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Let $\Gamma$ be a countable discrete group. Given any sequence $(f_n)_{n\geq1}$ of $\ell^p$-normalized functions ($p\in [1,2)$), consider the associatedpositive definite matrix coefficients $\langle f_n, \rho(\cdot) f_n\rangle$ ofthe right regular representation $\rho$. We construct an orthogonaldecomposition of the corresponding {\it Schur multipliers} on the reduced group$C^*$-algebra or the uniform Roe algebra of $\Gamma$. We identify thisdecomposition explicitly via the limit points of the orbits $(\widetildef_n)_{n\geq 1}$ in the group-invariant compactification of the quotient spaceconstructed by Varadhan and the first author in [14]. We apply this result anduse positive-definiteness to provide two (quite different) characterizations ofamenability of $\Gamma$ -- one via a variational approach and the other usinggroup-invariant percolation on Cayley graphs constructed by Benjamini, Lyons,Peres and Schramm [1]. These results underline, from a new point of view to thebest of our knowledge, the manner in which Schur multipliers capture geometricproperties of the underlying group $\Gamma$.

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