Measured Asymptotic Expanders and Rigidity for Roe Algebras

Abstract In this paper, we give a new geometric condition in terms of measured asymptotic expanders to ensure rigidity Roe algebras. Consequently, obtain the for all bounded geometry spaces that coarsely embed into some $L^p$-space $p\in [1,\infty )$. Moreover, also verify box constructed by Arzhantseva–Tessera and Delabie–Khukhro even though they do not any $L^p$-space. The key step our proof is showing block-rank-one (ghost) projection on sparse space $X$ belongs algebra $C^{\ast }(X)$ if only consists (ghostly) expanders. As by-product, deduce ghostly are sources counterexamples coarse Baum–Connes conjecture.