Tensor Products of Banach Algebras

Abstract

This paper is concerned with a generalization of some recent theorems of Hausner (1) and Johnson (4; 5). Their result can be summarized as follows: Let G be a locally compact abelian group, A a commutative Banach algebra, B 1 = B l (G,A) the (commutative Banach) algebra of A-valued, Bochner integrable junctions on G , 3m 1 the maximal ideal space of A , m 2 the maximal ideal space of L 1 (G) [the [commutative Banach] algebra of complex-valued, Haar integrable functions on G , m 3 the maximal ideal space of B 1 . Then m 3 and the Cartesian product m 1 X m 2 are homeomorphic when the spaces m i , i = 1, 2, 3, are given their weak* topologies. Furthermore, the association between m 3 and m 1 X m 2 is such as to permit a description of any epimorphism E 3 : B 1 → B 1 /m 3 in terms of related epimorphisms E 1 : A → A/M 1 and E 2 :L 1 (G) → L l (G)/M 2 , where M 1 is in m i i = 1, 2, 3.