Measured asymptotic expanders and rigidity for Roe algebras

TL;DR

This paper establishes new rigidity results for Roe algebras, showing that under certain geometric conditions, isomorphic Roe algebras imply coarse equivalence of the underlying spaces, with implications for the coarse Baum-Connes conjecture.

Contribution

It introduces a geometric condition involving ghostly measured asymptotic expanders that generalizes previous assumptions for rigidity of Roe algebras.

Findings

Rigidity holds for spaces coarsely embeddable into L^p spaces for p in [1,∞)

Rigidity verified for specific box spaces not coarsely embeddable into any L^p-space

Block-rank-one ghost projections relate to measured asymptotic expanders in Roe algebras

Abstract

Our main result about rigidity of Roe algebras is the following: if $X$ and $Y$ are metric spaces with bounded geometry such that their Roe algebras are $*$-isomorphic, then $X$ and $Y$ are coarsely equivalent provided that either $X$ or $Y$ contains no sparse subspaces consisting of ghostly measured asymptotic expanders. Note that this geometric condition generalises the existing technical assumptions used for rigidity of Roe algebras. Consequently, we show that the rigidity holds for all bounded geometry spaces which coarsely embed into some $L^p$-space for $p\in [1,\infty)$. Moreover, we also verify the rigidity for the box spaces constructed by Arzhantseva-Tessera and Delabie-Khukhro even though they do \emph{not} coarsely embed into any $L^p$-space. The key step towards our proof for the rigidity is to show that a block-rank-one (ghost) projection on a sparse space $X$ belongs to the Roe algebra $C^*(X)$ if and only if $X$ consists of (ghostly) measured asymptotic expanders. As a by-product, we also deduce that ghostly measured asymptotic expanders are new sources of counterexamples to the coarse Baum-Connes conjecture.