The Classification of Rokhlin Flows on $$\mathrm {C}^{*}$$-algebras

Abstract

We study flows on \(\mathrm {C}^{*}\)-algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on \(\mathrm {C}^{*}\)-algebras satisfying certain technical properties, which hold for many \(\mathrm {C}^{*}\)-algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimoto’s conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin flow on a separable, nuclear, \({\mathcal {O}}_\infty \)-absorbing \(\mathrm {C}^{*}\)-algebra is uniquely determined by its induced action on the prime ideal space. Secondly, we give a complete classification of Rokhlin flows on simple classifiable KK-contractible \(\mathrm {C}^{*}\)-algebras: Two Rokhlin flows on such a \(\mathrm {C}^{*}\)-algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate.