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.