Tensor products of group algebras

Abstract

Let C be a commutative Banach algebra. A commutative Banach algebra A is a Banach C-algebra if A is a Banach C-module and c (aa') = (c * a)a for all c e C, a, a c A. If A , *... , A are commutative Banach Calgebras, then the C-tensor product A1 C C * CAn--D is defined and is a commutative Banach C-algebra. The maximal ideal space 'DiD of D is identified with a closed subset of WA I X ... X 9)An in a natural fashion, yielding a generalization of the Gelbaum-Tomiyama characterization of the maximal ideal space of AI 8y ... (9yAn. If C = L (K) and A.= -L1(G ),for LCA groups K and G., i-=1, **,n, then the L (K)-tensor product D of L (G1), * L(Gn) is uniquely written in the form D = N D D,where N and D are closed ideals in D, L1(K) * N = 101, and el ~~~e D is the essential part of D, i.e D L'(K) * D. Moreover, if De I jO, then De is isometrically Ll(K)-isomorphic to L1(G 1 ?K .. 6KGn), where G1, Gn is a K-tensor product of G1, ... , G with respect to naturally induced actions of K on G1, ... I G . The above theorems are a significant generalization of the work of Gelbaum and Natzitz in characterizing tensor products of group algebras, since here the algebra actions are arbitrary. The Cohen theory of homomorphisms of group algebras is required to characterize the algebra actions between group algebras. Finally, the space of multipliers HomLl(K)(L1(G)2 L2' (H)) is characterized for all instances of algebra actions of L 1(K) on L 1(G) and L 1(H), generalizing the known result when K = G = H and the module action is given by convolution. If A1 and A2 are Banach modules over a Banach algebra C, then Rieffel [18] has defined and systematically studied the C-tensor product, A1 Xc A2, of Al and A2. If Al and A2, or more generally A1, . . ., An, are commutative Banach C-algebras for a commutative Banach algebra C, then the C-tensor product of A 1, . . . An, A1 ?c * * c A,n is naturally a commutative Banach Calgebra. In this paper we study this tensor algebra, characterize its structure Received by the editors December 5, 1971. AMS (MOS) subject classifications (1970). Primary 46H25, 46M10, 43A22; Secondary 22D20.