Cyclic cohomology of crossed products by algebraic groups
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.
cyclic cohomology of crossed products of Lie groups
Connes-Kasparov conjecture in cyclic cohomology
scientific article; zbMATH DE number 549675
scientific article