Related papers: On Rigidity of Roe algebras

In this memoir we develop a framework to study rigidity problems for Roe-like C*-algebras of countably generated coarse spaces. The main goal is to give a complete and self-contained solution to the problem of C*-rigidity for proper…

We prove that uniformly locally finite metric spaces with isomorphic Roe algebras must be coarsely equivalent. As an application, we also prove that the outer automorphism group of the Roe algebra of a metric space of bounded geometry is…

Given a coarse space $(X,\mathcal{E})$, one can define a $\mathrm{C}^*$-algebra $\mathrm{C}^*_u(X)$ called the uniform Roe algebra of $(X,\mathcal{E})$. It has been proved by J. \v{S}pakula and R. Willett that if the uniform Roe algebras of…

In this paper, we connect the rigidity problem and the coarse Baum-Connes conjecture for Roe algebras. In particular, we show that if $X$ and $Y$ are two uniformly locally finite metric spaces such that their Roe algebras are…

In this paper, we study the rigidity of uniform Roe algebras via the ideal structures. We showed that for given metric spaces X and Y with bounded geometry, if their uniform Roe algebras are isomorphic, then they are coarse equivalent.

Gong, Wang and Yu introduced a maximal, or universal, version of the Roe C*-algebra associated to a metric space. We study the relationship between this maximal Roe algebra and the usual version, in both the uniform and non-uniform cases.…

We show that if $X$ and $Y$ are uniformly locally finite metric spaces whose uniform Roe algebras, $\cstu(X)$ and $\cstu(Y)$, are isomorphic as \cstar-algebras, then $X$ and $Y$ are coarsely equivalent metric spaces. Moreover, we show that…

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…

We provide a construction of Roe (C*-)algebras of general coarse spaces in terms of coarse geometric modules. This extends the classical theory of Roe algebras of metric spaces and gives a unified framework to deal with either uniform or…

We generalize all known results on rigidity of uniform Roe algebras to the setting of arbitrary uniformly locally finite coarse spaces. For instance, we show that isomorphism between uniform Roe algebras of uniformly locally finite coarse…

For a discrete metric space (or more generally a large scale space) $X$ and an action of a group $G$ on $X$ by coarse equivalences, we define a type of coarse quotient space $X_G$, which agrees up to coarse equivalence with the orbit space…

In this paper, we investigate the rigidity problems for geometric ideals in uniform Roe algebras associated to discrete metric spaces of bounded geometry. These ideals were introduced by Chen and Wang, and can be fully characterised in…

We investigate the rigidity of the $\ell^p$ analog of Roe-type algebras. In particular, we show that if $p\in[1,\infty)\setminus\{2\}$, then an isometric isomorphism between the $\ell^p$ uniform Roe algebras of two metric spaces with…

The Roe algebra $C^*(X)$ is a non-commutative $C^*$-algebra reflecting metric properties of a space $X$, and it is interesting to understand relation between the Roe algebra of $X$ and the (uniform) Roe algebra of its discretization. Here…

We solve the rigidity problem for uniform Roe algebras, by showing that two uniformly locally finite metric spaces with isomorphic uniform Roe algebras are bijectively coarsely equivalent.

Given any quasi-countable, in particular any countable inverse semigroup $S$, we introduce a way to equip $S$ with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete…

Given a discrete and countable inverse semigroup $S$ one can study, in analogy to the group case, its geometric aspects. In particular, we can equip $S$ with a natural metric, given by the path metric in the disjoint union of its…

In this paper, we introduce a notion of strongly quasi-local algebras. They are defined for each discrete metric space with bounded geometry, and sit between the Roe algebra and the quasi-local algebra. We show that strongly quasi-local…

In this paper, we investigate *-homomorphisms between C*-algebras associated to \'etale groupoids. First, we prove that such a *-homomorphism can be described by closed invariant subsets, groupoid homomorphisms and cocycles under some…

We study the uniform Roe algebras associated to locally finite groups. We show that for two countable locally finite groups $\Gamma$ and $\Lambda$, the associated uniform Roe algebras $C^*_u(\Gamma)$ and $C^*_u(\Lambda)$ are $*$-isomorphic…