Restriction maps in equivariant $KK$-theory

TL;DR

This paper generalizes restriction map results from equivariant K-theory to bivariant KK-theory for compact Lie groups, showing vanishing conditions for KK-groups based on finite cyclic subgroups.

Contribution

It extends McClure's restriction map results to equivariant KK-theory, providing new vanishing criteria for KK-groups in the context of compact Lie groups.

Findings

If KK^{F}_{*}(A, B) = 0 for all finite cyclic subgroups F, then KK^{G}_{*}(A, B) = 0.

KK^{H}_{n}(A, B) is finitely generated over R(G) for all closed subgroups H.

The result applies to G-C*-algebras with finitely generated KK-groups.

Abstract

We extend McClure's results on the restriction maps in equivariant $K$-theory to bivariant $K$-theory: Let $G$ be a compact Lie group and $A$ and $B$ be $G$-$C^*$-algebras. Suppose that $KK^{H}_{n}(A, B)$ is a finitely generated $R(G)$-module for every $H \le G$ closed and $n \in \Z$. Then, if $KK^{F}_{*}(A, B) = 0$ for all $F \le G$ {\em finite cyclic}, then $KK^{G}_{*}(A, B) = 0$.