Bounded Derivations on Uniform Roe Algebras
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.
math.OA↗
arXiv subjects
Rufus Willett
Publications and source records attributed to Rufus Willett.
Cartan subalgebras in uniform Roe algebras
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.
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Injectivity, crossed products, and amenable group actions
This paper is motivated primarily by the question of when the maximal and reduced crossed products of a $G$-$C^*$-algebra agree (particularly inspired by results of Matsumura and Suzuki), and the relationships with various notions of amenability and injectivity. We give new connections between these notions. Key tools in this include the natural equivariant analogues of injectivity, and of Lance's weak expectation property: we also give complete characterizations of these equivariant properties, and some connections with injective envelopes in the sense of Hamana.
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Bott periodicity and almost commuting matrices
We give a proof of the Bott periodicity theorem for topological K-theory of C*-algebras based on Loring's treatment of Voiculescu's almost commuting matrices and Atiyah's rotation trick. We also explain how this relates to the Dirac operator on the circle; this uses Yu's localization algebra and an associated explicit formula for the pairing between the first K-homology and first K-theory groups of a (separable) C*-algebra.
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The maximal injective crossed product
A crossed product functor is said to be injective if it takes injective morphisms to injective morphisms. In this paper we show that every locally compact group $G$ admits a maximal injective crossed product $A\mapsto A\rtimes_{\inj}G$. Moreover, we give an explicit construction of this functor that depends only on the maximal crossed product and the existence of $G$-injective $C^*$-algebras; this is a sort of a `dual' result to the construction of the minimal exact crossed product functor, the latter having been studied for its relationship to the Baum-Connes conjecture. It turns out that $\rtimes_\inj$ has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property.
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Localization C*-algebras and K-theoretic duality
Based on the localization algebras of Yu, and their subsequent analysis by Qiao and Roe, we give a new picture of KK-theory in terms of time-parametrized families of (locally) compact operators that asymptotically commute with appropriate representations.
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Low dimensional properties of uniform Roe algebras
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 open. We also establish results about $K$-theory, showing that all $K_0$-classes come from the inclusion of the canonical Cartan in low dimensional cases, but not in general; in particular, our $K$-theoretic results answer a question of Elliott and Sierakowski about vanishing of $K_0$ groups for uniform Roe algebras of non-amenable groups. Along the way, we extend some results about paradoxicality, proper infiniteness of projections in uniform Roe algebras, and supramenability from groups to general metric spaces. These are ingredients needed for our $K$-theoretic computations, but we also use them to give new characterizations of supramenability for metric spaces.
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Roe C*-algebra for groupoids and generalized Lichnerowicz Vanishing theorem for foliated manifolds
We introduce the concept of Roe C*-algebra for a locally compact groupoid whose unit space is in general not compact, and that is equipped with an appropriate coarse structure and Haar system. Using Connes' tangent groupoid method, we introduce an analytic index for an elliptic differential operator on a Lie groupoid equipped with additional metric structure, which takes values in the K-theory of the Roe C*-algebra. We apply our theory to derive a Lichnerowicz type vanishing result for foliations on open manifolds.
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Dynamic Asymptotic Dimension: relation to dynamics, topology, coarse geometry, and $C^*$-algebras
We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact étale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lück, and Reich in the context of controlled topology, and its connections with Gromov's theory of asymptotic dimension. We also show that dynamic asymptotic dimension gives bounds on the nuclear dimension of Winter and Zacharias for C*-algebras associated to dynamical systems. Dynamic asymptotic dimension also has implications for K-theory and manifold topology: these will be drawn out in subsequent work.
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Exotic Crossed Products
An exotic crossed product is a way of associating a C*-algebra to each C*-dynamical system that generalizes the well-known universal and reduced crossed products. Exotic crossed products provide natural generalizations of, and tools to study, exotic group C*-algebras as recently considered by Brown-Guentner and others. They also form an essential part of a recent program to reformulate the Baum-Connes conjecture with coefficients so as to mollify the counterexamples caused by failures of exactness. In this paper, we survey some constructions of exotic group algebras and exotic crossed products. Summarising our earlier work, we single out a large class of crossed products --- the correspondence functors --- that have many properties known for the maximal and reduced crossed products: for example, they extend to categories of equivariant correspondences, and have a compatible descent morphism in KK-theory. Combined with known results on K-amenability and the Baum-Connes conjecture, this allows us to compute the K-theory of many exotic group algebras. It also gives new information about the reformulation of the Baum-Connes Conjecture mentioned above. Finally, we present some new results relating exotic crossed products for a group and its closed subgroups, and discuss connections with the reformulated Baum-Connes conjecture.
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Exotic crossed products and the Baum-Connes conjecture
We study general properties of exotic crossed-product functors and characterise those which extend to functors on equivariant C*-algebra categories based on correspondences. We show that every such functor allows the construction of a descent in KK-theory and we use this to show that all crossed products by correspondence functors of K-amenable groups are KK-equivalent. We also show that for second countable groups the minimal exact Morita compatible crossed-product functor used in the new formulation of the Baum-Connes conjecture by Baum, Guentner and Willett extends to correspondences when restricted to separable G-C*-algebras. It therefore allows a descent in KK-theory for separable systems.
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A non-amenable groupoid whose maximal and reduced $C^*$-algebras are the same
We construct a locally compact groupoid with the properties in the title. Our example is based closely on constructions used by Higson, Lafforgue, and Skandalis in their work on counterexamples to the Baum-Connes conjecture. It is a bundle of countable groups over the one point compactification of the natural numbers, and is Hausdorff, second countable and étale.
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Expanders, exact crossed products, and the Baum-Connes conjecture
We reformulate the Baum-Connes conjecture with coefficients by introducing a new crossed product functor for C*-algebras. All confirming examples for the original Baum-Connes conjecture remain confirming examples for the reformulated conjecture, and at present there are no known counterexamples to the reformulated conjecture. Moreover, some of the known expander-based counterexamples to the original Baum-Connes conjecture become confirming examples for our reformulated conjecture.
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A metric approach to limit operators
We extend the limit operator machinery of Rabinovich, Roch, and Silbermann from $\mathbb{Z}^N$ to (bounded geometry, strongly) discrete metric spaces. We do not assume the presence of any group structure or action on our metric spaces. Using this machinery and recent ideas of Lindner and Seidel, we show that if a metric space X has Yu's property A, then a band-dominated operator on X is Fredholm if and only if all of its limit operators are invertible. We also show that this always fails for metric spaces without property A.
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Geometric Property (T)
This paper discusses `geometric property (T)'. This is a property of metric spaces introduced in earlier work of the authors for its applications to K-theory. Geometric property (T) is a strong form of `expansion property': in particular for a sequence of finite graphs $(X_n)$, it is strictly stronger than $(X_n)$ being an expander in the sense that the Cheeger constants $h(X_n)$ are bounded below. We show here that geometric property (T) is a coarse invariant, i.e. depends only on the large-scale geometry of a metric space $X$. We also discuss the relationships between geometric property (T) and amenability, property (T), and various coarse geometric notions of a-T-menability. In particular, we show that property (T) for a residually finite group is characterised by geometric property (T) for its finite quotients.
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Random graphs, weak coarse embeddings, and higher index theory
This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse Baum-Connes assembly map is injective; the coarse Baum-Connes assembly map is not surjective; the maximal coarse Baum-Connes assembly map is an isomorphism. These results are closely tied to issues of expansion in graphs: in particular, we also show that such random sequences almost surely do not have geometric property (T), a strong form of expansion. The key geometric ingredients in the proof are due to Mendel and Naor: in our context, their results imply that a random sequence of graphs almost surely admits a weak form of coarse embedding into Hilbert space.
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On Rigidity of Roe algebras
Roe algebras are C*-algebras built using large-scale (or 'coarse') aspects of a metric space (X,d). In the special case that X=G is a finitely generated group and d is a word metric, the simplest Roe algebra associated to (G,d) is isomorphic to the reduced crossed product C*-algebra l^\infty(G)\rtimes G. Roe algebras are 'coarse invariants', in the sense that if X and Y are coarsely equivalent metric spaces, then their Roe algebras are isomorphic. Motivated in part by the coarse Baum-Connes conjecture, we ask if there is a converse to the above statement: that is, if X and Y are metric spaces with isomorphic Roe algebras, must X and Y be coarsely equivalent? We show that for very large classes of spaces the answer to this question, and some related questions, is 'yes'. This can be thought of as a 'C*-rigidity result': it shows that the Roe algebra construction preserves a large amount of information about the space, and is thus surprisingly 'rigid'. As an example of our results, in the group case we have that if G and H are finitely generated elementary amenable, hyperbolic, or linear, groups such that the crossed products l^\infty(G)\rtimes G and l^\infty(H)\rtimes H are isomorphic, then G and H are quasi-isometric.
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Ghostbusting and property A
We show that a bounded geometry metric space X has property A if and only if all ghost operators on X are compact.
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