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

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. We find in this way that there is a linear isometric copy of \(\ell _\infty (\kappa )\) in each of the following quotient spaces: \(\mathscr{WAP}_0(G)/C_0(G)\) 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. \(\mathscr{WAP}(G)/\mathscr {B}(G)\), 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 \(\kappa =\kappa (G)\). \(\mathscr{WAP}_0(G)/\mathscr {B}_0(G)\), when \(G\) and \(\kappa\) are as in the foregoing item. \(\mathscr{CB}(G)/\mathscr {LUC}(G)\), when \(G\) is any locally compact group that is neither compact nor discrete and \(\kappa =\kappa (G)\).