New algebras of functions on topological groups arising from $G$-spaces

E. Glasner ¹   ; M. Megrelishvili ²

¹ Department of Mathematics Tel-Aviv University Ramat Aviv, Israel
² Department of Mathematics Bar-Ilan University 52900 Ramat-Gan, Israel

Fundamenta Mathematicae, Tome 201 (2008) no. 1, pp. 1-51

Résumé

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.

Détail

DOI : 10.4064/fm201-1-1

Keywords: topological group introduce algebra suc strongly uniformly continuous functions suc contains algebra wap weakly almost periodic functions algebras asp locally equicontinuous asplund functions respectively polish groups order preserving homeomorphisms unit interval isometries urysohn space diameter suc trivial introduce notion fixed point class flows fpp study particular groups suc fpp study roelcke algebra right uniformly continuous functions suc compactifications groups mathbb permutations countable set homeomorphisms cantor set first group wap suc provide concrete description corresponding metrizable cantor semitopological semigroup compactification second group contrast suc properly contained deduce group does yield right topological semigroup compactification