Commutativity of automorphisms of subfactors modulo inner automorphisms

Abstract

Abstract. We introduce a new algebraic invariant χa(M, N) of a subfactor N ⊂ M . We show that this is an abelian group and that if the subfactor is strongly amenable, then the group coincides with the relative Connes invariant χ(M, N) introduced by Y. Kawahigashi. We also show that this group is contained in the center of Out(M, N) in many interesting examples such as quantum SU(n) k subfactors with level k (k ≥ n + 1), but not always contained in the center. We also discuss its relation to the most general setting of the orbifold construction for subfactors. §0. Introduction . These facts show that we can generalize this relative Connes invariant to properly infinite subfactors. In this paper we introduce a new algebraic invariant of a subfactor N ⊂ M . It is the intersection of the sets of non-strongly outer automorphisms and automorphisms with trivial Loi invariant modulo inner automorphisms arising from the normalizers. We define Recently Y. Kawahigashi introduced the relative Connes invariantwhere we denote the set of non-strongly outer automorphisms on the subfactor by Ψ (M, N ), the map assigning Loi's invariant by Φ and the set of normalizers of N in M , i.e., {u ∈ U(M)| uN u * = N } by N (M, N ). Note that we define

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

“…k=0 denotes the Loi invariant [25]. Let Cnt r (M, N) be a set of all non-strongly outer automorphisms [1], and χ a (M, N) = (Ker(Φ) ∩ Cnt r (M, N)) /Int(M, N) the algebraic χ-group [11]. Since N ⊂ M is strongly amenable, Ker(Φ) = Int(M, N) and Cnt(M, N) = Cnt r (M, N) hold.…”