Schur Multipliers of $C^*$-algebras, group-invariant compactification and applications to amenability and percolation

TL;DR

This paper develops a new decomposition of Schur multipliers for group $C^*$-algebras, linking them to group-invariant compactifications, and uses this to characterize amenability and percolation properties of groups.

Contribution

It introduces an explicit orthogonal decomposition of Schur multipliers via group-invariant compactification limits, providing novel characterizations of group amenability.

Findings

Decomposition of Schur multipliers on group $C^*$-algebras.

Characterizations of amenability via variational and percolation methods.

Connection between Schur multipliers and geometric group properties.

Abstract

Let $\Gamma$ be a countable discrete group. Given any sequence $(f_n)_{n\geq 1}$ of $\ell^p$-normalized functions ($p\in [1,2)$), consider the associated positive definite matrix coefficients $\langle f_n, \rho(\cdot) f_n\rangle$ of the right regular representation $\rho$. We construct an orthogonal decomposition of the corresponding {\it Schur multipliers} on the reduced group $C^*$-algebra or the uniform Roe algebra of $\Gamma$. We identify this decomposition explicitly via the limit points of the orbits $(\widetilde f_n)_{n\geq 1}$ in the group-invariant compactification of the quotient space constructed by Varadhan and the first author in [14]. We apply this result and use positive-definiteness to provide two (quite different) characterizations of amenability of $\Gamma$ -- one via a variational approach and the other using group-invariant percolation on Cayley graphs constructed by Benjamini, Lyons, Peres and Schramm [1]. These results underline, from a new point of view to the best of our knowledge, the manner in which Schur multipliers capture geometric properties of the underlying group $\Gamma$.