The Product Between Three Banach Algebras

Abstract

. Let A , B , and C be Banach algebras, (cid:11) 2 Hom ( A ; B ) and (cid:12) 2 Hom ( C ; B ) , and ∥ (cid:11) ∥(cid:20) 1 , ∥ (cid:12) ∥(cid:20) 1 . IN this paper we define the Banach algebra A(cid:2) (cid:11) B(cid:2) (cid:12) C by new product on A(cid:2)B(cid:2)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.