Approximate biprojectivity of certain semigroup algebras

Abstract

In this paper, we investigate the notion of approximate biprojectivity for semigroup algebras and for some Banach algebras related to semigroup algebras. We show that ℓ 1 (S) is approximately biprojective if and only if ℓ 1 (S) is biprojective, provided that S is a uniformly locally finite inverse semigroup. Also for a Clifford semigroup S, we show that approximate biprojectivity ℓ 1 (S) * * gives pseudo amenability of ℓ 1 (S). We give a class of Banach algebras related to semigroup algebras which is not approximately biprojective.

Approximate Biprojectivity of ℓ1‐Munn Banach Algebras

“…Tus, ψ φ ∈ Δ(UP(I, P, A)). Since A is unital by Lemma 2, UP(I, P, A) has a right approximate identity and by [15] Teorem 3.9, we can conclude that UP(I, P, A) is approximately right ψ φ -amenable. Defne a closed ideal in UP(I, P, A) by…”

Generalized notion of amenability for a class of matrix algebras

“…G is compact and also we show that L 1 (G, w) is approximately biprojective if and only if G is compact, provided that w ≥ 1 is a continuous weight function, see [21] and [23].…”

“…Define ψ ∈ ∆(U P (I, A)) by ψ(a) = φ(a in,in ) for every a = (a i,j ) ∈ U P (I, A). By [23,Theorem 3.9] approximate biprojectivity of U P (I, A) implies that U P (I, A) is left ψ-contractible, then the rest is similar to the proof of Theorem 2.1.…”

On Some Homological Properties of Hypergroup Algebras with Relation to Their Character Spaces

“…Indeed, a Banach algebra A is called approximately biprojective if there exists a net (ρ α ) of continuous A-bimodule morphism from A into A ∧ ⊗ A such that π A °ρα (a) ⟶ a for every a ∈ A, where π A : A ∧ ⊗ A ⟶ A is the diagonal operator defined by π A (a ⊗ b) � ab. For recent works about this concept, refer to [8]. roughout the paper, Δ(A) stands for the set of all nonzero multiplicative linear functionals on A. Kaniuth et al [9] introduced the notion of left φ-amenable Banach algebras (φ ∈ Δ(A)) as a generalization of the notion of amenable Banach algebras introduced by Johnson in [10].…”