Periodic cyclic homology of Iwahori-Hecke algebras

We determine the periodic cyclic homology of the Iwahori-Hecke algebras \Hecke_q, for q \in \CC^* not a ``proper root of unity.'' (In this paper, by a {\em proper root of unity} we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a ``weakly spectrum preserving'' morphism of finite type algebras induces an isomorphism in periodic cyclic homology. The concept of a weakly spectrum preserving morphism is defined in this paper, and most of our work is devoted to understanding this class of morphisms. Results of Kazhdan--Lusztig and Lusztig show that, for the indicated values of $q$, there exists a weakly spectrum preserving morphism \phi_q : \Hecke_q \to J, to a fixed finite type algebra $J$. This proves that $\phi_q$ induces an isomorphism in periodic cyclic homology and, in particular, that all algebras \Hecke_q have the same periodic cyclic homology, for the indicated values of $q$. The periodic cyclic homology groups of the algebra \Hecke_1 can then be determined directly, using results of Karoubi and Burghelea, because it is the group algebra of an extended affine Weyl group.Comment: 24 pages, LaTe

K-theory for group C*-algebras

These notes are based on a lecture course given by the first author in the Sedano Winter School on K-theory held in Sedano, Spain, on January 22-27th of 2007. They aim at introducing K-theory of C*-algebras, equivariant K-homology and KK-theory in the context of the Baum-Connes conjectur

Deflation at Turnaround for Oscillatory Cosmology

It is suggested that dark energy in a brane world can help reconcile an infinitely cyclic cosmology with the second law of thermodynamics. A cyclic cosmology is described, in which dark energy with constant equation of state leads to a turnaround at finite future time, when entropy is decreased by a huge factor equal to the inverse of its enhancement during the initial inflation. Thermodynamic consistency of cyclicity requires the arrow of time to reverse during contraction. Entropy reduction in the contracting phase is infinitesimally smaller than entropy increase during expansion.Comment: 11 pages late

On the Equivalence of Geometric and Analytic K-Homology

We give a proof that the geometric K-homology theory for finite CW-complexes defined by Baum and Douglas is isomorphic to Kasparov's K-homology. The proof is a simplification of more elaborate arguments which deal with the geometric formulation of equivariant K-homology theory.Comment: 29 pages, v4: corrected definition of E in proof of Prop 3.

Local-global principle for the Baum-Connes conjecture with coefficients

We establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group $GL(2,F)$ where $F$ is any global field