Cartan subalgebras in uniform Roe algebras

Abstract

In this paper we study structural and uniqueness questions for Cartan subalgebras of uniform Roe algebras. We characterise when an inclusion $B\subseteq A$ of $C^{\ast }$-algebras is isomorphic to the canonical inclusion of ${\ell }^{\infty }(X)$ inside a uniform Roe algebra $C_u^{\ast }(X)$ associated to a metric space of bounded geometry. We obtain uniqueness results for “Roe Cartans” inside uniform Roe algebras up to automorphism when $X$ coarsely embeds into Hilbert space, and up to inner automorphism when $X$ has property A.