Schur multipliers of C⁎-algebras, group-invariant compactification and applications to amenability and percolation

Let Γ be a countable discrete group. Given any sequence (fn)n≥1 of ℓp-normalized functions (p∈[1,2)), consider the associated positive definite matrix coefficients 〈fn,ρ(⋅)fn〉 of the right regular representation ρ. We construct an orthogonal decomposition of the corresponding Schur multipliers on the reduced group C⁎-algebra or the uniform Roe algebra of Γ. We identify this decomposition explicitly via the limit points of the orbits (f˜n)n≥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 Γ – 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 Γ.