The range of invariant means on locally compact groups and semigroups

Abstract

This paper extends the results of Granirer and Chou concerning the range of a left invariant mean on a discrete semigroup to the case when S is any Borel subsemigroup of a locally compact group. 0. Introduction. Granirer has shown in [2] that for an infinite, right cancellation, left amenable discrete semigroup S, other than what he calls an AB group, there exists a nested family of left almost convergent subsets of S on which any left invariant mean (LIM) attains all values of the closed interval [0, 1]. That is, there exists a family {A(t)|t E [0, 1]} of subsets of S for which (i) s<t implies A(s)c A(t), and (ii) p(%A(t))=t for any LIM q on m(S). Chou, in [1], partially extended this result to the case when S is a group and obtained the following theorem thereby proving a conjecture made in [2]. THEOREM (CHOU). If S is an infinite right cancellation left amenable semigroup then the range of each LIM on m(S) is the whole [0, 1] interval. In this paper we extend these results to locally compact topological groups and obtain the following main theorems: THEOREM A. Let S be an infinite Borel subsemigroup of positive Haar measure in a locally compact group. If q is a LIM on L'(S) then there exists a nested family {A(t)lt E [0, 1 } of Borel subsets of S stuch that cp(A(t))=t for all t E [0, 1]. THEOREM B. Let S be an infinite Borel subsemigroup in a locally compact group and let A denote the algebra of bounded Borel measurable Received by the editors March 7, 1972. AMS (MOS) subject classifications (1969). Primary 2875, 2220; Secondary 4250, 4655. 1 The results in this paper constitute a portion of the author's Ph.D. thesis written under the direction of Dr. E. A. Granirer at the University of British Columbia. The author wishes to thank Professor Granirer for his aid and advice and as well the Canada Council for its financial support. ? American Mathematical Society 1973