Functional analytic properties of topological semigroups and $n$-extreme amenability

Abstract:

Let $S$ be a topological semigroup, $\operatorname {LUC} (S)$ be the space of left uniformly continuous functions on $S$, and $\Delta (S)$ be the set of multiplicative means on $\operatorname {LUC} (S)$. If $( \ast )\operatorname {LUC} (S)$ has a left invariant mean in the convex hull of $\Delta (S)$, we associate with $S$ a unique finite group $G$ such that for any maximal proper closed left translation invariant ideal $I$ in $\operatorname {LUC} (S)$, there exists a linear isometry mapping $\operatorname {LUC} (G)/I$ one-one onto the set of bounded real functions on $G$. We also generalise some recent results of T. Mitchell and E. Granirer. In particular, we show that $S$ satisfies $( \ast )$ iff whenever $S$ is a jointly continuous action on a compact hausdorff space $X$, there exists a nonempty finite subset $F$ of $X$ such that $sF = F$ for all $s \in S$. Furthermore, a discrete semigroup $S$ satisfies $( \ast )$ iff whenever $\{ {T_s};s \in S\}$ is an antirepresentation of $S$ as linear maps from a norm linear space $X$ into $X$ with $||{T_s}|| \leqq 1$ for all $s \in S$, there exists a finite subset $\sigma \subseteq S$ such that the distance (induced by the norm) of $x$ from ${K_X} = \text {linear span}$ of $\{ x - {T_s}x;x \in X,s \in S\}$ in $X$ coincides with distance of $O(\sigma ,x) = \{ (1/|\sigma |)\sum \nolimits _{a \in \sigma } {{T_{at}}(x);t \in S\} }$ from 0 for all $x \in X$.