On certain products of Banach algebras with applications to harmonic analysis
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.
Monfared, Mehdi S.. (2007). On certain products of Banach algebras with applications to harmonic analysis. Studia Mathematica, 178 (3).