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

TL;DR

This paper introduces a general method using interpolation sets to estimate the size of quotients of function algebras on locally compact groups, unifying previous approaches and revealing large subspaces in these quotients.

Contribution

It develops a unified framework based on interpolation sets for estimating quotient sizes of function algebras on locally compact groups, extending and generalizing prior methods.

Findings

WAP(G)/B(G) contains a linearly isometric copy of _((G)) for certain groups

CB(G)/LUC(G) contains a linearly isometric copy of _((G)) for non-compact, non-discrete groups

The method applies to various classes of locally compact groups, including IN and nilpotent groups.

Abstract

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 sets and unifies the approaches followed by many authors to obtain particular cases. Among the applications we find, we obtain that the quotients WAP(G)/B(G) (G being a locally compact group in the class [IN] or a nilpotent locally compact group) and CB(G)/LUC(G) (G being any non-compact non-discrete locally compact group) contain a linearly isometric copy of \ell_\infty(\kappa(G)) where \kappa(G) is the compact covering number of G, and WAP(G), B(G) and LUC(G) refer, respectively, to the algebra of weakly almost periodic functions, the uniform closure of the Fourier-Stieltjes algebra and the bounded right uniformly continuous functions.