Schur Multipliers of $C^{\ast }$-algebras, group-invariant compactification and applications to amenability and percolation

Let $\mathrm{\Gamma }$ be a countable discrete group. Given any sequence $(f_n{)}_{n\ge 1}$ of ${\mathrm{\ell }}^p$-normalized functions ($p\in [1,2)$), consider the associated positive definite matrix coefficients $⟨f_n,\rho (\cdot )f_n⟩$ of the right regular representation $\rho$. We construct an orthogonal decomposition of the corresponding {\it Schur multipliers} on the reduced group $C^{\ast }$-algebra or the uniform Roe algebra of $\mathrm{\Gamma }$. We identify this decomposition explicitly via the limit points of the orbits $({\tilde{f}}_n{)}_{n\ge 1}$ 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 $\mathrm{\Gamma }$ – 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 $\mathrm{\Gamma }$.