Classification of Approximately Inner Actions of Discrete Amenable Groups on Strongly Amenable Subfactors

Abstract

We give the classification theorem of approximately inner actions of discrete amenable groups on strongly amenable subfactor of type II 1 by means of the characteristic invariant and ν invariant. To prove this theorem, we also give the classification theorem when the inner part and the centrally trivial part of actions coincide.

Unified approach to the classification of actions of discrete amenable groups on injective factors

“…Take α ∈ Ker(Φ) and σ ∈ Cnt r (M, N). Let 0 = a ∈ M k be an element such that σ(x)a = ax holds for all x ∈ M. Then there exists a unitary u(α, σ) ∈ N such that α(a) = u(α, σ)a [28], which does not depend on a. This u(α, σ) satisfies α • σ • α −1 = Ad u(α, σ) • σ.…”

“…In [28] and [26], we show Inv(α) = (H, [λ, µ], ν) is a complete cocycle conjugacy invariant for approximately inner actions of discrete amenable groups on subfactors with conditions (1) and (2) under some restrictions, e.g, the triviality of the κ-invariant. However if we modify the argument in the previous sections in a suitable way, we can get rid of these restrictions.…”

Classification of actions of compact abelian groups on subfactors with index less than 4

“…In Section 3, we will construct the invariants for the classification. This part is essentially the same as Section 5 of Loi [13] and Section 4 of Masuda [15]. In Section 4, we will show a classification theorem for actions of discrete abelian groups on some inclusions of von Neumann algebras.…”

“…Group actions on subfactors themselves have been studied by many hands. Constructing invariants for actions of discrete amenable groups and classifying them by these invariants have intensively been studied in 1990's and early 2000's (See Loi [13], Kawahigashi [9], Popa [21] and Toshihiko Masuda [15]). Hence as a next problem, we consider a classification of actions of continuous groups.…”

“…In order to classify actions of compact abelian groups on subfactors, we need to investigate actions of discrete abelian groups on inclusions of von Neumann algebras which may not be factors. It is possible to construct invariants by the same method as that in the case of actions of discrete amenable groups on subfactors (Section 4 of Masuda [15]). We will try to prove the classification theorem by the same line as that of the case of single factors (Jones-Takesaki [5], Kawahigashi-Takesaki [10]).…”