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Abstract

Given Banach algebras $A$ and $B$ with spectrum $\sigma (B)\ne \emptyset$, and given $\theta \in \sigma (B)$, we define a product $A\mathbin{\times_{\theta}} B$, 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 $\mathcal{C}_0(X)$ whose spectrum can be identified with $X$, we apply our results to harmonic analysis, and study the question of spectral synthesis, and primary ideals.