Cartan subalgebras in uniform Roe algebras

TL;DR

This paper investigates the structure and uniqueness of Cartan subalgebras within uniform Roe algebras, providing characterizations and conditions for their uniqueness based on properties of the underlying metric space.

Contribution

It characterizes when subalgebras are isomorphic to the canonical $ ext{l}^ ext{infty}$ inclusion in uniform Roe algebras and establishes uniqueness results under coarse embedding and property A.

Findings

Characterization of when an inclusion is isomorphic to the canonical inclusion.

Uniqueness of Roe Cartans up to automorphism when the space coarsely embeds into Hilbert space.

Uniqueness up to inner automorphism when the space has property A.

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 $\mathrm{C}^*$-algebras is isomorphic to the canonical inclusion of $\ell^\infty(X)$ inside a uniform Roe algebra $C^*_u(X)$ 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.