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.     

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l¹(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ?¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ?¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ?¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.     

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.     

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.     

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.     

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.     

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.     

Approximate amenability of Banach category algebras with application to semigroup algebras

Let C be a small category. Then we consider ℓ ¹(C) as the ℓ ¹ algebra over the morphisms of C, with convolution product and also consider as the ℓ ¹ algebra over the objects of C, with pointwise multiplication. The main purpose of this paper is to show that approximate amenability of ℓ ¹(C) implies of and clearly this implies that C has only finitely many objects. Some applications are given, the main one is the characterization of approximate amenability for ℓ ¹(S), where S is a Brandt semigroup, which corrects a result of Lashkarizadeh Bami and Samea (Semigroup Forum 71:312–322, 2005).     

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.     

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.     

A mixed topology over the space of measures with continuous translations on a locally compact semigroup

In this paper, we deal with the semigroup algebra M _a (S) for a locally compact semigroup S under the topology β ¹(S) which has been recently defined in Maghsoudi, Nasr-Isfahani and Rejali (Semigroup Forum 73:367–376, 2006). We first show that the topology β ¹(S) can be considered a natural mixed topology. Then, using this fact, we obtain new results about the topology β ¹(S). As an application of our results, we show that (M _a (S),β ¹(S)), for a wide class of locally compact semigroups, is a complete semi-topological algebra with the convolution multiplication.     

(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.     

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.     

On pseudo-amenability of commutative semigroup algebras and their second duals

In this paper, we first characterize pseudo-amenability of semigroup algebras \(\ell ^1(S),\) for a certain class of commutative semigroups S,  the so-called archimedean semigroups. We show that for an archimedean semigroup S,  pseudo-amenability, amenability and approximate amenability of \(\ell ^1(S)\) are equivalent. Then for a commutative semigroup S,  we show that pseudo-amenability of \(\ell ^{1}(S)\) implies that S is a Clifford semigroup. Finally, we give some results on pseudo-amenability and approximate amenability of the second dual of a certain class of commutative semigroup algebras \(\ell ^1(S)\).     

Regular Semigroups Each of Whose Least Completely Simple Congruence Classes Has a Greatest Element

A regular semigroup S is termed locally F-regular, if each class of the least completely simple congruence ξ contains a greatest element with respect to the natural partial order. It is shown that each locally F-regular semigroup S admits an embedding into a semidirect product of a band by S/ξ. Further, if ξ satisfies the additional property that for each s ∈ S and each inverse (sξ)′ of sξ in S/ξ the set (sξ)′ ∩ V(s) is not empty, we represent S both as a Rees matrix semigroup over an F-regular semigroup as well as a certain subsemigroup of a restricted semidirect product of a band by S/ξ.     

Zero-Semidistributive Inverse Semigroups

An inverse semigroup S is said to be 0-semidistributive if its lattice ?F (S) of full inverse subsemigroups is 0-semidistributive. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a simple inverse semigroup S is 0-semidistributive if and only if (1) S is E-unitary, (2) S is aperiodic, (3) for any a,b ∈ S/σ with ab ≠ 1, there exist nonzero integers n and m such that (ab) ^m  = a ⁿ or (ab) ^m  = b ⁿ , where σ is the minimum group congruence on S.     

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.     

A semilattice structure for Arens products

Let (A,?) be a Banach algebra. Then for n∈?, A ⁽²ⁿ⁾ has 2 ⁿ Arens products. In this paper we study the relations between the Arens products on A ⁽²ⁿ⁾. Moreover, if P _n (A) denotes the set of all Arens products on A ⁽²ⁿ⁾, for n∈?, we show that $P(A)=\bigcup_{n=1}^{\infty} P_{n}(A)$ is a ∧-semilattice. Also, we study P(A) as an infinite commutative semigroup and P(A)?{?} as a free semigroup generated by two elements. Then we investigate amenability and weak amenability for their semigroup Banach algebras.