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

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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.

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Notes

  1. For convenience we restrict to the unital case for the moment; see Definition 2.8 for the general formal definition.

  2. This means that there exists an affine homeomorphism \(\gamma \) on the topological cone T(A) of lower-semicontinuous extended traces respecting the scale \(\Sigma _A\) such that \(\gamma (\tau \circ \alpha _t)=\gamma (\tau )\circ \beta _t\) for all \(\tau \in T(A)\) and \(t\in {\mathbb {R}}\).

  3. Indeed \(A_{\infty ,\alpha }\) coincides with the \(\mathrm {C}^{*}\)-subalgebra of elements in \(A_\infty \) on which the induced algebraic \({\mathbb {R}}\)-action is continuous; see [7, Theorem 2].

  4. In fact, just insert \(a={\mathbf {1}}\) in the previous calculations, although some steps then become redundant.

  5. The superscripts attached to flows in this context are not to be confused with taking powers of the involved automorphisms, which we will never consider throughout this proof.

  6. This case will only use functional calculus and not the assumption that \(A\cong A\otimes {\mathcal {O}}_\infty \).

  7. Note here that the general version of [30, Lemma 14.8] also involves a constant \(\sigma >0\), a finite set \({\mathcal {P}}{\subset \!\!\!\subset }\underline{K}(A)\) and a subgroup \(G_u\subset K_0(A)\), all depending on the pair \((\varepsilon ,{\mathcal {F}})\). The constant \(\sigma \) is redundant due to our assumption that u is in the closed commutator subgroup, and both \({\mathcal {P}}\) and \(G_u\) are redundant because we assume A to be KK-trivial.

  8. The actual construction of the path begins on the line after (e14.131) on page 110 in [30].

  9. Note that any flow acts trivially on the K-theory of A, so the existence of a trace-scaling flow on A implies that the pairing between \(K_0(A)\) and T(A) must be trivial.

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Acknowledgements

I have been partially supported by the following sources, either while carrying out the research, or while writing or revising this manuscript: EPSRC grant EP/N00874X/1; the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92); the European Union’s Horizon 2020 research and innovation programme under the grants MSCA-IF-2016-746272-SCCD and MSCA-RISE-2015-691246-QUANTUM DYNAMICS; the start-up grant STG/18/019 of KU Leuven; the research project C14/19/088 funded by the research council of KU Leuven; the research project G085020N funded by the Research Foundation Flanders (FWO). I would like to thank the following colleagues for valuable discussions, remarks or other interactions that have benefited this paper in one way or another: Akitaka Kishimoto, Marius Dadarlat, Norio Nawata, James Gabe, Huaxin Lin, Masaki Izumi, Selçuk Barlak, Guihua Gong, George Elliott, David Kerr, Aaron Tikuisis, and Hannes Thiel.

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    Gábor Szabó

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Szabó, G. The Classification of Rokhlin Flows on \(\mathrm {C}^{*}\)-algebras. Commun. Math. Phys. 382, 2015–2070 (2021). https://doi.org/10.1007/s00220-020-03812-2

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