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In recent years, a large class of nuclear $C^\ast$-algebras have been classified, modulo an assumption on the Universal Coefficient Theorem (UCT). We think this assumption is redundant and propose a strategy for proving it. Indeed, following the original proof of the classification theorem, we propose bridging the gap between reduction theorems and examples. While many such bridges are possible, various approximate ideal structures appear quite promising., Comment: 13 pages; to appear in the Rocky Mountain Journal of Mathematics

In this work we introduce and study a new notion of amenability for actions of locally compact groups on $C^*$-algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and permanence properties analogous to those known in the discrete case. For example, for actions on commutative $C^*$-algebras, we show that our notion of amenability is equivalent to measurewise amenability. Combined with a recent result of Alex Bearden and Jason Crann, this also settles a long standing open problem about the equivalence of topological amenability and measurewise amenability for a second countable $G$-space $X$. We use our new notion of amenability to study when the maximal and reduced crossed products agree. One of our main results generalizes a theorem of Matsumura: we show that for an action of an exact locally compact group $G$ on a locally compact space $X$ the full and reduced crossed products $C_0(X)\rtimes_\max G$ and $C_0(X)\rtimes_{\operatorname{red}} G$ coincide if and only if the action of $G$ on $X$ is amenable. We also show that the analogue of this theorem does not hold for actions on noncommutative $C^*$-algebras. Finally, we study amenability as it relates to more detailed structure in the case of $C^*$-algebras that fibre over an appropriate $G$-space $X$, and the interaction of amenability with various regularity properties such as nuclearity, exactness, and the (L)LP, and the equivariant versions of injectivity and the WEP., Comment: Some small changes (mostly in the introduction) and an adjustment to the AMS book style. This is the final version which will appear in the Memoirs of the AMS

We introduce the concept of coarse groupoid cohomology for a groupoid equipped with a coarse structure. We construct a Connes-Chern character map from the coarse groupoid cohomology to the cyclic cohomology of the smooth coarse Roe algebra of the groupoid. As an example, we construct a cyclic coarse cocycle on the holonomy groupoid associated to a flat principal $G$-bundle for a Lie group $G$.

We introduce a notion of approximate ideal structure for a $C^*$-algebra, and use it as a tool to study $K$-theory groups. The notion is motivated by the classical Mayer-Vietoris sequence, by the theory of nuclear dimension as introduced by Winter and Zacharias, and by the theory of dynamical complexity introduced by Guentner, Yu, and the author. A major inspiration for our methods comes from recent work of Oyono-Oyono and Yu in the setting of controlled $K$-theory of filtered C*-algebras; we do not, however, use that language in this paper. We give two main applications. The first is a vanishing result for $K$-theory that is relevant to the Baum-Connes conjecture. The second is a permanence result for the K\"{u}nneth formula in $C^*$-algebra $K$-theory: roughly, this says that if $A$ can be decomposed into a pair of subalgebras $(C,D)$ such that $C$, $D$, and $C\cap D$ all satisfy the K\"{u}nneth formula, then $A$ itself satisfies the K\"{u}nneth formula., Comment: 65 pages

We introduce a notion of topological property (T) for ��tale groupoids. This simultaneously generalizes Kazhdan's property (T) for groups and geometric property (T) for coarse spaces. One main goal is to use this property (T) to prove the existence of so-called Kazhdan projections in both maximal and reduced groupoid $C^*$-algebras, and explore applications of this to exactness, K-exactness, and the Baum-Connes conjecture. We also study various examples, and discuss the relationship with other notions of property (T) for groupoids and with a-T-menability., 56 pages. Version 2 makes (fairly minor) corrections and clarifications

In this paper, we introduce a property of topological dynamical systems that we call finite dynamical complexity. For systems with this property, one can in principle compute the $K$-theory of the associated crossed product $C^*$-algebra by splitting it up into simpler pieces and using the methods of controlled $K$-theory. The main part of the paper illustrates this idea by giving a new proof of the Baum-Connes conjecture for actions with finite dynamical complexity. We have tried to keep the paper as self-contained as possible: we hope the main part will be accessible to someone with the equivalent of a first course in operator $K$-theory. In particular, we do not assume prior knowledge of controlled $K$-theory, and use a new and concrete model for the Baum-Connes conjecture with coefficients that requires no bivariant $K$-theory to set up., Comment: Second version has extensive revisions: chiefly, the main argument has been reworked based on a new notion of `finite dynamical complexity'. This is partly to correct an error, and partly as the new version seemed simpler and more natural. A new appendix has been added comparing finite dynamical complexity to other properties. Third version corrects a minor error in Definition 7.4

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.

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 \'{e}tale., Comment: Version two has some clarifications, minor fixes, and additional background and references

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.