Invariant means on almost periodic functions and equicontinuous actions

Lau, Anthony To Ming

doi:10.1090/S0002-9939-1975-0367551-5

Invariant means on almost periodic functions and equicontinuous actions
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Proc. Amer. Math. Soc. 49 (1975), 379-382 Request permission

Abstract:

Let $S$ be a topological semigroup such that the almost periodic functions on $S$ have a left invariant mean (this is the case, for example, when $S$ has finite intersection property for closed right ideals). Then whenever $S$ acts equicontinuously on a compact Hausdorff space $X$, there exists a compact group $G$ of homeomorphisms acting equicontinuously on a retract $Y$ of $X$ such that $S$ has a common fixed point in $X$ if and only if $G$ has a common fixed point in $Y$. This result generalises some recent work of T. Mitchell. As an application, we show that whenever $S$ acts equicontinuously on the closed unit interval $I$, then $I$ contains a common fixed point for $S$.

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