On certain products of Banach algebras with applications to harmonic analysis

Document Type

Article

Publication Date

2007

Publication Title

Studia Mathematica

Volume

178

Issue

3

Abstract

Given Banach algebras A and B with spectrum sigma(B) not equal theta, and given theta is an element of or sigma(B), we define a product A x (theta) 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.

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