New algebras of functions on topological groups arising from $G$-spaces
Volume 201 / 2008
Abstract
For a topological group $G$ we introduce the algebra $SUC(G)$ of strongly uniformly continuous functions. We show that $SUC(G)$ contains the algebra $WAP(G)$ of weakly almost periodic functions as well as the algebras $LE(G)$ and Asp$(G)$ of locally equicontinuous and Asplund functions respectively. For the Polish groups of order preserving homeomorphisms of the unit interval and of isometries of the Urysohn space of diameter 1, we show that $SUC(G)$ is trivial. We introduce the notion of fixed point on a class~P of flows (${\rm P}$-${\rm fpp}$) and study in particular groups with the SUC-fpp. We study the Roelcke algebra (= $UC(G)$ = right and left uniformly continuous functions) and SUC compactifications of the groups $S({\mathbb N})$, of permutations of a countable set, and $H(C)$, of homeomorphisms of the Cantor set. For the first group we show that $WAP(G)=SUC(G)=UC(G)$ and also provide a concrete description of the corresponding metrizable (in fact Cantor) semitopological semigroup compactification. For the second group, in contrast, we show that $SUC(G)$ is properly contained in $UC(G)$. We then deduce that for this group $UC(G)$ does not yield a right topological semigroup compactification.
