Approximate amenability of certain inverse semigroup algebras

In this article, the approximate amenability of semigroup algebra ?¹(S) is investigated, where (S) is a uniformly locally finite inverse semigroup. Indeed, we show that for a uniformly locally finite inverse semigroup (S), the notions of amenability, approximate amenability and bounded approximate amenability of ?¹(S) are equivalent. We use this to give a direct proof of the approximate amenability of ?¹(S) for a Brandt semigroup (S). Moreover, we characterize the approximate amenability of ?¹(S), where S is a uniformly locally finite band semigroup.     

Pseudo-amenability of certain semigroup algebras

In this paper, we investigate the pseudo-amenability of semigroup algebra ? ¹(S), where S is an inverse semigroup with uniformly locally finite idempotent set. In particular, we show that for a Brandt semigroup \(S={\mathcal{M}}^{0}(G,I)\), the pseudo-amenability of ? ¹(S) is equivalent to the amenability of G.     

Pseudo-contractibility and pseudo-amenability of semigroup algebras

In this paper, we characterize pseudo-contractibility of ℓ ¹(S), where S is a uniformly locally finite inverse semigroup. As a consequence, we show that for a Brandt semigroup S=M⁰(G,I),{S={\mathcal{M}}^{0}(G,I),} the semigroup algebra ℓ ¹(S) is pseudo-contractible if and only if G and I are finite. Moreover, we study the notions of pseudo-amenability and pseudo-contractibility of a semigroup algebra ℓ ¹(S) in terms of the amenability of S.     

Amenability constants for semilattice algebras

For any finite commutative idempotent semigroup S, a semilattice, we show how to compute the amenability constant of its semigroup algebra ℓ ¹(S). This amenability constant is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We also give example of a commutative Clifford semigroups G _n whose semigroup algebras ℓ ¹(G _n ) admit amenability constants of the form 41+4(n−1)/n. We also show there is no commutative semigroup whose semigroup algebra has an amenability constant between 5 and 9. N. Spronk’s research was supported by NSERC Grant 312515-05.     

Module character amenability of Banach algebras

In this paper we introduce the notion of module character amenable Banach algebras and show that they possess module character virtual (approximate) diagonals. As a basic example, we show that for an inverse semigroup S with the set of idempotents E, the semigroup algebra ? ¹(S) is module character amenable as an ? ¹(E)-module if only if S is amenable.     

Irregular abelian semigroups with weakly amenable semigroup algebra

In this paper we give counterexamples for the open problem, posed by Blackmore (Semigroup Forum 55:359–377, 1987) of whether weak amenability of a semigroup algebra ℓ ¹(S) implies complete regularity of the semigroup S. We present a neat set of conditions on a commutative semigroup (involving concepts well known to those working with semigroups, e.g. the counterexamples are nil and 0-cancellative) which ensure that S is irregular (in fact, has no nontrivial regular subsemigroup), but ℓ ¹(S) is weakly amenable. Examples are then given.     

Weighted Rees matrix semigroup algebras and their applications

In this work, we will describe the weighted semigroup algebra ? ¹(S, ω), where S is a regular Rees matrix semigroup and ω ≥  1. Then as an application, we investigate the amenability of the semigroup algebra ? ¹(S, ω) and its second dual for an arbitrary semigroup S.     

Super module amenability of inverse semigroup algebras

In this paper we compare the notions of super amenability and super module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. We find conditions for the two notions to be equivalent. In particular, we study arbitrary module actions of l ¹(E _S ) on l ¹(S) for an inverse semigroup S with the set of idempotents E _S and show that under certain conditions, l ¹(S) is super module amenable if and only if S is finite. We also study the super module amenability of l ¹(S)^?? and module biprojectivity of l ¹(S), for arbitrary actions.     

(Super) module amenability, module topological centre and semigroup algebras

For a Banach algebra $\mathcal{A}For a Banach algebra A\mathcal{A} which is also an \mathfrakA\mathfrak{A}-bimodule, we study relations between module amenability of closed subalgebras of A"\mathcal{A}', which contains A\mathcal{A}, and module Arens regularity of A\mathcal{A} and the role of the module topological centre in module amenability of A"\mathcal{A}'. Then we apply these results to A=l¹(S)\mathcal{A}=l^{1}(S) and \mathfrakA=l¹(E)\mathfrak{A}=l^{1}(E) for an inverse semigroup S with subsemigroup E of idempotents. We also show that l ¹(S) is module amenable (equivalently, S is amenable) if and only if an appropriate group homomorphic image of S, the discrete group \fracS ? \frac{S}{\approx}, is amenable. Moreover, we define super module amenability and show that l ¹(S) is super module amenable if and only if \fracS ? \frac{S}{\approx} is finite.     

Amenability of the Second Dual of Hypergroup Algebras

The amenability of the Banach algebra L ¹(G), the measure algebra M(G) and their second duals of a locally compact group have been considered by a number of authors. During these investigations it has been shown that L ¹(G)^** is amenable if and only if G is finite. If LUC (G)^*, the dual of the space of left uniformly continuous functions on G, is amenable, then G is compact and M(G) is amenable. Finally, if M(G)^** is amenable, then G is finite. The aim of this paper is to generalize all of the above results to the locally compact hypergroups.     

Amenability and weak amenability of the Fourier algebra

Let G be a locally compact group. We show that its Fourier algebra A(G) is amenable if and only if G has an abelian subgroup of finite index, and that its Fourier–Stieltjes algebra B(G) is amenable if and only if G has a compact, abelian subgroup of finite index. We then show that A(G) is weakly amenable if the component of the identity of G is abelian, and we prove some partial results towards the converse.Research supported by NSERC under grant no. 90749-00.Research supported by NSERC under grant no. 227043-00.     

Homological properties of modules over semigroup algebras

Let S be a semigroup. In this paper we investigate the injectivity of ?¹(S) as a Banach right module over ?¹(S). For weakly cancellative S this is the same as studying the flatness of the predual left module c₀(S). For such semigroups S, we also investigate the projectivity of c₀(S). We prove that for many semigroups S for which the Banach algebra ?¹(S) is non-amenable, the ?¹(S)-module ?¹(S) is not injective. The main result about the projectivity of c₀(S) states that for a weakly cancellative inverse semigroup S, c₀(S) is projective if and only if S is finite.     

Semiprimitivity of Orthodox Semigroup Algebras

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R₀[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.     

Preduals of semigroup algebras

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C ₀(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak^*-continuous. Given a discrete semigroup S, the convolution algebra ℓ ¹(S) also carries a coproduct. In this paper we examine preduals for ℓ ¹(S) making both the product and the coproduct weak^*-continuous. Under certain conditions on S, we show that ℓ ¹(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ ¹(S) when S is either ℤ₊×ℤ or (ℕ,⋅).     

A class of unary semigroups admitting a Rees matrix representation

In Billhardt et al. (Semigroup Forum 79:101–118, 2009) the authors introduced the notion of an associate inverse subsemigroup of a regular semigroup, extending the concept of an associate subgroup of a regular semigroup, first presented in Blyth et al. (Glasgow Math. J. 36:163–171, 1994). The main result of the present paper, Theorem 2.15, establishes that a regular semigroup S with an associate inverse subsemigroup S ^? satisfies three simple identities if and only if it is isomorphic to a generalised Rees matrix semigroup M(T;A,B;P), where T is a Clifford semigroup, A and B are bands, with common associate inverse subsemigroup E(T) satisfying the referred identities, and P is a sandwich matrix satisfying some natural conditions. If T is a group and A, B are left and right zero semigroups, respectively, then the structure described provides a usual Rees matrix semigroup with normalised sandwich matrix, thus generalising the Rees matrix representation for completely simple semigroups.     

A note on the approximate amenability of semigroup algebras

It is known that the bicyclic semigroup S ₁ is an amenable inverse semigroup. In this note we show that the convolution semigroup algebra ℓ ¹(S ₁) is not approximately amenable.     

N(2,2,0)代数的E-反演半群

本文研究了N(2,2,0)代数(S,*,△,0)的E-反演半群.利用N(2,2,0)代数的幂等元,弱逆元,中间单位元的性质和同宇关系,得到了N(2,2,0)代数的半群(S,*)构成E-反演半群的条件及元素α的右伴随非零零因子唯一,且为α的弱逆元等结论,这些结果进一步刻画了N(2,2,0)代数的结构.     

Morita equivalence of inverse semigroups

We describe how to construct all inverse semigroups Morita equivalent to a given inverse semigroup S. This is done by taking the maximum inverse images of the regular Rees matrix semigroups over S where the sandwich matrix satisfies what we call the McAlister conditions.     

Biflatness of semigroup algebras

We shall study the biflatness of the convolution algebra ℓ ¹(S) for a semigroup S. We show that for any semigroup S such that ℓ ¹(S) is biflat the canonical partial ordering on the idempotents must be uniformly locally finite. We use this to characterize the biflatness of ℓ ¹(S) for an inverse semigroup S.     

On Incidence Algebras of Combinatorial Inverse Monoids

We study the incidence algebra of the reduced standard division category of a combinatorial bisimple inverse monoid [with (E(S), ≤) locally finite], and we describe semigroups of poset type (i.e., a combinatorial inverse semigroup for which the corresponding Möbius category is a poset) as being combinatorial strict inverse semigroups. Up to isomorphism, the only Möbius-division categories are the reduced standard division categories of combinatorial inverse monoids.