KK-theory of circle actions with the Rokhlin property

We investigate the structure of circle actions with the Rokhlin property, particularly in relation to equivariant $KK$ -theory. Our main results are $\mathbb {T}$ -equivariant versions of celebrated results of Kirchberg: any Rokhlin action on a separable, nuclear C*-algebra is $KK<^>{\mathbb {T}}$ -equivalent to a Rokhlin action on a Kirchberg algebra; and two circle actions with the Rokhlin property on a Kirchberg algebra are conjugate if and only if they are $KK<^>{\mathbb {T}}$ -equivalent.In the presence of the Universal Coefficient Theorem (UCT), $KK<^>{\mathbb {T}}$ -equivalence for Rokhlin actions reduces to isomorphism of a K-theoretical invariant, namely of a canonical pure extension naturally associated with any Rokhlin action, and we provide a complete description of the extensions that arise from actions on nuclear $C<^>*$ -algebras. In contrast with the non-equivariant setting, we exhibit an example showing that an isomorphism between the $K<^>{\mathbb {T}}$ -theories of Rokhlin actions on Kirchberg algebras does not necessarily lift to a $KK<^>{\mathbb {T}}$ -equivalence; this is the first example of its kind, even in the absence of the Rokhlin property.