Persistence Approximation Property for Maximal Roe Algebras

Persistence approximation property was introduced by Herv\''e Oyono-Oyono and Guoliang Yu. This property provides a geometric obstruction to Baum-Connes conjecture. In this paper, the authors mainly discuss the persistence approximation property for maximal Roe algebras. They show that persistence approximation property of maximal Roe algebras follows from maximal coarse Baum-Connes conjecture. In particular, let $X$ be a discrete metric space with bounded geometry, assume that $X$ admits a fibred coarse embedding into Hilbert space and $X$ is coarsely uniformly contractible, then $C^{*}_{\rm max}(X)$ has persistence approximation property. The authors also give an application of the quantitative $K$-theory to the maximal coarse Baum-Connes conjecture.