Finite-dimensional approximation properties for uniform Roe algebras

Abstract

We study property A for metric spaces $X$ with bounded geometry introduced by Guoliang Yu. Property A is an amenability-type condition, which is less restrictive than amenability for groups. The property has a connection with finite-dimensional approximation properties in the theory of operator algebras. It has been already known that property A of a metric space $X$ with bounded geometry is equivalent to nuclearity of the uniform Roe algebra C${}_u^{\ast }(X)$. We prove that exactness and local reflexivity of C${}_u^{\ast }(X)$ also characterize property A of $X$.

22 pages, simpler proof than v1, title changed, to appear in JLMS

References in corpus (2)

• Amenable actions and exactness for discrete groups
• Property A and the operator norm localization property for discrete metric spaces

Cited by in corpus (2)

• The uniform Roe algebra of an inverse semigroup
• ${\mathrm{C}}^{\ast }$-exactness and property A for group actions