Journal of the Australian Mathematical Society - Series A Vol. 64 Part 2 (1998) Ergodicity and differences of functions on semigroups Bolis Basit Department of Mathematics Monash University Clayton, VIC 3168 Australia e-mail: bbasit(ajpryde)@vaxc.cc.monash.edu.au and A. J. Pryde Abstract: Iseki [11] defined a general notion of ergodicity suitable for functions $\varphi : J \rightarrow X$ where J is an arbitrary abelian semigroup and X is a Banach space. In this paper we develop the theory of such functions, showing in particular that it fits the general framework established by Eberlein [9] for ergodicity of semigroups of operators acting on X. Moreover, let $\cal{A}$ be a translation invariant closed subspace of the space of all bounded functions from J to X. We prove that if $\cal{A}$ contains the constant functions and $\varphi$is an ergodic function whose differences lie in $\cal{A}$ then $\varphi \in \cal{A}$. This result has applications to spaces of sequences facilitating new proofs of theorems of Gelfand and Katznelson-Tzafriri [12]. We also obtain a decomposition for the space of ergodic vectors of a representation $T : J \rightarrow L(X)$ generalizing results known for the case $J = \Bbb{Z}^+$. Finally, when J is a subsemigroup of a locally compact abelian group G, we compare the Iseki integrals with the better known Cesàro integrals. PDF file size: 99K © Copyright 1998, Australian Mathematical Society