Group Algebra Modules. II

Abstract

The present paper began as a natural outgrowth of our first paper, where we characterized the module homomorphisms from group algebras into a fairly restrictive class of group algebra modules. We now investigate module homomorphisms from group algebras into a more general class of group algebra modules. Although the two papers are thus related, they can be read quite independently.Section 2 contains our extension, Theorem 2.1, of P. J. Cohen's theorem on factorization in Banach algebras (1). Our extension is to Banach modules over Banach algebras equipped with an approximate identity. We should mention first that J.-K. Wang observed the existence of such a generalization, and secondly, that our proof requires no ideas different from those in Cohen's proof. Nevertheless, we include a proof that condenses the original proof considerably.

Factorization in Banach algebras and the general strict topology

“…For example, Rieffel [12] showed, by means of a simple device, that if Z is a sequence of elements in X that converges to zero, then Theorem 2.1 isa corollary of a result found by Hewitt [7], Curtis and Figá-Talamanca [5], and Gulick, Liu, and van Rooij [6]. More recently, Collins and Summers [3] introduced an auxiliary left Banach 5-module and then applied Hewitt's factorization theorem to improve the method of proof of our Theorem 2.1.…”

On the Continuity of Certain Intertwining Operators, Centralizers, and Positive Linear Functionals

“…This theorem is an immediate consequence of a lemma found independently by Varopoulos [9] and Johnson [5] [2], and Gulick, Liu and van Rooij [4]. Furthermore, a method for simplifying the proof of Cohen's theorem has been found by Koosis [ó], and, as he remarks, this method can be used equally well to simplify the proof of the theorem of Hewitt et al In this way we obtain a quite short proof of the lemma of Varopoulos and Johnson, and so of the three continuity results mentioned above.…”

A generalized Fourier transformation for L1(G)-Modules

“…for locally almost all x G A'(see [1]). If we can prove that g G L t (m), then JU and gm are bounded regular measures and (k, ju.)…”