Jan Spakula
We prove that uniform Roe C*-algebras associated to some expander graphs coming from discrete groups with property (τ) are not K-exact. In particular, we show that this is the case for the expander obtained as Cayley graphs of a sequence of alternating groups (with appropriately chosen generating sets).
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It was shown recently by M. Lorentz and R. Willett that all bounded derivations of the uniform Roe algebras of metric spaces of bounded geometry are inner. Here we calculate the space of outer derivations of the uniform Roe algebras with coefficients in uniform Roe bimodules related to various metrics on the two copies of the given space. We also give some results on the higher Hochschild cohomology with coefficients in uniform Roe bimodules.
We show that if $C_u^*(X)$ is a uniform Roe algebra associated to a bounded geometry metric space X, then all bounded derivations on $C^*_u(X)$ are inner.
In Rufus Willett's and the authors paper "Bounded Derivations on Uniform Roe Algebras" we showed that all bounded derivations on a uniform Roe algebra $C^*_u(X)$ associated to a bounded geometry metric space $X$ are inner. This naturally leads to the question of whether or not the higher dimensional Hochschild cohomology groups of the uniform Roe algebra vanish also. While we cannot answer this question completely, we are able to give necessary and sufficient conditions for the vanishing of $H^n_c(C_u^*(X),C_u^*(X))$. Lastly, we show that if the norm continuous Hochschild cohomology of a unifo…
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 spaces whose uniform Roe algebras contain only compact ghost projections implies that the base spaces are coarsely equivalent. Moreover, if one of the spaces has property A, then the base spaces are bijectively coarsely equivalent. We also provide a characterization for the existence of an embedding onto hereditary subalgebra in terms of the underlying spaces…
Let $Γ$ be a countable discrete group. We show that $Γ$ has the approximation property if and only if $Γ$ is exact and for any operator space $S \subseteq \K(H)$ we have $\Cu(Γ)^Γ \otimes S = (\Cu(Γ) \otimes S)^Γ$, where $\Cu(Γ)$ is the uniform Roe algebra with the right adjoint $Γ$-action. This answers a question of J. Zacharias. We also show that characterisations of several properties of $Γ$ in terms of the reduced group \cast-algebra apply to the invariant uniform Roe algebra $\Cu(Γ)^Γ$.
We study the uniform Roe algebras associated to locally finite groups. We show that for two countable locally finite groups $Γ$ and $Λ$, the associated uniform Roe algebras $C^*_u(Γ)$ and $C^*_u(Λ)$ are $*$-isomorphic if and only if their $K_0$ groups are isomorphic as ordered abelian groups with units. This can be seen as a non-separable non-simple analogue of the Glimm-Elliott classification of UHF algebras. To the best of our knowledge, this is the first classification result for a class of non-separable unital $C^*$-algebras. Along the way we also obtain a rigidity result: two countable lo…
We work on $\ell_p$ uniform Roe algebras associated to metric spaces, and on their mutual embedding. We generalize results of I. Farah and the authors to mutual embeddings of uniform Roe algebras of operators on $\ell_p$ spaces. Simultaneously, we obtain rigidity results for the classic uniform Roe $\mathrm{C}^*$-algebras which depend only on their Banach algebra structure.
Rigidity for geometric ideals in uniform Roe algebras
本文研究有界几何离散度量空间的一致 Roe 代数中几何理想的刚性问题，证明了稳定同构的几何理想对应粗等价的粗空间，并探讨了 ghostly ideals 情形与开放问题。
Isomorphism rigidity of uniform Roe algebras over arbitrary uniformly locally finite coarse spaces
研究任意一致局部有限粗空间上一致 Roe 代数的同构刚性问题，通过证明\(C^*\) - 代数同构能推出空间双射粗等价，解决了该同构刚性问题。
In this paper, we study embeddings of uniform Roe algebras. Generally speaking, given metric spaces $X$ and $Y$, we are interested in which large scale geometric properties are stable under embedding of the uniform Roe algebra of $X$ into the uniform Roe algebra of $Y$.
A quantization of coarse spaces and uniform Roe algebras
本文提出了粗空间与一致Roe代数的量子化理论，基于N. Weaver的量子关系并依赖表示的von Neumann代数，发展了量子粗空间间映射的基础理论及其对量子一致Roe代数的推论，并引入了新的支撑扩张C*-代数类。
We consider two versions of the uniform Roe algebra for uniformly discrete spaces without bounded geometry and discuss some of their properties.
In this paper, we investigate the ideal structure of uniform Roe algebras for general metric spaces beyond the scope of Yu's property A. Inspired by the ideal of ghost operators coming from expander graphs and in contrast to the notion of geometric ideal, we introduce a notion of ghostly ideal in a uniform Roe algebra, whose elements are locally invisible in certain directions at infinity. We show that the geometric ideal and the ghostly ideal are respectively the smallest and the largest element in the lattice of ideals with a common invariant open subset of the unit space of the coarse group…
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.
$K$-Theoretic Comparison of Roe and Quasi-Local Algebras via Projections
本文针对稀疏度量空间，证明了关于Roe代数与准局部代数K-理论的三项结果，给出了二者K-理论不同的首个否定实例。
The rigidity problem for uniform Roe algebras
本文解决了一致罗代数的刚性问题，证明同构的一致罗代数对应的度量空间是双射粗等价的。
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. 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.
A uniform Roe corona is the quotient of the uniform Roe algebra of a metric space by the ideal of compact operators. Among other results, we show that it is consistent with ZFC that isomorphism between uniform Roe coronas implies coarse equivalence between the underlying spaces, for the class of uniformly locally finite metric spaces which coarsely embed into a Hilbert space. Moreover, for uniformly locally finite metric spaces with property A, it is consistent with ZFC that isomorphism between the uniform Roe coronas is equivalent to bijective coarse equivalence between some of their cofinite…
Embeddings of matrix algebras into the universal Roe algebra
本文研究万有Roe代数的嵌入性质，证明其可嵌入$\prod_n\mathbb{M}_n$，否定Ozawa定理的推广，并得到置换算子凸包刻画及可分拟对角代数的嵌入结果。
Embeddings of matrix algebras into uniform Roe algebras and quasi-local algebras
本文解决了矩阵代数的$\ell_\infty$-直和能否嵌入一致Roe代数与拟局部代数的问题，证明前者不可嵌入而后者可嵌入，从而说明一致Roe代数到拟局部代数的包含关系可为真包含。
In this paper we study structural and uniqueness questions for Cartan subalgebras of uniform Roe algebras. We characterise when an inclusion $B\subseteq A$ of $\mathrm{C}^*$-algebras is isomorphic to the canonical inclusion of $\ell^\infty(X)$ inside a uniform Roe algebra $C^*_u(X)$ associated to a metric space of bounded geometry. We obtain uniqueness results for `Roe Cartans' inside uniform Roe algebras up to automorphism when $X$ coarsely embeds into Hilbert space, and up to inner automorphism when $X$ has property A.
An example of a continuous field of Roe algebras
本文在最简单的非平凡例子$X=\mathbb R$中，通过在$[0,1]$上构造$C^*$-代数的连续场，探索了空间Roe代数与其离散化一致Roe代数之间的关系。
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. The main result is that if a (uniformly discrete, bounded geometry) metric space X coarsely embeds in a Hilbert space, then the canonical map between the maximal and usual (uniform) Roe algebras induces an isomorphism on K-theory. We also give a simple proof that if X has property A, then the maximal and usual (uniform) Roe algebras are the same. These two re…
Bijective rigidity of uniform Roe algebras and injectivity of the comparison map
本文研究了均匀罗氏代数同构的均匀度量空间之间的双射粗等价与HK猜想中0阶比较映射单射性之间的等价关系，并证明了0阶比较映射无条件单射性及在空间粗连通时的分裂单射性。
We introduce a generalization of expander graphs, which is called a weak expander sequence. It is proved that a uniform Roe algebra of a weak expander sequence is not locally reflexive. It follows that uniform Roe algebras of expander graphs are not exact. We introduce the notion of a generalized box space to discuss box spaces and expander sequences in a unified framework. Key tools for the proof are amenable traces and measured groupoids associated to generalized box spaces.
Let $X$ be a discrete metric space with bounded geometry. We show that if $X$ admits an "A-by-CE coarse fibration", then the canonical quotient map $λ: C^*_{\max}(X)\to C^*(X)$ from the maximal Roe algebra to the Roe algebra of $X$, and the canonical quotient map $λ: C^*_{u, \max}(X)\to C^*_u(X)$ from the maximal uniform Roe algebra to the uniform Roe algebra of $X$, induce isomorphisms on $K$-theory. A typical example of such a space arises from a sequence of group extensions $\{1\to N_n\to G_n\to Q_n\to 1\}$ such that the sequence $\{N_n\}$ has Yu's property A, and the sequence $\{Q_n\}$ adm…
Generalizing the case of an infinite discrete metric space of finite diameter, we say that a discrete metric space $(X,d)$ is a Kuiper space, if the group of invertible elements of its uniform Roe algebra is norm-contractible. Various sufficient conditions on $(X,d)$ to be or not to be a Kuiper space are obtained.
On Spectral Invariant Dense Subalgebras of Uniform Roe Algebras with Subexponential Growth
针对满足性质 P 的次指数增长群，构造 R^∞(G) 子代数，并利用容许权证明其在一致 Roe 代数 C_u^*(G) 中的谱不变性，从而把相关结果推广到多项式增长之外。
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 coarse equivalence between $X$ and $Y$ is equivalent to Morita equivalence between $\cstu(X)$ and $\cstu(Y)$. As an application, we obtain that if $Γ$ and $Λ$ are finitely generated groups, then the crossed products $\ell_\infty(Γ)\rtimes_rΓ$ and $ \ell_\infty(Λ)\rtimes_rΛ$ are isomorphic if and only if $Γ$ and $Λ$ are bi-Lipschitz equivalent.
A tautological continuous field of Roe bimodules
本文将C*-代数的连续场推广为Hilbert C*-双模，通过构造特定的紧致化空间γP，得到一个塔尔斯基连续场，研究了粗等价类与Roe双模的关系。
We study property A for metric spaces $X$ with bounded geometry introduced by Guoliang Yu. Property A is an amenability-type condition, which is less restrictive than amenability for groups. The property has a connection with finite-dimensional approximation properties in the theory of operator algebras. It has been already known that property A of a metric space $X$ with bounded geometry is equivalent to nuclearity of the uniform Roe algebra C$^*_u(X)$. We prove that exactness and local reflexivity of C$^*_u(X)$ also characterize property A of $X$.
We study embeddings of uniform Roe algebras which have "large range" in their codomain and the relation of those with coarse quotients between metric spaces. Among other results, we show that if $Y$ has property A and there is an embedding $Φ:\mathrm{C}^*_u(X)\to \mathrm{C}^*_u(Y)$ with "large range" and so that $Φ(\ell_\infty(X))$ is a Cartan subalgebra of $\mathrm{C}^*_u(Y)$, then there is a bijective coarse quotient $X\to Y$. This shows that the large scale geometry of $Y$ is, in some sense, controlled by the one of $X$. For instance, if $X$ has finite asymptotic dimension, so does $Y$.
The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups
本文利用 Proietti--Yamashita 谱序列，计算了由有限秩自由阿贝尔子群生成的粗结构的一致 Roe 代数的 K-群，并将其表示为群同调。
Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups
本文为拟可数逆半群构造典范真右次不变度量并定义一致 Roe 代数，证明渐近维数0等价于局部有限及强拟对角性，并刻画更弱的有限 Roe 代数条件。
Let $\widetilde{X}$ be a smooth Riemannian manifold equipped with a proper, free, isometric and cocompact action of a discrete group $Γ$. In this paper we prove that the analytic surgery exact sequence of Higson-Roe for $\widetilde{X}$ is isomorphic to the exact sequence associated to the adiabatic deformation of the Lie groupoid $\widetilde{X}\times_Γ\widetilde{X}$. We then generalize this result to the context of smoothly stratified manifolds. Finally, we show, by means of the aforementioned isomorphism, that the $\varrho$-classes associated to a metric with positive scalar curvature defined…
A continuous field of Roe algebras
本文研究了度量测度空间中受控 Delone 子集的紧致空间，通过赋予其一致 Roe 代数与 Spakula 版本的 Roe 代数，构建了重言式 $C^*$-代数族，并证明了在无孤立点的有界几何空间上可形成连续场。
The goal of this paper is to study when uniform Roe algebras have certain $C^*$-algebraic properties in terms of the underlying space: in particular, we study properties like having stable rank one or real rank zero that are thought of as low dimensional, and connect these to low dimensionality of the underlying space in the sense of the asymptotic dimension of Gromov. Some of these results (for example, on stable rank one, cancellation, strong quasidiagonality, and finite decomposition rank) give definitive characterizations, while others (on real rank zero) are only partial and leave a lot o…
We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated to a uniformly locally finite metric space $X$. Under weak assumptions, these $\mathrm{C}^*$-algebras contain embedded copies of $\prod_{k}\mathrm{M}_{n_k}(\mathbb C)$ for any \emph{bounded} countable (possibly finite) collection $(n_k)_k$ of natural numbers; we aim to show that they cannot contain any other von Neumann algebras. One of our main results shows that $L_\infty[0,1]$ does not embed into any of those algebras, even by a not-necessarily-normal $*$-homomorphism. In partic…
The goal of this paper is to solve the problem of existence of an $\ell^2$ relative eta morphism on the Higson-Roe structure group. Using the Cheeger-Gromov $\ell^2$ eta invariant, we construct a group morphism from the Higson-Roe maximal structure group constructed in [HiRo:10] to the reals. When we apply this morphism to the structure class associated with the spin Dirac operator for a metric of positive scalar curvature, we get the spin $\ell^2$ rho invariant. When we apply this morphism to the structure class associated with an oriented homotopy equivalence, we get the difference of the $\…
Very recently, Špakula and Tikuisis provide a new characterisation of (uniform) Roe algebras via quasi-locality when the underlying metric spaces have straight finite decomposition complexity. In this paper, we improve their method to deal with the $L^p$-version of (uniform) Roe algebras for any $p\in [1,\infty)$. Due to the lack of reflexivity on $L^1$-spaces, some extra work is required for the case of $p=1$.
Morita equivalence of two $\ell^p$ Roe-type algebras
本文证明有界几何度量空间的 $\ell^p$ 一致 Roe 代数与 $\ell^p$ 一致代数在 Lafforgue 意义下 Morita 等价，从而具有相同 $K$-理论，并构造了不总满射的 $\ell^p$ 一致粗装配映射。
Roe algebras of coarse spaces via coarse geometric modules
本文通过引入粗几何模与粗支集，将度量空间的Roe代数推广到一般粗空间，统一处理一致、非一致Roe代数及拟局部算子代数，并证明Roe代数的K-理论对粗空间具有函子性。
We define the concept of a partial translation structure T on a metric space X and we show that there is a natural C*-algebra C*(T) associated with it which is a subalgebra of the uniform Roe algebra C*_u(X). We introduce a coarse invariant of the metric which provides an obstruction to embedding the space in a group. When the space is sufficiently group-like, as determined by our invariant, properties of the Roe algebra can be deduced from those of C*(T). We also give a proof of the fact that the uniform Roe algebra of a metric space is a coarse invariant up to Morita equivalence.
Mapping graph homology to $K$-theory of Roe algebras
该研究针对图的同调理论与顶点度量空间的（一致）Roe代数$K$-理论，构建了二者间的自然映射及其一致版本，并证明当图为整数集的Cayley图时，该映射为同构。
Amenability for groups can be extended to metric spaces, algebras over commutative fields and $C^*$-algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra $C_u^*(X)$ over a metric space $(X,d)$ with bounded geometry. In particular, we show that the following conditions are equivalent: (1) $(X,d)$ is amenable; (2) the translation algebra generating $C_u^*(X)$ is algebraically amenable (3) $C_u^*(X)$ has a tracial state; (4) $C_u^*(X)$ is not pro…
We introduce the dual Roe algebras for proper étale groupoid actions and deduce the expected Higson-Roe short exact sequence. When the action is cocompact, we show that the Roe $C^*$-ideal of locally compact operators is Morita equivalent to the reduced $C^*$-algebra of our groupoid, and we further identify the boundary map of the associated periodic six-term exact sequence with the Baum-Connes map, via a Paschke-Higson map for groupoids. For proper actions on continuous families of manifolds of bounded geometry, we associate with any $G$-equivariant Dirac-type family, a coarse index class whi…
The rigidity problem for uniform Roe algebras was recently positively solved. Before its solution was found, there were positive solutions under the assumption of certain technical geometric conditions. In this paper, we introduce weaker versions of the operator norm localization property (ONL) which turn out to characterize those technical geometric conditions. We use this to obtain new rigidity results for nonmetrizable coarse spaces. As an application, we provide a novel partial answer to a question of White and Willett about Cartan subalgebras of uniform Roe algebras. We also study embeddi…
KMS states on uniform Roe algebras
本文研究一致Roe代数上一类自然流的KMS态，证明其通过对角算子分解，并揭示了强连续态的唯一性及其与Higson冠的联系，最后应用于$n$-分支树。
A Lichnerowicz Vanishing Theorem for the Maximal Roe Algebra
本文证明了在适当条件下，Dirac算子的极大高阶指标在极大等变Roe代数的K理论中消没，其关键是在极大等变一致Roe代数中建立泛函演算。
Let X be a proper metric space, which has finite asymptotic dimension in the sense of Gromov (or more generally, straight finite decomposition complexity of Dranishnikov and Zarichnyi). New descriptions are provided of the Roe algebra of X: (i) it consists exactly of operators which essentially commute with diagonal operators coming from Higson functions (that is, functions on X whose oscillation tends to 0 at infinity) and (ii) it consists exactly of quasi-local operators, that is, ones which have finite epsilon propogation (in the sense of Roe) for every epsilon>0. These descriptions hold bo…
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 bounded geometry yields a bijective coarse equivalence between the underlying metric spaces, while a stable isometric isomorphism yields a coarse equivalence. We also obtain similar results for other $\ell^p$ Roe-type algebras. In this paper, we do not assume that the metric spaces have Yu's property A or finite decomposition complexity.
Transfinite Extension of Nuclear Dimension
本文引入$C^*$-代数的超限核维数概念，证明均匀Roe代数的超限核维数受底层空间超限渐近维数约束，并由此推出渐近性质C空间的均匀Roe代数具有corona分解性质。
We prove the following two results for a given uniformly locally finite metric space with Yu's property A: 1) The group of outer automorphisms of its uniform Roe algebra is isomorphic to its group of bijective coarse equivalences modulo closeness. 2) The group of outer automorphisms of its Roe algebra is isomorphic to its group of coarse equivalences modulo closeness. The main difficulty lies in the latter. To prove that, we obtain several uniform approximability results for maps between Roe algebras and use them to obtain a theorem about the `uniqueness' of Cartan masas of Roe algebras. We fi…
We construct a slant product $/ \colon \mathrm{S}_p(X \times Y) \times \mathrm{K}_{1-q}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{S}_{p-q}(X)$ on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer. The latter is the domain of the co-assembly map $μ^\ast \colon \mathrm{K}_{1-\ast}(\mathfrak{c}^{\mathrm{red}}Y) \to \mathrm{K}^\ast(Y)$. We obtain such products on the entire Higson--Roe sequence. They imply injectivity results for external product maps. Our results apply to products with aspherical manifolds whose fundamental groups adm…
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 Schützenberger graphs. This graph, which we denote by $Λ_S$, inherits much of the structure of $S$. In this article we compare the C*-algebra $\mathcal{R}_S$, generated by the left regular representation of $S$ on $\ell^2(S)$ and $\ell^\infty(S)$, with the uniform Roe algebra over the metric space, namely $C^*_u(Λ_S)$. This yields a chacterization of when $\mathcal{R}…
Homotopy type of the unitary group of the uniform Roe algebra on $\mathbb{Z}^n$
本文研究 $\mathbb{Z}^n$ 上均匀 Roe 代数酉群的同伦型，证明其稳定化映射是同伦等价，并在 $n=1,2$ 时将其同伦型分解为 Roe 代数酉群与有理 Eilenberg--MacLane 空间的乘积。
We construct an isomorphism between the geometric model and Higson-Roe's analytic surgery group, reconciling the constructions in the previous papers in the series on "Realizing the analytic surgery group of Higson and Roe geometrically" with their analytic counterparts. Following work of Lott and Wahl, we construct a Chern character on the geometric model for the surgery group; it is a "delocalized Chern character", from which Lott's higher delocalized $ρ$-invariants can be retrieved. Following work of Piazza and Schick, we construct a geometric map from Stolz' positive scalar curvature seque…
C*-rigidity of bounded geometry metric spaces
本文证明了一致局部有限度量空间若Roe代数同构则粗等价，并得出其Roe代数外自同构群典范同构于空间粗等价群的结论。
A nonstanadard analysis approach to limit operators and Fredholmness in Roe-like algebras
本文利用非标准分析定义遥远星系上的极限算子，给出一致 Roe 或拟局部代数中广义 Fredholm 性及性质 A 下 Fredholm 性的可逆性刻画，并以更少极限算子强化既有结果。
The goal of this paper is to study band-dominated operators on Banach spaces with Schauder basis with respect to uniformly locally finite metric spaces as well as the Banach algebras generated by them: the so called uniform Roe algebras. We investigate several kinds of isomorphisms between those Banach spaces (e.g., isomorphisms preserving norm, order, algebraic structure, etc) and prove several rigidity results on when a certain kind of isomorphism between the uniform Roe algebras implies that the base metric spaces are (bijectively) coarsely equivalent.
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