Cyclic cohomology of crossed products by algebraic groups
type: Item
instance of: scholarly article
title: Cyclic cohomology of crossed products by algebraic groups
zbMATH Open document ID: 0799.46078
DOI: 10.1007/BF01232449
author: Victor Nistor
published in: Inventiones Mathematicae
publication date: 1994-11-17T00:00:00Z
full work available at URL: https://eudml.org/doc/144116
review text: From the author's Introduction: It is the purpose of this paper to study the cyclic cohomology of crossed products of Lie groups. The main result is the Connes-Kasparov conjecture in cyclic cohomology. Let \(G\) be a Lie group acting smoothly on a locally convex algebra \(A\) over \(\mathbb C\). Then \(C_ c^ \infty (G,A)\) with convolution product becomes a locally convex algebra denoted by \(A \rtimes G\). We now indicate our main result: Let \(G\) be a real algebraic group, \(k\) a maximal compact subgroup of \(G\) and \(C^ \infty_{\text{inv}} (G)\) the ring of smooth \(Ad_ G\)-invariant functions on \(G\). The periodic cyclic homology groups \(PHC_ *(A \rtimes G)\) and \(PHC_ *(A \rtimes k)\) are modules over \(C^ \infty_{\text{inv}} (G)\) such that \[ PHC_ *(A \rtimes G)_ m \simeq PHC_ *(A \rtimes k)_ m \] for any maximal ideal \(m\) of \(C^ \infty_{\text{inv}} (G)\). So in particular \(PHC_ *(A \rtimes G)_ m\) vanishes if \(m\) is not elliptic.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46M20
zbMATH DE Number: 549675
zbMATH Keywords: cyclic cohomology of crossed products of Lie groups
zbMATH Keywords: Connes-Kasparov conjecture in cyclic cohomology
MaRDI profile type: Publication
cites work: Q5606652
cites work: Cyclic homology and the Selberg principle
cites work: Q5607044
cites work: Q3885948
cites work: Q3317281
cites work: Non-commutative differential geometry
cites work: An analogue of the Thom isomorphism for crossed products of a C* algebra by an action of R
cites work: Cyclic cohomology for one-parameter smooth crossed products
cites work: Q3915179
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Q4720245
cites work: Q3781596
cites work: Equivariant KK-theory and the Novikov conjecture
cites work: Cyclic homology and the Lie algebra homology of matrices
cites work: Q3999621
cites work: Q3348215
cites work: Cyclic cohomology of crossed products with \({\mathbb{Z}}\)
cites work: Group cohomology and the cyclic cohomology of crossed products
cites work: A bivariant Chern character for \(p\)-summable quasihomomorphisms
cites work: A bivariant Chern-Connes character
cites work: KK-groups of crossed products by groups acting on trees
cites work: Q3930248
cites work: Q3316025
cites work: Q3667036
cites work: Q3228642
cites work: Q5682517
description: scientific article; zbMATH DE number 549675