Invariant Means on Locally Compact Semigroups

Abstract

Let $G$ be a locally compact semigroup (jointly continuous semigroup operation), $M(G)$ the algebra of all bounded regular Borel measures on $G$ (with convolution as multiplication), $E$ a separated locally convex space and $S$ a compact convex subset of $E$. We show that there is a left invariant mean on the space ${\text {LUC}}(G)$ of all bounded left uniformly continuous functions on $G$ iff $G$ has the following fixed point property: For any bilinear mapping $T:M(G) \times E \to E$ (denoted by $(\mu ,s) \to {T_\mu }(s)$) such that (a) ${T_\mu }(S) \subset S$ for any $\mu \geqq 0,||\mu || = 1$, (b) ${T_{\mu \ast \nu }} = {T_\mu } \circ {T_\nu }$ for any $\mu ,\nu \in M(G)$, (c) ${T_\mu }:S \to S$ is continuous for any $\mu \geqq 0,||\mu || = 1$, and ${\text {(d)}}\mu \to {T_\mu }(s)$ is continuous for each $s \in S$ when $M(G)$ has the topology induced by the seminorms ${p_f}(\mu ) = |\int {fd\mu |} ,f \in {\text {LUC}}(G)$, there is some ${s_0} \in S$ such that ${T_\mu }({s_0}) = {s_0}$ for any $\mu \geqq 0,||\mu || = 1$.