Suppose that*G* is a second countable compact Lie group and that*A* and*B* are commutative*G-C**-algebras. Then the Kasparov group*KK*_{*}^{G}
(*A, B*) is a bifunctor on*G*-spaces. It is computed here in terms of equivariant stable homotopy theory. This result is a consequence of a more general study of equivariant Spanier-Whitehead duality and uses in an essential way the extension of the Kasparov machinery to the setting of σ-*G-C**-algebras. As a consequence, we show that if (*X, x*_{0}) is a based separable compact metric*G*-ENR (such as a smooth compact*G*-manifold) and (*Y, y*_{0}) is a based countable*G*-CW-complex then there is a natural isomorphism
$$KK_*^G (C(X,x_0 ),C(Y,y_0 )) \cong K_G^* (Y \wedge FX)$$
where F*X* is the functional equivariant Spanier-Whitehead dual of*X*. This specializes when*Y* is trivial to yield a natural isomorphism
$$KK_*^G (C(X,x_0 ),\mathbb{F}) \cong {}^sK_*^G (X)$$
where^{s}*K*_{*}^{G}
(−) denotes equivariant Steenrod*K*-homology theory. This result is new even for*X* a finite*G*-CW-complex, in which case Steenrod*K*-homology coincides with the usual topological equivariant*K*-homology*K*_{*}^{G}
(*X*).