Poly- $${\mathbb {Z}}$$ Z group actions on Kirchberg algebras II
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Poly-Z group actions on Kirchberg algebras II Masaki Izumi1 · Hiroki Matui2
Received: 11 July 2019 / Accepted: 30 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This is the second part of our serial work on the classification of polyZ group actions on Kirchberg algebras. Based on technical results obtained in our previous work, we completely reduce the problem to the classification of continuous fields of Kirchberg algebras over the classifying spaces. As an application, we determine the number of cocycle conjugacy classes of outer Zn -actions on the Cuntz algebras. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . 3 Cohomology vanishing via homotopy fixed points 3.1 First cohomology vanishing . . . . . . . . . . 3.2 Second cohomology vanishing . . . . . . . . 4 Proof of Theorem 1.9 . . . . . . . . . . . . . . . 4.1 Step I . . . . . . . . . . . . . . . . . . . . . 4.2 Step II . . . . . . . . . . . . . . . . . . . . . 4.3 Step III . . . . . . . . . . . . . . . . . . . . . 5 Dynamical realization theorem . . . . . . . . . .
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M. Izumi: Supported in part by JSPS KAKENHI Grant No. JP15H03623. H. Matui: Supported in part by JSPS KAKENHI Grant No. JP18K03321.
B Masaki Izumi
[email protected]
1
Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
2
Graduate School of Science, Chiba University, Inage-ku, Chiba 263-8522, Japan
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M. Izumi, H. Matui 6 Primary obstructions . . . . . . . . . . . . . . . . . . . . . . . 7 The Cuntz algebra case . . . . . . . . . . . . . . . . . . . . . 8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Proof of Theorem 2.6 . . . . . . . . . . . . . . . . . . . . 8.2 The map A,X comes from a weak homotopy equivalence References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction This is a continuation of our previous work [19] on the classification of discrete amenable group actions on Kirchberg algebras. Kirchberg algebras are surely the most prominent class among classifiable amenable C ∗ -algebras, and it should be the first place to work on in order to establish classification theory of amenable group actions comparable to the case of von Neumann algebras, where satisfactory classification theory is known. The reader is referred to [41] and [44] for the classification of Kirchberg algebr
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