Pseudo-amenability and pseudo-contractibility of restricted semigroup algebra

Abstract

In this article the pseudo-amenability and pseudo-contractibility of restricted semigroup algebra $l_r^1(S)$ and semigroup algebra, l1(Sr) on restricted semigroup, Sr are investigated for different classes of inverse semi-groups such as Brandt semigroup, and Clifford semigroup. We particularly show the equivalence between pseudo-amenability and character amenability of restricted semigroup algebra on a Clifford semigroup and semigroup algebra on a restricted semigroup. Moreover, we show that when S = M0(G, I)is a Brandt semigroup, pseudo-amenability of l1(Sr) is equivalent to its pseudo-contractibility.

Counter examples for pseudo-amenability of some semigroup algebras

“…In light of the above discussion, [11,Propositon 4.10] can not be valid. In other words, when 1 (S) is pseudo-amenable, then by [11,Proposition 3.5], S is amenable (not finite) and in general case, amenability of S does not imply amenability or character amenability of 1 (S). This happen when, S is an abelian or left (right) cancellative unital semigroup.…”

“…Since N ∧ and N ∨ are inverse semigroups, by Theorem 3, N ∧ and N ∨ must be finite which is impossible. It is shown in the proof of [11,Proposition 3.5] that every amenable inverse semigroup is finite which is not true. For instance, every bicyclic inverse semigroup is amenable but not finite and hence one side of the mentioned proposition is not true.…”

“…Recently [11] come to our attention. The paper is devoted to study the pseudoamenability of some semigroup algebras related to inverse and restricted semigroups.…”