Centrally trivial automorphisms and an analogue of Connes’s χ(M) for subfactors

April, 1992 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.