The Product Between Three Banach Algebras

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.