Means for the bounded functions and ergodicity of the bounded representations of semi-groups

Abstract

1. A mean on a semi-group 2 is a positive linear functional of norm one on the space m(z) of bounded, real-valued functions on 1. A bounded semigroup S of linear operators from a Banach space B to itself is called ergodic if there exists a system cd of averages A such that for every S in s limA (AS-A)=limA (SA -A) =0; we have three strengths of ergodicity of S according as uniform, strong, or weak convergence is used in the operator algebra. The first part of this paper deals with the relationship between existence of invariant means and ergodicity of bounded representations. In Theorem 2 it is shown that weak ergodicity of every bounded representation of I is equivalent to weak ergodicity of the right and left representations of f by right and left translations on m(2), and equivalent to the existence of a mean on m(z) invariant under right and left translations. These conditions, in turn, are equivalent to existence of a directed system of finite means on mn(z) converging weakly to two-sided invariance under all right and left translations of m(z). Uniform ergodicity is similarly related to existence of finite means converging in the norm of m(z2)* to two-sided invariance (Theorem 4). The second part of this paper gives some sufficient conditions for existence of invariant means or of finite means converging in norm to invariance. For the former, the Markoff method of proof by fixed-point arguments is applied (?5) to semi-groups and groups, and to semi-groups which are the union of expanding directed systems of sub-semi-groups with means. (For example, a group G such that every finite subset generates a finite subgroup has an invariant mean.) It is also proved by a direct construction (Theorem 6) that if G is a normal subgroup of a group H and if G and H/G have two-sided invariant means, so has H. In ?6 a parallel result is proved for finite means converging in norm to invariance. These results greatly increase the family of groups known to have invariant means. Solvable groups formed the only such class previously known; ?6 now shows that a solvable group satisfies a stronger property; it has a system of finite means converging in norm to invariance.