Subalgebras of group algebras

Abstract

I. Let G be a locally compact group and m its Haar measure. For any m-measurable subset S of G, let L(S) be the subspace of L1(G) consisting of elements f such that fG\S If I dm =0. If S is a subsemigroup then L(S) is a subalgebra of L1(G). Various papers ([4], [5] and [7]) have been devoted to the study of L(S) and to the question of whether there is a subsemigroup T such that L(S) = L(T) whenever L(S) is an algebra. In [5] it is shown that this is the case whenever S is contained in a a-compact subset. A related problem is the following. Let dS be the set of all x in G such that each measurable neighborhood of x meets S in a set of positive measure. Whenever L(S) is an algebra, dS is a subsemigroup [7], but it need not be true that L(S) =L(dS). In this paper we show that in certain cases L(S) =L(dS). Using this we give very easy proofs of some of the results in [4] and [7]. Let M(G) be the Banach *-algebra of bounded regular Borel measures on G. (We follow [3] in the definition of Borel subsets etc.) For a Borel subset S of G, let M(S) be the set of A.EM(G) with I A I (G\S) = 0. Suppose that S is a measurable subsemigroup of G so that L(S) is a subalgebra of L1(G). Let L(S)T be the algebra of left multipliers of L(S), i.e. the algebra of bounded linear maps 7r of L(S) into itself such that 7r(f * g) = wrf * g. Let St = { x&G: xS\S is locally null }. In this paper we show that if G is abelian then St is closed and that M(Sr) CL(S)T, and under certain additional hypotheses M(ST) =L(S)r. This question has been considered by Birtel in [1] and T. A. Davis in [2]. The question is of some interest since Davis (op. cit.) showed that for abelian G, the Wiener-Pitt phenomenon occurs for measures in L(S)T.