On the size of quotients of function spaces on a topological group

Ahmed Bouziad¹ ; Mahmoud Filali²

¹ Département de Mathématiques Université de Rouen UMR CNRS 6085 Avenue de l'Université, BP.12 F-76801 Saint-Étienne-du-Rouvray, France
² Department of Mathematical Sciences University of Oulu 90014 Oulu, Finland

Studia Mathematica, Tome 202 (2011) no. 3, pp. 243-259

Résumé

For a non-precompact topological group $G,$ we consider the space $C(G)$ of bounded, continuous, scalar-valued functions on $G$ with the supremum norm, together with the subspace ${\it LMC}(G)$ of left multiplicatively continuous functions, the subspace ${\it LUC}(G)$ of left norm continuous functions, and the subspace ${\it WAP}(G)$ of weakly almost periodic functions.We establish that the quotient space ${\it LUC}(G)/{\it WAP}(G)$ contains a linear isometric copy of $\ell_\infty,$ and that the quotient space $C(G)/{\it LMC}(G)$ (and a fortiori $C(G)/{\it LUC}(G)$) contains a linear isometric copy of $\ell_\infty$ when $G$ is a normal non-$P$-group. When $G$ is not a $P$-group but not necessarily normal we prove that the quotient is non-separable. For non-discrete $P$-groups, the quotient may sometimes be trivial and sometimes non-separable. When $G$ is locally compact, we show that the quotient space ${\it LUC}(G)/{\it WAP}(G)$ contains a linear isometric copy of $\ell_\infty(\kappa(G)),$ where $\kappa(G)$ is the minimal number of compact sets needed to cover $G.$ This leads to the extreme non-Arens regularity of the group algebra $L^1(G)$ when in addition either $\kappa(G)$ is greater than or equal to the smallest cardinality of an open base at the identity $e$ of $G$, or $G$ is metrizable. These results are improvements and generalizations of theorems proved by various authors along the last 35 years and until very recently.

Détail

DOI : 10.4064/sm202-3-3

Keywords: non precompact topological group consider space bounded continuous scalar valued functions supremum norm together subspace lmc multiplicatively continuous functions subspace luc norm continuous functions subspace wap weakly almost periodic functions establish quotient space luc wap contains linear isometric copy ell infty quotient space lmc fortiori luc contains linear isometric copy ell infty normal non p group p group necessarily normal prove quotient non separable non discrete p groups quotient may sometimes trivial sometimes non separable locally compact quotient space luc wap contains linear isometric copy ell infty kappa where kappa minimal number compact sets needed cover nbsp leads extreme non arens regularity group algebra addition either kappa greater equal smallest cardinality base identity metrizable these results improvements generalizations theorems proved various authors along years until recently