arXiv subjects
Publications and source records attributed to Paul Baum.
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.
We conjecture the existence of a simple geometric structure underlying questions of reducibility of parabolically induced representations of reductive p-adic groups.
We prove that if the classical Baum-Connes conjecture in complex K-theory is true (for a given discrete group G), then the conjecture is also true in the real case (for the same group G). The essential ingredients of the proof are the descent theorem in topological K-theory (math.KT/0509396) and a Paschke duality for C*-algebras proved by John Roe (Q.J. Math. 55 (2004) 325-331). According to S. Stolz, this result is linked with the existence of a Riemannian metric of positive scalar curvature on a compact connected spin manifold with G as fundamental group.
Let H(G) be the Hecke algebra of a reductive p-adic group G. We formulate a conjecture for the ideals in the Bernstein decomposition of H(G). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig's conjecture on the asymptotic Hecke algebra. We prove our conjecture for SL(2) and GL(n). We also prove part (1) of our conjecture for the Iwahori ideals of the groups PGL(n) and SO(5).
We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C*-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C*-algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts.
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.
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 $ϕ_q : \Hecke_q \to J$, to a fixed finite type algebra $J$. This proves that $ϕ_q$ 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.
Let F be a global field, A its ring of adeles, G a reductive group over F. We prove the Baum-Connes conjecture for the adelic group G(A).