Poly-ℤ Group Actions on Kirchberg Algebras I

Izumi, Masaki; Matui, Hiroki

doi:10.1093/imrn/rnz140

Cited by 20 publications

(31 citation statements)

References 49 publications

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“…This deep theorem states that Kirchberg algebras are completely classified by KK-theory. This motivates further study on Kirchberg algebras, especially on their symmetrical structure and inclusions (see e.g., [41], [23], [24], [26]). We point out that this subject also has a strong connection to the index theory (see e.g., [21], [22]).…”

Section: Introductionmentioning

confidence: 97%

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Suzuki

2019

Commun. Math. Phys.

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Practically and intrinsically, inclusions of operator algebras are of fundamental interest. The subject of this paper is intermediate operator algebras of inclusions. There are two previously known theorems which naturally and completely describe all intermediate operator algebras: the Galois Correspondence Theorem and the Tensor Splitting Theorem. Here we establish the third, new complete description theorem which gives a canonical bijective correspondence between intermediate operator algebras and intermediate extensions of dynamical systems. One can also regard this theorem as a crossed product splitting theorem, analogous to the Tensor Splitting Theorem. We then give concrete applications, particularly to maximal amenability problem and a new realization result of intermediate operator algebra lattice.2000 Mathematics Subject Classification. Primary 46L05, 46L10, Secondary 46L55, 54H20.

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Section: Introductionmentioning

confidence: 97%

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Suzuki

2019

Commun. Math. Phys.

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“…Results of Matui [14,15] and Izumi-Matui [11] have previously shown that the above is true for Z d -actions on all the known strongly self-absorbing C *algebras except for the Jiang-Su algebra Z. Sato [22] has shown such a uniqueness for Z-actions on Z, and Matui-Sato [16] have extended this also to Z 2 -actions on Z. We note that the uniqueness for actions of the Klein bottle group Z ⋊ −1 Z is also known by further work of Matui-Sato [17] on UHF algebras as well as Z; this was the first classification result for actions of non-abelian infinite groups on stably finite C * -algebras.…”

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confidence: 99%

“…It seems natural to expect that the known uniqueness results for Z dactions from [14,15,11] could be reproved abstractly within the common framework of semi-strongly self-absorbing actions, and without requiring the UCT assumption. It also seems plausible that this should in fact be possible for not necessarily abelian acting groups.…”

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confidence: 99%

Strongly self-absorbing C⁎-dynamical systems, III

Szabó

2017

Advances in Mathematics

102

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In this paper, we accomplish two objectives. Firstly, we extend and improve some results in the theory of (semi-)strongly selfabsorbing C * -dynamical systems, which was introduced and studied in previous work. In particular, this concerns the theory when restricted to the case where all the semi-strongly self-absorbing actions are assumed to be unitarily regular, which is a mild technical condition. The central result in the first part is a strengthened version of the equivariant McDuff-type theorem, where equivariant tensorial absorption can be achieved with respect to so-called very strong cocycle conjugacy.Secondly, we establish completely new results within the theory. This mainly concerns how equivariantly Z-stable absorption can be reduced to equivariantly UHF-stable absorption with respect to a given semistrongly self-absorbing action. Combining these abstract results with known uniqueness theorems due to Matui and Izumi-Matui, we obtain the following main result. If G is a torsion-free abelian group and D is one of the known strongly self-absorbing C * -algebras, then strongly outer G-actions on D are unique up to (very strong) cocycle conjugacy. This is new even for Z 3 -actions on the Jiang-Su algebra.

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“…Roughly speaking, a tensorial factorization of the form A ∼ = A ⊗ D -for a given C *algebra A and a strongly self-absorbing C * -algebra D -provides sufficient space to perform non-trival manipulations on elements inside A, which often gives rise to structural properties of particular interest for classification. The underlying motivation behind [82,83,80] is the idea that this kind of phenomenon should persist at the level of C * -dynamics if one is interested in classification of group actions up to cocycle conjugacy; in fact some much earlier work [55,56,40,29,63,65,39] has (sometimes implicitly) used this idea to reasonable success. It was further demonstrated in [80,81] how this approach can indeed give rise to new insights about classification or rigidity of group actions on certain C * -algebras, in particular strongly self-absorbing ones.…”

Section: Introductionmentioning

confidence: 99%

“…Initially appearing in works of Herman-Jones [30] and Herman-Ocneanu [31], the Rokhlin property for single automorphisms and its applications for classification were perfected in works of Kishimoto and various collaborators [8,50,5,52,21,53,54,18,6,68]. Further work pushed these techniques to actions of infinite higher-rank groups as well [67,44,60,61,62,40,39]. The case of finite groups was treated in work of Izumi [37,38], where it was shown that such actions with the Rokhlin property have a particularly rigid theory; see also [75,28,22,23,24,1,2].…”

Section: Introductionmentioning

confidence: 99%

Rokhlin dimension: absorption of model actions

Szabó

2019

Analysis & PDE

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In this paper, we establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly selfabsorbing C * -algebras. Namely, as to be made precise in the paper, let G be a well-behaved locally compact group. If D is a strongly selfabsorbing C * -algebra, and α : G A is an action on a separable, Dabsorbing C * -algebra that has finite Rokhlin dimension with commuting towers, then α tensorially absorbs every semi-strongly self-absorbing Gactions on D. In particular, this is the case when α satisfies any version of what is called the Rokhlin property, such as for G = R or G = Z k . This contains several existing results of similar nature as special cases. We will in fact prove a more general version of this theorem, which is intended for use in subsequent work. We will then discuss some nontrivial applications. Most notably it is shown that for any k ≥ 1 and on any strongly self-absorbing Kirchberg algebra, there exists a unique R k -action having finite Rokhlin dimension with commuting towers up to (very strong) cocycle conjugacy.

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Poly-ℤ Group Actions on Kirchberg Algebras I

Cited by 20 publications

References 49 publications

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Strongly self-absorbing C⁎-dynamical systems, III

Rokhlin dimension: absorption of model actions

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