Spectrum, homomorphisms and multipliers of Lau product of Banach algebras

Abstract

Given Banach algebras $A$, $B$ and a continuous homomorphism $\theta:B\longrightarrow A$ with $\Vert\theta\Vert \leq1$, we obtain characterization of spectrum, homomorphisms and multipliers of $A\times_{\theta}B$, which is a strongly splitting Banach algebra extension of $B$ by $A$. Also we characterize the semisimplicity of these algebras.