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We study the C*-algebra crossed product $C_0(X)\rtimes G$ of a locally compact group $G$ acting properly on a locally compact Hausdorff space $X$. Under some mild extra conditions, which are automatic if $G$ is discrete or a Lie group, we…

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…

We prove definable versions of the Universal Coefficient Theorems of Eilenberg--Mac Lane expressing the (Steenrod) homology groups of a compact metrizable space in terms of its integral cohomology groups, and the (\v{C}ech) cohomology…

We introduce the spatial Rokhlin property for actions of coexact compact quantum groups on $\mathrm{C}^*$-algebras, generalizing the Rokhlin property for both actions of classical compact groups and finite quantum groups. Two key…

We establish a new connection between local and large-scale structure in compactly generated totally disconnected locally compact (t.d.l.c.) groups $G$, finding a sufficient condition for $G$ to have more than one end in terms of its…

Let X be a compact almost complex manifold with an action of a finite group G. We compute the algebra of G^n coinvariants of the stringy cohomology (math.AG/0104207) of X^n with an action of a wreath product of G. We show that it is…

Many geometric learning problems require invariants on heterogeneous product spaces, i.e., products of distinct spaces carrying different group actions, where standard techniques do not directly apply. We show that, when a group $G$ acts…

We prove that if $G$ is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action of $G$ on ${\underset g \in G \to \Pi} (X_0, \mu_0)_g$ for $(X_{0},\mu_{0})$ an arbitrary…

Given a pseudo-free self-similar action of a countable group $G$ on a countable directed graph $E$ with amenable stabilizers of the vertices, we identify the exact conditions under which these stabilizers do not contribute to the ideal…

Given a separable metrisable space X, and a group G of homeomorphisms of X, we introduce a topological property of the action of G on X which is equivalent to the existence of a G-invariant compatible metric on X. This extends a result of…

Let the groupoid $G$ with unit space $G^0$ act via a representation $\rho$ on a $C^*$-correspondence ${\mathcal H}$ over the $C_0(G^0)$-algebra $A$. By the universal property, $G$ acts on the Cuntz-Pimsner algebra ${\mathcal O}_{\mathcal…

Let $\Gamma$ be a countable group that admits an essential measurable splitting (for instance, any group measure equivalent to a free product of nontrivial groups). We show: (1) for any two nontrivial countable groups $B$ and $C$ that are…

The aim of the article is to provide characterizations of the Haage-rup property for locally compact, second countable groups in terms of approximations of some non-ergodic invariant states by mixing ones for actions on unital…

We introduce a new construction, the isotropy groupoid, to organize the orbit data for split $\Gamma$-spaces. We show that equivariant principal $G$-bundles over split $\Gamma$-CW complexes $X$ can be effectively classified by means of…

Let $G$ be a compact connected Lie group and $K$ a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of $G$ and $K$ is invertible in a given principal ideal domain $k$. It is known that in…

In this paper we give a short, direct proof, using only properties of the Haagerup tensor product, that if an operator algebra A possesses a diagonal in the Haagerup tensor product of A with itself, then A must be isomorphic to a finite…

Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a…

If $X$ is a locally compact space which admits commuting free and proper actions of locally compact groups $G$ and $H$, then the Brauer groups $\Br_H(G/X)$ and $\Br_G(X/H)$ are naturally isomorphic.

We prove that for a compact subgroup $H$ of an almost connected locally compact Hausdorff group $G$, the following properties are mutually equivalent: (1) $H$ is a maximal compact subgroup of $G$, (2) $G/H$ is contractible, (3) $G/H$ is…

Let G be a locally compact group, let X be a universal proper G-space, and let Z be a G-equivariant compactification of X that is H-equivariantly contractible for each compact subgroup H of G. Let W be the resulting boundary. Assuming the…