Tensor products over Banach algebras

Abstract

0. Introduction. In [4], [5], [6] the structures A1 0 ) A2, where A1 and A2 are Banach algebras, are discussed. Actually, a proper parallel to the algebraic situation is: three commutative Banach algebras A, B, C, where A and B are C-bimodules (in the sense described below), and some Banach-algebraic version of A 03cB. In the first part of the following, we shall give a general discussion of A ?cB, a natural Banach-algebraic version of the (algebraic) tensor product of A and B over C. Thereafter, we shall discuss a special case in which A, B, C are group algebras of locally compact abelian groups. Finally we shall handle the even more special problem in which A and B are group algebras of locally compact abelian groups and C is a group algebra of a compact abelian group. In the last two parts, particularly the last, a connection will be established between the theory of tensor products and the theory of group extensions. In this connection, the author thanks L. Auslander for indicating the point in question. The author is also indebted to Dr. B. Natzitz for pointing out errors and the need for clarifying several items of an earlier draft of this paper.