On certain products of Banach algebras with applications to harmonic analysis

Abstract

Abstract. Given Banach algebras A and B with spectrum σ(B) = ∅, and given θ ∈ σ(B), we define a product A × θ B, which is a strongly splitting Banach algebra extension of B by A. We obtain characterizations of bounded approximate identities, spectrum, topological center, minimal idempotents, and study the ideal structure of these products. By assuming B to be a Banach algebra in C 0 (X) whose spectrum can be identified with X, we apply our results to harmonic analysis, and study the question of spectral synthesis, and primary ideals.

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

“…Monfared [16] showed that A × θ B is amenable if and only if both A and B are amenable. Moreover, he proved that weak amenability of A and B implies weak amenability of A × θ B, but in general case the converse of this statement is not true.…”

“…The θ-Lau product A × θ B of two Banach algebras A and B, for some nonzero characters θ on B, was introduced and investigated by Monfared [16]. 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].…”

“…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].…”

Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

“…The first important paper related to this construction is Lau's paper [16] which he defined a new algebra product on the Cartesian product space A×B for the case where B is the predual of a van Neumann algebra M such that the identity of M is a multiplicative linear functional on B. Later on, Monfared [18] extended the Lau product to arbitrary Banach algebras. This construction has many applications in different contexts, see for examples [9,15,19,20].…”

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

“…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.…”