Interpolation sets and the size of quotients of function spaces on a locally compact group

Abstract

<div lang="en" class="abs"><p class="abs_header">Abstract</p><p>We devise a fairly general method for estimating the size of quotients between algebras of functions on a locally compact group. This method is based on the concept of interpolation set we introduced and studied recently and unifies the approaches followed by many authors to obtain particular cases.</p><p>We find in this way that there is a linear isometric copy of \(\ell _\infty (\kappa )\) in each of the following quotient spaces: </p><ul><li>\(\mathscr{WAP}_0(G)/C_0(G)\) whenever \(G\) contains a subset \(X\) that is an \(E\)-set (see the definition in the paper) and \(\kappa =\kappa (X)\) is the minimal number of compact sets required to cover \(X\). In particular, \(\kappa =\kappa (G)\) when \(G\) is an \(SIN\)-group.</li><li>\(\mathscr{WAP}(G)/\mathscr {B}(G)\), when \(G\) is any locally compact group and \(\kappa =\kappa (Z(G))\) and \(Z(G)\) is the centre of \(G\), or when \(G\) is either an \(IN\)-group or a nilpotent group and \(\kappa =\kappa (G)\).</li><li>\(\mathscr{WAP}_0(G)/\mathscr {B}_0(G)\), when \(G\) and \(\kappa\) are as in the foregoing item.</li><li>\(\mathscr{CB}(G)/\mathscr {LUC}(G)\), when \(G\) is any locally compact group that is neither compact nor discrete and \(\kappa =\kappa (G)\).</li></ul></div>