On the size of quotients of function spaces on a topological group
Volume 202 / 2011
Abstract
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.
