Some notions of amenability for certain products of Banach algebras

Abstract

For two Banach algebras A and B, an interesting product A × θ B, called the θ-Lau product, was recently introduced and studied for some nonzero characters θ on B. Here, we characterize some notions of amenability as approximate amenability, essential amenability, n-weak amenability and cyclic amenability between A and B and their θ-Lau product. Introduction. LetA and B be two Banach algebras and θ ∈ σ(B), the spectrum of B of all nonzero characters on B. Then the θ-Lau product of A and B, denoted by A × θ B, is defined as the space A × B equipped with the multiplication (a, b)(a , b ) = (aa + θ(b)a + θ(b )a, bb ), and the norm (a, b) = a + b , for all a, a ∈ A and b, b ∈ B. The θ-Lau product A × θ B is a Banach algebra.This product was first introduced by Lau [L1] for Lau algebras; recall that a Lau algebra is a Banach algebra which is the predual of a von Neumann algebra for which the identity of the dual is a multiplicative linear functional. The study of this large class of Banach algebras originated with a paper published in 1983 by Lau [L1] in which he referred to them as "F-algebras"; see also Lau [L2]. Later on, in his useful monograph Pier [Pi] introduced the name "Lau algebra". Examples of Lau algebras include the group algebra and the measure algebra of a locally compact group or hypergroup (see Lau [L1]), and also the Fourier algebra and

Some notes on amenability and weak amenability of Lau product of Banach algebras defined by a Banach algebra morphism

“…Then by [3, Proposition 2.8.63 (i)], D| A = 0. Similar to [5], consider X as a closed linear subspace of A (2n) × T (2n) B (2n) spanned by…”

“…for all (a, b), (a , b ) ∈ A × θ B. Amenability and weak forms of amenability of A × θ B studied in [5,9]. Let T : B → A be an algebra homomorphism, and A be a commutative Banach algebra.…”

Pseudo-amenability and pseudo-contractibility for certain products of Banach algebras

“…We recently have shown in [6] that if also A × θ B is approximately amenable, then A and B are approximately amenable. We have proved that the converse of this statement in general is not true; but, A × θ B is approximately amenable if A is amenable and B is approximately amenable.…”

“…Recently, several notions of amenability were first studied on A × θ B by Monfared [16] and then pursued by Vishki and Khoddami [5] and the authors [6].…”

Some Notes on Semidirect Products of Banach Algebras

“…11. If A is a weakly amenable commutative Banach algebra, then for any commutative Banach A-bimodule X we have H 1 (A, X ) = (0).…”