A Rohlin Property for One-parameter Automorphism Groups of the Hyperfinite $\mathrm{II}_1$ Factor

Kawamuro, Keiko

doi:10.2977/prims/1195142813

Cited by 12 publications

(12 citation statements)

References 6 publications

(7 reference statements)

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“…Before delving deeper on the C * -algebraic side, it is worthwhile to take a look back into the theory of von Neumann algebras. Based on Kishimoto's ideas for C * -algebras and building on some preliminary work of Kawamuro [42], Masuda-Tomatsu [64] have recently introduced the Rokhlin property for flows on von Neumann algebras. With it, they gave a complete classification of Rokhlin flows on a von Neumann algebra with separable predual.…”

Section: Definition Let α : Rmentioning

confidence: 99%

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

Szabó

2020

Commun. Math. Phys.

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We study flows on 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 C * -algebras satisfying certain technical properties, which hold for many 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, O∞-absorbing 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 C * -algebras: Two Rokhlin flows on such a C * -algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate. Contents 19 4. O ∞ -absorbing C * -algebras 30 5. Simple KK-contractible algebras 42 References 54

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Section: Definition Let α : Rmentioning

confidence: 99%

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

Szabó

2020

Commun. Math. Phys.

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“…However, the problem is not so simple, compared with the problem for discrete group actions. As a candidate for appropriate outerness, the Rohlin property was introduced by Kishimoto [16] and Kawamuro [15]. In the following, we will explain the definition.…”

Section: 2mentioning

confidence: 99%

“…It seems to be reasonable to think of flows with full Connes spectra to be outer. However, the problem is that these outernesses do not coincide (See Example 2.3 of Kawamuro [15], which is based on Kawahigashi [12], [13]). Thus we need to clarify what appropriate outerness is.…”

Section: Introductionmentioning

confidence: 99%

A classification of flows on AFD factors with faithful Connes–Takesaki modules

Shimada

2015

Trans. Amer. Math. Soc.

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We completely classify flows on approximately finite dimensional (AFD) factors with faithful Connes-Takesaki modules up to cocycle conjugacy. This is a generalization of the uniqueness of the trace-scaling flow on the AFD factor of type II∞, which is equivalent to the uniqueness of the AFD factor of type III 1 . In order to achieve this, we show that a flow on any AFD factor with faithful Connes-Takesaki module has the Rohlin property, which is a kind of outerness for flows introduced by Kishimoto and Kawamuro. 1 2 KOICHI SHIMADA(Theorem 1.2.5 of Connes [3]), which is derived from outerness of actions of Z. Actually, Kishimoto's definition is for flows on C * -algebras. After Kishimoto's work, Kawamuro [15] introduced the Rohlin property for flows on von Neumann algebras. Recently, Masuda-Tomatsu [25] have presented a classification theorem for Rohlin flows. Thus the Rohlin property is now considered to be appropriate outerness. However, there is a problem. In general, it is not easy to see whether a given flow has the Rohlin property or not. Moreover, the Rohlin property is not written by "standard invariants" for flows. This can be an obstruction for the complete classification of flows on AFD factors. Hence it is important to characterize the Rohlin property in an appropriate way. At this point, it is conjectured that a flow on an AFD factor has the Rohlin property if and only if it has full Connes spectrum and is centrally free at each non-trivial point. Now, we explain the relation between our main theorem and this characterization program. First of all, the uniqueness of the trace-scaling flow on the AFD factor of type II ∞ is deeply related to its having the Rohlin property. Indeed, by the results of Connes [2] and Haagerup [18], it is possible to see that any trace-scaling flow on the AFD factor of type II ∞ has the Rohlin property (See Theorem 6.18 of Masuda-Tomatsu [25]). The uniqueness follows from the classification theorem of Rohlin flows. Thus it is expected that flows have the Rohlin property under our generalized assumption, that is, having faithful Connes-Takesaki modules (See Problem 8.5 of Masuda-Tomatsu [25]). In this paper, we actually show that flows on any AFD factor with faithful Connes-Takesaki modules have the Rohlin property, and obtain the main theorem by using Masuda-Tomatsu's theorem. Hence it is possible to think of our main theorem as a partial answer to the characterization problem of the Rohlin property. The theorem means that if a flow is "very outer" at any non-trivial point, then it is globally "very outer". Our main theorem provides interesting examples of Rohlin flows, and we believe that it is a useful observation for the characterization problem.Hence the difficult point of the proof of the main theorem is to show the Rohlin property. In order to show the Rohlin property of a flow α on a factor M , we need to find good unitaries of M . To achieve this, we consider the continuous decomposition of M . The dual action θ of a modular flow of M and the canonical extentionα of ...

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“…One prescription of that is to focus on the much smaller subalgebra M ω,α , which consists of (α, ω)-equicontinuous sequences (see Definition 3.4). Then the Rohlin property, which has been introduced by Kishimoto to flows on C * -algebras [37] and later by Kawamuro to flows on finite von Neumann algebras [35], can be a candidate of "outerness". This property means that we can find out a unitary eigenvector in M ω,α with the eigenvalue p for any p ∈ R.…”

Section: Introductionmentioning

confidence: 99%

Rohlin flows on von Neumann algebras

Masuda

Tomatsu

2016

Memoirs of the AMS

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We will introduce the Rohlin property for flows on von Neumann algebras and classify them up to strong cocycle conjugacy. This result provides alternative approaches to some preceding results such as Kawahigashi's classification of flows on the injective type II 1 factor, the classification of injective type III factors due to Connes, Krieger and Haagerup and the non-fullness of type III 0 factors. Several concrete examples are also studied.

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A Rohlin Property for One-parameter Automorphism Groups of the Hyperfinite $\mathrm{II}_1$ Factor

Cited by 12 publications

References 6 publications

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

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

A classification of flows on AFD factors with faithful Connes–Takesaki modules

Rohlin flows on von Neumann algebras

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