Quasi-local Algebras and Asymptotic Expanders

Abstract

In this paper, we study the relation between the uniform Roe algebra and the uniform quasi-local algebra associated to a metric space of bounded geometry. In the process, we introduce a weakening of the notion of expanders, called asymptotic expanders. We show that being a sequence of asymptotic expanders is a coarse property under certain connectedness condition, and it implies non-uniformly local amenability. Moreover, we also analyse some $C^{\ast }$-algebraic properties of uniform quasi-local algebras. In particular, we show that a uniform quasi-local algebra is nuclear if and only if the underlying metric space has Property A.

arXiv commentAccepted by Groups, Geometry and Dynamics

References in corpus (6)

• Some notes on property A
• Ideal structure and pure infiniteness of ample groupoid $C^{\ast }$-algebras
• Amenability and uniform Roe algebras
• Relative commutant pictures of Roe algebras
• A generalization of expander graphs and local reflexivity of uniform Roe algebras
• Structure and $K$-theory of ${\ell }^p$ uniform Roe algebras

Cited by in corpus (3)

• A Markovian and Roe-algebraic approach to asymptotic expansion in measure
• Asymptotic expansion in measure and strong ergodicity
• Strongly Quasi-local algebras and their $K$-theories