Various notions of module amenability on weighted semigroup algebras

1 Introduction and preliminaries

The story of amenable Banach algebras was commenced by Johnson in [1], where he linked the amenability of Banach algebras and groups. One of the fundamental results was that the group algebra L 1 ( G ) is an amenable Banach algebra if and only if G is an amenable locally compact group. The notion of approximately (contractible) amenable Banach algebras was initiated and studied by Ghahramani and Loy in [2] for the first time. They characterized the structure of approximately (contractible) amenable Banach algebras through several ways and showed that the group algebra L 1 ( G ) of a locally compact group G is approximately (contractible) amenable if and only if G is amenable. The mentioned Johnson’s amenability and approximate amenability theorems fail for semigroups. The concepts of module amenability [3] and module approximate amenability [4] were introduced to resolve those gaps for semigroups. Here, we indicate some definitions and notations for the module version.

Let A and A be Banach algebras such that A is a Banach A -bimodule with compatible actions, that is

for all a , b ∈ A , α ∈ A . When A is a A -module, it does not necessarily satisfy the compatible actions (1) in general.

Let X be a Banach A -bimodule and a Banach A -bimodule with compatible actions, that is,

for all a ∈ A , α ∈ A , x ∈ X and similarly for the right and two-sided actions. Then, we say that X is a Banach A - A -module. If α ⋅ x = x ⋅ α for all α ∈ A and x ∈ X , then X is called a commutative A - A -module. Note that when A acts on itself by algebra multiplication, it is not in general a Banach A - A -module. Indeed, if A is a commutative A -module and acts on itself by multiplication from both sides, then it is also a Banach A - A -module.

Let A and A be as above and X be a Banach A - A -module. A ( A -)module derivation is a bounded A -module map D : A ⟶ X satisfying

Note that D is not assumed to be C -linear and that D : A ⟶ X is bounded if there exists M > 0 such that ‖ D ( a ) ‖ ≤ M ‖ a ‖ , for each a ∈ A [3]. However, the boundedness of D still implies its continuity, as it preserves subtraction. When X is commutative, each x ∈ X defines a module derivation D x ( a ) = a ⋅ x − x ⋅ a for all a ∈ A . These are called inner A -module derivations. A Banach algebra A is called module amenable (as an A -module) if for any commutative Banach A - A -module X , each A -module derivation D : A ⟶ X ∗ is inner. This notion was initiated by Amini [3] for a class of Banach algebras, which could be considered as a generalization of the Johnson’s results when the C -module structure is replaced by a Banach algebra module structure [1].

An inverse semigroup is a semigroup S so that, for each s ∈ S , there exists a unique element s ∗ ∈ S such that s s ∗ s = s and s ∗ s s ∗ = s ∗ . The element s ∗ is termed the inverse of s . The set E ( S ) (or briefly, E ) of idempotents of S is a commutative subsemigroup; it is ordered by e ≤ f if and only if e f = e . With this ordering, E ( S ) is a meet semilattice with the meet given by the product; see [5, Theorem 5.1.1]. We recall that a semigroup S is a semilattice if S is commutative and E = S . The order on E extends to S as the so-called natural partial order by putting s ≤ t if s = e t for some idempotent e (or equivalently s = t f for some idempotent f ). This is equivalent to s = t s ∗ s or s = s s ∗ t . If e ∈ E , then the set G e = { s ∈ S ∣ s s ∗ = e = s ∗ s } is a group, called the maximal subgroup of S at e .

For an inverse semigroup S with the set of idempotents E , the semigroup algebra l 1 ( S ) is module (approximately) amenable, as a Banach module over l 1 ( E ) , if and only if S is amenable [3,4].

Motivated by ϕ -amenability and character amenability, which were studied in [6] and [7], the Bodaghi and Amini [8] introduced the concept of module ( ϕ , φ ) -amenability for Banach algebras and investigated a module character amenable Banach algebra. They proved that l 1 ( S ) is module character amenable if and only if the inverse semigroup S is amenable; see [9] for some generalized notions of module character amenability.

The Beurling (weighted) algebra L 1 ( G , ω ) of a locally compact group G was studied extensively ( ω is a weight on G ); for instance [10]. It was shown in [1, Theorem 3.2] that the algebra l 1 ( G , ω ) is amenable if and only if there exists a positive left invariant mean on l ∞ ( G , ω − 1 ) . Moreover, Ghahramani et al. [11] showed that for a locally compact group G , if L 1 ( G , ω ) is approximately amenable ( ω is bounded away from 0), then G is amenable. In addition, Mewomo and Maepa [12] proved that for a locally compact group G , L 1 ( G , Ω ) is character amenable if and only if L 1 ( G , ω ) is amenable, where Ω ( g ) ≔ ω ( g ) ω ( g − 1 ) for all g ∈ G . Furthermore, Mewomo [13] showed that the amenability of L 1 ( G , ω ) and its character amenability are equivalent when the weight ω is symmetric.

Asgari et al. in [14] studied the module amenability of the weighted semigroup algebras and showed that for an inverse semigroup S equipped to the corresponding weight ω with the set of idempotents E , when l 1 ( E ) acts on l 1 ( S , ω ) trivially from left and by multiplication from right, the weighted semigroup algebra l 1 ( S , ω ) is l 1 ( E ) -module amenable if and only if S is amenable and sup { Ω ( s ) : s ∈ S } < ∞ , where Ω ( s ) = ω ( s ) ω ( s ∗ ) for all s ∈ S .

In this paper, we give some basic properties of module derivations and module amenable Banach algebras. We also present some characterization of module approximate amenability (contractibility) and module character amenability for the weighted inverse semigroup algebra l 1 ( S , ω ) . As a typical example, we show that for every Brandt inverse semigroup S = ℳ ( G , I ) , l 1 ( S , ω ) is module character amenable but not amenable for any weight ω .

2 Main results

A left Banach A -module X is called left essential if the linear span of A ⋅ X = { a ⋅ x : a ∈ A , x ∈ X } is dense in X . Right essential A -modules and two-sided essential A -bimodules are defined similarly. It is shown in [4] that if A is a left (right) essential A -module, then every A -module derivation is also C -linear. There are plenty of known examples of non linear, additive derivations (see for instance [15]), and some of these are also module derivations (at least for the trivial action). On the other hand, when A is unital (or even has a bounded approximate identity) then each module derivation is automatically linear [3, Proposition 2.1] and so every amenable Banach algebra is module amenable. Note that there are some module amenable Banach algebras which are not amenable and are available in [3,16] and other sources. In the next result, we weaken the condition having bounded approximate identity to that A 2 is dense in A , where A 2 = { a b , a , b ∈ A } .

Consider the projective tensor product A ⊗ ˆ A . It is well known that A ⊗ ˆ A is a Banach algebra with respect to the canonical multiplication map defined by

and extended by linearity and continuity. Then, A ⊗ ˆ A is a Banach A - A -module with canonical actions (2). Let I A be the ideal of the projective tensor product A ⊗ ˆ A generated by elements of the form

The product map on A extends to a map π : A ⊗ ˆ A ⟶ A , determined by π ( a ⊗ b ) = a b , for all a , b ∈ A . Let J A be the closed ideal of A generated by elements of the form

Then, A / J A is a Banach A - A -module with the compatible actions when A acts on A / J A canonically. We shall denote I A and J A by I and J , respectively, when no confusion can arise.

Throughout this paper, S is a discrete semigroup. A function ω from a semigroup S to ( 0 , ∞ ) is called a weight function if ω ( s t ) ≤ ω ( s ) ω ( t ) for all s , t ∈ S . Let us consider the spaces

where δ s is the characteristic function of the singleton { s } . The dual space of l 1 ( S , ω ) under the canonical pairing is

For a discrete group G , we have the same definition for the weight function with further condition that ω ( e ) = 1 , where e is the identity element of G . A weight ω on G is said to be symmetric if ω ( g ) = ω ( g − 1 ) for all g ∈ G .

Let ω be a weight on S . Then, for each e ∈ E , we have ω ( e ) = ω ( e 2 ) ≤ ω ( e ) 2 and so ω ( e ) ≥ 1 . Here and subsequently, we assume that ω is a weight on the discrete inverse semigroup S such that ω ( e ) = 1 for all e ∈ E unless otherwise stated explicitly. Therefore, the Banach algebras l 1 ( E , ω ) and l 1 ( E ) coincide.

Let S be a inverse semigroup with the set of idempotents E . We recall that the subsemigroup E of S is a semilattice, and so l 1 ( E ) could be regarded as a commutative subalgebra of l 1 ( S , ω ) . Thus, l 1 ( S , ω ) is a Banach algebra and a Banach l 1 ( E ) -module with compatible actions; refer to [3]. We consider the actions of l 1 ( E ) on l 1 ( S , ω ) as follows:

In other words, for f = ∑ e ∈ E f ( e ) δ e ∈ l 1 ( E ) and g ∈ l 1 ( S , ω ) , the left action in (4) can be extended as f ⋅ g = ∑ e ∈ E f ( e ) g . In this case, the ideal J = J l 1 ( S , ω ) is the closed linear span of { δ set − δ s t : s , t ∈ S , e ∈ E } .

Recall from [16] that a Banach algebra A acts trivially on A from left [resp., right] if for each α ∈ A and a ∈ A , α ⋅ a = f ( α ) a [resp., a ⋅ α = f ( α ) a ], where f is a fixed and continuous linear functional on A .

Let ϕ be the augmentation character on l 1 ( E ) , that is, ϕ ( δ e ) = 1 for all e ∈ E . For each f = ∑ e ∈ E f ( e ) δ e ∈ l 1 ( E ) and g = ∑ s ∈ S g ( s ) δ s ∈ l 1 ( S , ω ) , similar to [4], one can show that f ⋅ g = ϕ ( f ) g . Therefore, the action from left in (4) is trivial. We consider an equivalence relation on S such that s ≈ t if and only if δ s − δ t ∈ J for s , t ∈ S . It is shown in [17] that the quotient S / ≈ is a discrete group (see also [16]). Indeed, S / ≈ is homomorphic to the maximal group homomorphic image G S of S [18,19]. Moreover, S is amenable if and only if G S = S / ≈ is amenable [18,20]. Suppose that ω is a weight on inverse semigroup S such that inf s ∈ S ω ( s ) > 0 . It is shown in [14] that the function ω ¯ : G S ⟶ ( 0 , ∞ ) defined by ω ¯ ( [ s ] ) ≔ inf { ω ( t ) ∣ t ≈ s } is a weight on G S , where [ s ] is the equivalence class of s .

From now on, for every inverse semigroup S by a weight on G S , we mean that it is ω ¯ , which is induced by the weight ω on S as mentioned earlier. We recall from [14, Proposition 2.2] that l 1 ( S , ω ) / J ≅ l 1 ( G S , ω ¯ ) . We use the last equivalency in the rest of paper. In light of the aforementioned discussions, we consider the following weighted group algebra.

Here, l 1 ( G S , ω ¯ ) is a commutative l 1 ( E ) -bimodule with the following actions:

For an inverse semigroup S , we say a weight is symmetric if ω ( s ) = ω ( s ∗ ) for all s ∈ S . It is shown in [14] that there are some weights on inverse semigroups for which their restrictions to idempotents are constant function 1. Here, we show that some of them can be symmetric.

3 Conclusion

In this paper, we have presented new results for studying the module approximate amenability (contractibility) and module character amenability of the weighted semigroup algebra l 1 ( S , ω ) , where S is an inverse semigroup with the set of idempotents E . It will be of interest to mathematicians in Banach algebras and harmonic analysis.