The paper defines asymptotic expanders, characterizes quasi-locality of averaging projections via them, and proves the uniform quasi-local algebra is nuclear iff the space has Property A.
Cartan subalgebras in uniform Roe algebras
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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.
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Quasi-local Algebras and Asymptotic Expanders
The paper defines asymptotic expanders, characterizes quasi-locality of averaging projections via them, and proves the uniform quasi-local algebra is nuclear iff the space has Property A.