  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 $KK^{G}_{*}(A, B) = 0$.
