Group algebra modules. III, IV

Abstract

Let T be a locally compact group, Cl a measurable subset of Y, and let Ln denote the subspace of Ll(Y) consisting of all functions vanishing off fi. Assume that La is a subalgebra of L1^). We discuss the collection SRn(J0 of all module homomorphisms from La into an arbitrary Banach space K which is simultaneously a left LHr) module. We prove that na(K) = na(K0) © *a(.K*0s), where K0 is the collection of all k e K such that fk = 0, for all/eLl(r), and where K^bs consists of all elements of K which can be factored with respect to the module composition. We prove that 5"n(*'o) is the collection of linear continuous maps from La to K0 which are zero on a certain measurable subset of X. We reduce the determination of 9til(Ä_at,3) to the determination of Kr(Ä"alis). Denoting the topological conjugate space of K by K*, we prove that (A"ab3)* is isometrically isomorphic to SKn(.fv*). Finally, we discuss module homomorphisms R from Ln into L1(X) such that for each/e La, /(/vanishes off Y.

On some group algebra modules related to Wiener’s algebraM1

“…The following theorem is essentially due to Rieffel [6] and is proved in [2] A Radon measure on G is a linear functional μ: C oo ~> C such that for every compact set CcG there exists a number c such that…”