Abstract. We complete classification of discrete abelian or finite group actions on injective type III1 factors up to cocycle conjugacy. We also give a proof for Connes ’ characterization of the Ker (mod) and Cnt(M) for an injective factor M of type III. §0 Introduction. The purpose of this paper is to give a proof of Connes ’ announcement on approx-imately inner automorphisms and centrally trivial automorphisms of an injective

Abstract. We study a class of centrally trivial automorphisms for subfactors, and get an upper bound for the order of the group they make (modulo normalizers) in terms of the “dual ” principal graph for AFD type II1 subfactors with trivial relative commutant, finite index and finite depth. We prove that this upper bound is attained for many known subfators. We also introduce χ(M,N) for subfactors N ⊂ M as the relative version of Connes ’ invariant χ(M), and compute this group for many AFD type II1 subfactors with finite index and finite depth including all the cases with index less than 4 and many Hecke algebra subfactors of Wenzl. In these finite depth cases, the group χ(M,N) is always finite and abelian, and we realize all the finite abelian groups as χ(M,N). Analogy between this topic and modular structure of type III factors is also discussed. As an application, we give some classification results for Aut(M,N). For example, for the subfactors of type A2n+1, there are two and only two outer actions of Z2. One is of the “standard” form and the other is given by the “orbifold ” action arising from the paragroup symmetry. As preliminaries, we also prove several statements on central sequence subfactors announced by A. Ocneanu.

and Longo, we will construct an AFD type III0 factor with uncountably many non-conjugate subfactors such that (i) each subfactor has the same flow of weights as the ambient factor, and (ii) the principal and the dual principal graphs are of a specific form. We will deal with two cases: (a) the graphs are described by the Dynkin diagram A4m−3, and (b) the graphs are the ones given by a pair of a group and its subgroup (see Kosaki and Yamagami) which are simultaneous semi-direct products. Subfactors are distinguished by looking at the dual action on the type II graphs. It is also possible to distinguish subfactors by investigating automorphisms appearing in the irreducible decomposition of the relevant sector (or bimodule). 1. Introduction. Classification of subfactors in the Powers factor of type IIIλ (0 <λ<1) with small indices is known to be closely related to that for the AFD II1-factor

Abstract. We show the uniqueness of minimal actions of a compact Kac algebra with the amenable dual on the AFD factor of type II1. This particularly implies the uniqueness of minimal actions of a compact group. Our main tools are a Rohlin type theorem, the 2-cohomology vanishing theorem, and the Evans-Kishimoto type intertwining argument.