The Product Between Three Banach Algebras

Abstract

Let A, B, and C be Banach algebras, α ∈ Hom(A, B) and β ∈ Hom(C, B), and ∥ α ∥≤ 1, ∥β ∥≤ 1. IN this paper we define the Banach algebra A×α B× β C by new product on A×B×C which is a strongly splitting extension of C by B. Then we show that these products from a large class of Banach algebras which contains all module extensions and triangular Banach algebras. Finally we consider spectrum, Arens regularity, amenability and weak amenability of these products.