Let G be a connected, real semisimple Lie group. Let K < G be maximal compact, and let Gamma < G be discrete and such that Gamma\G has finite volume. If the real rank of G is 1 and P is torsion-free, then Barbasch and Moscovici obtained an index theorem for Dirac operators on the locally symmetric space Gamma\G/K. We obtain a higher version of this, using an index of Dirac operators on G/K in the K-theory of an algebra on which the conjugation-invariant terms in Barbasch and Moscovici's index theorem define continuous traces. The resulting index theorems also apply when P has torsion. The cases of these index theorems for traces defined by semisimple orbital integrals extend to Song and Tang's higher orbital integrals, and yield nonzero and computable results even when rank(G) > rank(K), or the real rank of G is larger than 1.
This note provides a counterexample showing that the assumptions that Chabert and Echterhoff have imposed in their permanence property of the Baum-Connes conjecture for group extensions cannot be simplified.
The reciprocality in this paper means a duality in Kirchberg algebras between strong extension groups and K-theory groups. We describe a construction of the reciprocal dual algebra A for a Kirchberg algebra A with finitely generated K-groups via K-theoretic duality for extensions. In particular, we may concretely construct and realize the reciprocal algebra O-A for a simple Cuntz-Krieger algebra O-A in several different ways. As a result, the algebra O-A is realized as a unital simple purely infinite universal C-& lowast;-algebra generated by a family of partial isometries subject to certain operator relations. We will finally study the gauge action on the reciprocal algebra O-A and prove that there exists an isomorphism between the fundamental groups pi(1)(Aut(O-A)) and pi(1)(Aut(O-A)) preserving their gauge actions.
This paper explores the cup and cap products within the cohomology and homology groups of ample groupoids, focusing on their applications and fundamental properties. Ample groupoids, which are étale groupoids with a totally disconnected unit space, play a crucial role in the study of topological dynamical systems and operator algebras. We introduce the cup product, which defines a bilinear map on cohomology classes, providing a graded ring structure, and the cap product, which defines a bilinear map relating homology and cohomology. The paper aims to make these concepts accessible to a broader mathematical audience, offering clear definitions and detailed explanations. We also demonstrate an application of the cap product in the analysis of automorphisms of groupoid C^*-algebras. Specifically, we show how it helps determine the asymptotic innerness of automorphisms. Our results include the first explicit computations of cup products in the cohomology of tiling spaces, which may pave the way for new research in this area.
Given the maximal compact subalgebra (A) of a split-real Kac-Moody algebra g(A) of type A, we study certain finite-dimensional representations of (A) that do not lift to the maximal compact subgroup K(A) of the minimal Kac-Moody group G(A) associated to g(A) but only to its spin cover Spin(A) described in [11]. Currently, four elementary of these so-called spin representations are known. We study their (ir)reducibility, semisimplicity, and lift to the group level. The interaction of these representations with the spin-extended Weyl group is used to derive a partial parametrization result of the representation matrices by the real roots of g(A).
