invariant mean on . Note . This generates a contractive projection from onto . On the other hand, the mapping () defines a Banach algebra isometry : L^{1}(G_{1})\hbox{\widehat{\otimes}}L^{1}(G_{2})\to L^{1}(G_{1}\times G_{2}) which maps into , where denotes the function
[TABLE]
Since is complemented in , the inclusion mappings
[TABLE]
induce a norm preserving Banach algebra embedding:
[TABLE]
We warn here that, in general for closed subspaces of Banach spaces and embeddings : (), : B_{1}\hbox{\widehat{\otimes}}B_{2}\to A_{1}\hbox{\widehat{\otimes}}A_{2} is not necessarily an embedding; it may not be even injective (see [32]). Denote the inclusion mapping by . Then one can easily verify that
[TABLE]
Lemma 2.1**.**
Let and be locally compact [FC]- groups. Then, as Banach algebras,
[TABLE]
Proof.
Consider the following chain:
[TABLE]
From the above discussion the composition of the chain provides a Banach algebra isomorphism (in fact, isometric isomorphism) from ZL^{1}(G_{1})\hbox{\widehat{\otimes}}ZL^{1}(G_{2}) onto . ∎
Here we note that Lemma 2.1 generalizes former known results for compact and discrete cases in [2] and [1].
Now we consider the weighted case. If is a continuous weight on the [FC]- group , then for each there is a constant such that
[TABLE]
Assume that there is an upper bound for all . We then have the following.
Proposition 2.2**.**
Let , be [FC]- groups, and be a weight on satisfying () (), where is a constant. Then, as Banach algebras,
[TABLE]
Proof.
Up to equivalence we may assume and to be continuous. If is a continuous weight on an [FC]- group such that (), we can still consider the mapping on defined by (1). Let . Then we have
[TABLE]
By the Fubini Theorem we obtain
[TABLE]
These are true for all , which is dense in . So extends to a bounded linear mapping from to . Similar to the non-weighted case, we have . So is a continuous projection from onto and . Then we follow the same argument for Lemma 2.1 to get the isomorphic relation (2). ∎
As is well-known, [FC]- groups are amenable [IN] groups [25]. For general amenable [IN] groups we have the following result.
Proposition 2.3**.**
Let and be amenable [IN] groups, and let and be weights on them, respectively. Then is weakly amenable if and only if both and are weakly amenable.
Proof.
Again, we may assume that the weights are continuous.
Since and are commutative, The sufficiency follows from [7, Proposition 2.8.71].
For the converse, we first note that, if is an amenable [SIN] group and is a weight on , then there is a character (namely, a bounded multiplicative linear functional) on from [30, Lemma 1]. Restricting to , is clearly non-trivial (note that has a central bounded approximate identity). Now let be an amenable [IN] group. Then it is well-known that there is a compact normal subgroup of such that [SIN] (see [16, Theorem 1]) and is still amenable. Define a weight on by
[TABLE]
Then there is a standard Banach algebra homomorphism from onto ([26, Theorem 3.7.13]). In fact, is precisely formulated by
[TABLE]
Clearly, maps Z\hbox{L^{1}(G,\omega)} onto . As we have shown, there is a character on which is nontrivial on . Then the composition gives a character on which is nontrivial on Z\hbox{L^{1}(G,\omega)}. Apply this to and assume is weakly amenable. Then the mapping
[TABLE]
generates a Banach algebra homomorphism from onto . Whence is weakly amenable by [7, Proposition 2.8.64]. Similarly, is weakly amenable. ∎
Consider the special case , where is an [FC]- group and is a compact group. Let be a continuous weight on . Define
[TABLE]
where is the unit of . Then is a weight on , and is equivalent to the weight on . Therefore, is a Banach algebra isomorphic to . Now assume satisfies () for some constant . Since is weakly amenable (see [31, Proposition 5.1]), by Propositions 2.2 and 2.3 we see Z\hbox{L^{1}(G,\omega)} is weakly amenable if and only if is weakly amenable. This, in particular, leads us to the following extension of [31, Theorem 3.1], where since is abelian.
Proposition 2.4**.**
Suppose that , is an abelian group and is a compact group. Let be a weight on . Then is weakly amenable if and only if there is no non-trivial continuous group homomorphism such that
[TABLE]
Proof.
One only needs to note that there is a non-trivial continuous group homomorphism such that (3) holds if and only if there is a non-trivial continuous group homomorphism such that
[TABLE]
So the conclusion follows from [31, Theorem 3.1]. ∎
Remark 2.5**.**
According to [14, Theorem 4.3], if is a connected [SIN] group, then for some (abelian) vector group and a compact group . So Proposition 2.4 is valid in particular for this kind of group .
In fact, the necessity part of Proposition 2.4 remains true for general [FC]- groups. To prove this we first consider a general [IN] group.
Lemma 2.6**.**
Let be an [IN] group, be a weight on , and be an open set of with a compact closure and invariant under inner automorphisms of . Suppose that there exists a continuous group homomorphism non-trivial on and such that
[TABLE]
Then there is a nontrivial continuous derivation from into L^{\infty}\bigl{(}G,\frac{1}{\omega}\bigr{)}. Consequently, Z\hbox{L^{1}(G,\omega)} is not weakly amenable.
Proof.
Since is the center of , L^{\infty}\bigl{(}G,\frac{1}{\omega}\bigr{)} is a symmetric Banach -bimodule. We construct a non-trivial continuous derivation : ZL^{1}(G,\omega)\to\hbox{L^{\infty}\bigl{(}G,\frac{1}{\omega}\bigr{)}}. Whence this is done, it follows from the definition of weak amenability for a commutative Banach algebra given in [3] that is not weakly amenable.
We define as follows
[TABLE]
First we note that is non-trivial. To see this we consider the function , where is the characteristic function of and is the conjugate of . Since is a group homomorphism and is invariant under we have that . Moreover,
[TABLE]
This shows that for in a neighbourhood of the identity of because on some open subset of when is near . Hence, is non-trivial. Using the method of [29, Theorem 2.2] one can show that the formula (4) defines a bounded derivation even from the whole into L^{\infty}\bigl{(}G,\frac{1}{\omega}\bigr{)}. So, it also defines a (non-trivial) continuous derivation from into L^{\infty}\bigl{(}G,\frac{1}{\omega}\bigr{)}. ∎
We will need some elementary property of an [FC]- group, which we state as follows.
Lemma 2.7**.**
Let be an [FC]- group. Then for every there exists an open precompact neighbourhood of in that is invariant under inner automorphisms of .
Proof.
It is known that an [FC]- group belongs to [IN]. Let be a precompact open invariant neighbourhood of and let be the conjugacy class of (which is also precompact and invariant). Then satisfies our requirement. ∎
Proposition 2.8**.**
Let be a locally compact [FC]- group and be a weight on . Suppose that there exists a non-trivial continuous group homomorphism such that
[TABLE]
Then is not weakly amenable.
Proof.
Since is non-trivial, there exists such that . Applying Lemma 2.7, we get an open neighbourhood of that is invariant under inner automorphisms and has compact closure. Therefore Lemma 2.6 applies. ∎
We wonder whether the converse of Proposition 2.8 remains true as in the commutative case. We raise it here as an open problem. We note that, in many cases, Z\hbox{L^{1}(G,\omega)} is isomorphic to a weighted commutative hypergroup algebra. So our question links to the general open problem of characterizing weak amenability of (commutative) hypergroup algebras. In particular, it would be of great interest if one can characterize weak amenability of Z^{B}\hbox{L^{1}(G,\omega)} for and being a closed subgroup of with (see [19] for definitions).
3. Central Beurling algebras on [FD]- groups
In this section we consider [FD]- groups and aim to establish some sufficient conditions for to be weakly amenable. We recall first that is an [FD]- group if and only if there exists a compact normal subgroup of such that the quotient is abelian.
The following structural result, which is [24, Lemma 1] for , is crucial in the sequel.
Lemma 3.1**.**
Let be an [FD]- group and a compact normal subgroup of such that is abelian. Let be a weight on satisfying
[TABLE]
for all , and let be the induced weight on defined by
[TABLE]
Then may be written as the closure of the linear span of a family of complemented ideals, each of which is isomorphic to a Beurling algebra of the form for some open normal subgroup of .
We need also the following well-known result.
Lemma 3.2**.**
Let be a commutative Banach algebra and be a family of closed subalgebras of such that . If each is weakly amenable, then so is .
We note that in Lemma 3.1 is a commutative Beurling algebra. So [31, Theorem 3.1] applies for the weak amenability of it. This leads to the following result.
Theorem 3.3**.**
Let be an [FD]- group and be a continuous weight on satisfying
[TABLE]
Then is weakly amenable.
Proof.
First we show that (6) implies that for every . Since , it suffices to prove that
[TABLE]
Fix and let be arbitrary. Because , there exists such that for every . Using the assumption (6) and the inequality , we can find such that
[TABLE]
For any there exist and such that . Using the weight inequality for , we can make the following estimates
[TABLE]
where
[TABLE]
is a constant that does not depend on . It follows that
[TABLE]
Since was arbitrary, we obtain that , as desired.
So, the condition of Lemma 3.1 is satisfied. Then there exists a family of complemented ideals of such that and for each there exists an open subgroup of for which . Let be a non-trivial continuous group homomorphism. Choose so that . Then
[TABLE]
Let be a representative of , i.e. . We note that, for each ,
[TABLE]
In particular, and (). Combining this with condition (6) and (7), we obtain
[TABLE]
According to [31, Theorem 3.1], this implies that is weakly amenable. Then Lemma 3.2 applies. ∎
We now apply Theorem 3.3 to compactly generated [FC]- groups, which are, in fact, [FD]- groups according to [14, Theorem 3.20].
Let be a compactly generated locally compact group. Then there is an open symmetric neighbourhood of the identity in with compact closure and satisfying . Following [24], we consider the length function defined by
[TABLE]
It is readily checked that (), and for every the corresponding polynomial weight () is, indeed, an upper semicontinuous weight on . As addressed in the introduction section, is equivalent to a continuous weight.
Theorem 3.4**.**
Let be a compactly generated non-compact [FC]- group and be the weight on defined as above. Then is weakly amenable if and only if .
Proof.
From the definition of the length function we have and (, ). If , then
[TABLE]
This is still true if is replaced by a continuous equivalent weight. Therefore, is weakly amenable by Theorem 3.3.
To prove the converse we let be a compact subgroup of such that is abelian. The quotient group is clearly still compactly generated. By the structure theorem for compactly generated locally compact abelian groups [15, Theorem II.9.8], is topologically isomorphic to \mathbb{R}^{m}\times\hbox{\mathbb{Z}}^{n}\times F for some integers and and some compact abelian group . Since is not compact, neither is . Then either or is a quotient group of . Thus there is a non-trivial continuous group homomorphism : . Then : is a non-trivial continuous group homomorphism, where : is the quotient map. If , this satisfies the inequality (5) with . In fact, for there is a smallest k\in\hbox{\mathbb{N}} such that . We have and
[TABLE]
where which is finite since is compact. This leads to inequality (5) for (and also for any continuous equivalent to ) since . Hence is not weakly amenable due to Proposition 2.8. ∎
Remark 3.5**.**
Consider again the general [FD]- group . Let be a compact normal subgroup of it such that is commutative. If there is a continuous non-trivial group homomorphism : such that (5) holds, then Z\hbox{L^{1}(G,\omega)} is not weakly amenable from Proposition 2.8. If there is no such , then there is no such for with the weight . Then is weakly amenable due to Theorem [31, Theorem 3.1]. We want to know whether is weakly amenable for any open subgroup of containing . If this is true we will then obtain a characterization for the weak amenability of Beurling algebras on an [FD]- group. We note that is an open subgroup of . However, in general weak amenability of a Beurling algebra does not pass to the Beurling algebra on a subgroup (see section 5 of [29]).
The situation is simple when is isomorphic to or .
Proposition 3.6**.**
Suppose that is a locally compact group and has a compact normal subgroup such that or . Let be a weight on . Then is weakly amenable if and only if there is no non-trivial continuous group homomorphism such that
[TABLE]
Proof.
It suffices to show the sufficiency. If there is no non-trivial continuous group homomorphism for which (8) holds, then, as is known, is weakly amenable. This in turn implies that
[TABLE]
due to [31, Corollary 3.7]. Since for , where
[TABLE]
the last condition leads to
[TABLE]
Then, applying Theorem 3.3, we conclude that is weakly amenable. ∎
The authors are grateful to the referees for their thoughtful comments and suggestions.
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