On the rigidity of uniform Roe algebras over uniformly locally finite coarse spaces

Abstract

Given a coarse space (X, E), one can define a C * -algebra C * u (X) called the uniform Roe algebra of (X, E). It has been proved by J. Špakula and R. Willett that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this paper, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.

Cartan subalgebras in uniform Roe algebras

“…In this short section we prove Corollary D. This is a reasonably straightforward consequence of the results of the previous section combined with the main results of [6] and a theorem of Whyte [49,Theorem 4.1]. First, we give a slight variation of [49,Theorem 4.1]; this is probably well-known to experts.…”

“…the 0-cycle defined by the constant function on X with value one everywhere. From the discussion around [5, Proposition 2.1], if f : X Ñ Y is a coarse embedding 6 , then f induces a map f ˚: H uf ˚pX q Ñ H uf ˚pY q. Whyte proves in [49,Theorem 4.1] that if f : X Ñ Y is a quasi-isometry between uniformly discrete 7 , bounded geometry metric spaces with f ˚rX s " rY s, then there is a bi-Lipschitz map X Ñ Y that is close to f . Let us sketch why Whyte's arguments also imply the result in the statement.…”

“…The equivalence of (i), (ii), and (iii) in the above is fairly well-known: it seems to have been observed independently by several people. We are not sure if it has explicitly appeared in the literature before: see [6,Theorem 8.1] for a closely related, and overlapping, result. The content of the corollary is the equivalence of these with (iv) when X has property A.…”

“…If these conjectures have a positive answer, one can apply powerful analytic tools (positivity and the spectral theorem) to the study of X, and thus deduce important consequences in topology and geometry. This latter motivation has been made particularly stark by recent results of Braga and Farah [6], who show that possible failure of rigidity is intimately tied to the existence of so-called ghost operators that are also known to cause problems for the coarse Baum-Connes conjecture (see [22,Section 6] and [51,).…”

“…It uses both Proposition A and Theorem B above as ingredients in its proof. Other key ingredients include the rigidity results from [47], a recent criterion for detecting when an operator lies in a uniform Roe algebra under the hypothesis of property A due to Špakula and Zhang [48] (which builds on work of Špakula and Tikuisis [46]), the operator norm localisation property of [10], and results of Braga and Farah [6].…”