KK-THEORY OF CIRCLE ACTIONS WITH THE ROKHLIN PROPERTY

Journal article, 2025

We investigate the structure of circle actions with the Rokhlin property, particularly in relation to equivariant KK-theory. Our main results are T-equivariant versions of celebrated results of Kirchberg: any Rokhlin action on a separable, nuclear C*-algebra is KKT-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 KKT-equivalent. In the presence of the UCT, KKT-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, we exhibit an example showing that an isomorphism between the KT-theories of Rokhlin actions on Kirchberg algebras does not necessarily lift to a KKT-equivalence; this is the first example of its kind, even in the absence of the Rokhlin property.