On the range of an invariant mean

Abstract

Introduction. Let S be a discrete semigroup, m(S) the space of bounded real functions on S with the usual sup. norm, and m(S)* the conjugate Banach space of m(S). An element b e m(S)* is a mean if O(f) >0 whenever f>0 and #(1) =1, where 1 denotes also the constant one function on S. S is said to be left [right] amenable if there is a mean b E m(S)* which is in addition left [right] invariant, i.e., satisfies c(fa) = off) [5b(fa) = b(f)] for each f n m(S) and a e S (where fa(S) =f(as) and fa(S) =f(sa) for f in m(S) and a, s E S). S is amenable if there is a mean 0 E m(S)* which is left and right invariant. If Ac S, then 1A will be the function which is one on A and zero otherwise. We shall write 1 instead of ls and ?>(A) instead of k(lA) (if E m(S)*), sometimes. The range of an element f E m(S) is the set of numbers {+(A), where A ranges over all subsets of S}. It is clear that the range of a mean is a subset of [0, 1] ={x; O<x< l}. If S is a left amenable semigroup, define the following relation between elements of S: a b iff as = bs for some s in S. The relation is an equivalence relation which is two-sidedly stable (or a congruence), i.e., if a -.b, then asrbs and sasb for any s in S (since S is left amenable) (see [3, p. 371] and Ljapin [1, p. 39]). If s e S, let s' stand for the equivalence class to which s belongs and let S'={s'; s e S}. A multiplication between the elements of S' can be defined by s't'= (st)'. This multiplication is well defined and associative, rendering thus S' a semigroup (since is a congruence. See Ljapin [1, pp. 265-266]). Furthermore, S' has right cancellation (and coincides with S if S has right cancellation). It has been shown by this author in [4] (Corollary to Lemma 1) that, if S has right cancellation, is left amenable and contains an element of infinite order, then the range of any invariant mean on m(S) contains the set of rationals in [0, 1]. Moreover, if 0 < r < 1 is a rational number, then there is a set A (-S such that +(A) = r for any left invariant mean 0 on m(S). T. Mitchell has even suggested orally a proof to show that, for the above considered semigroup S, the range of each left invariant mean ? on m(S) is the whole [0, 1] interval. We would like to prove in this paper the following: CONJECTURE. Let S be a left amenable semigroup. Then the range of each left invariant mean on m(S) is the whole [0, 1] interval if and only if S' is infinite.