## 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.