Some Notes on Semidirect Products of Banach Algebras

Abstract

Let ${\mathcal {A}}$ and ${\mathcal {U}}$ be Banach algebras and ${\mathcal {A}} < imes {\mathcal {U}}$ be the semidirect product of Banach algebras ${\mathcal {A}}$ and ${\mathcal {U}}$ where ${\mathcal {A}} < imes {\mathcal {U}}={\mathcal {A}}\oplus {\mathcal {U}}$ as direct sum of Banach spaces and ${\mathcal {A}}$ is closed subalgebra and ${\mathcal {U}}$ is a closed ideal of ${\mathcal {A}} < imes {\mathcal {U}}$. The semidirect product ${\mathcal {A}} < imes {\mathcal {U}}$ is a block of Banach algebras that covers the important classes of Banach algebras such as Lau product Banach algebras and triangular Banach algebras. In this paper we consider this question that under what conditions the semidirect product Banach algebra ${\mathcal {A}} < imes {\mathcal {U}}$ is a Lau product Banach algebra or a triangular Banach algebra? We find these conditions and answer this question which form the main results of our paper.