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; any Rokhlin action on an exact separable C*-algebra embeds equivariantly into $\mathcal{O}_2$ (with its unique Rokhlin action); 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 UCT, $KK^\mathbb{T}$-equivalence for Rokhlin actions reduces to isomorphism of a $K$-theoretical invariant, namely of a canonical pure extension naturally associated to 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, an isomorphism between the $K^\mathbb{T}$-theories of Rokhlin actions on Kirchberg algebras does not necessarily lift to a $KK^\mathbb{T}$-equivalence.