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

Abstract

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 ...

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

“…Thirdly, under what reasonable assumptions can one expect the Rokhlin property to hold automatically? Is it enough to assume that the flow acts sufficiently non-trivially on the classification invariant in the style of [78]? Does it hold for trace-scaling flows [55,56]?…”

Locally Compact Separable Abelian Group Actions on Factors with the Rokhlin Property

“…There exist "many" Rohlin actions on AFD factors of type III 0 . The following is a generalization of a part of [24]. Proof.…”

“…Proof. The proof is the same as that of Case ker = 0 of Lemma 6 of [24]. In the proof of Lemma 6 of [24], we use Rohlin's lemma for actions of R 2 .…”

“…The proof is the same as that of Case ker = 0 of Lemma 6 of [24]. In the proof of Lemma 6 of [24], we use Rohlin's lemma for actions of R 2 . Rohlin's lemma for actions of G × R also holds, which is shown by the same argument as in Lemma of Lind [16] and Theorem 1 of [6].…”

“…(See Theorem 5 of[24]) Let α be an action of a locally compact abelian group G on an AFD factor M and (θ, Z) be the flow of weights of M . Assume that an action {mod(α g ) • θ t } (g,t)∈G×R of G × R is faithful.…”