Commutativity of automorphisms of subfactors modulo inner automorphisms

We introduce a new algebraic invariant of a subfactor . 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 introduced by Y. Kawahigashi. We also show that this group is contained in the center of in many interesting examples such as quantum subfactors with level , but not always contained in the center. We also discuss its relation to the most general setting of the orbifold construction for subfactors.

     

Pairs of II1 factors

A class of objects—that are best described as being actions ofgroup-like objects of von Neumann algebras—is axiomatised and it is shown that there exists a bijective correspondence between isomorphism classes of suchcovariant systems and isomorphism classes of pairs of II₁ factors (M, N) satisfyingN⊂M, [M:N]<∞ andM∮N′=C.     

Jones index theory by Hilbert C-bimodules and K-theory

In this paper we introduce the notion of Hilbert -bimodules, replacing the associativity condition of two-sided inner products in Rieffel's imprimitivity bimodules by a Pimsner-Popa type inequality. We prove Schur's Lemma and Frobenius reciprocity in this setting. We define minimality of Hilbert -bimodules and show that tensor products of minimal bimodules are also minimal. For an - bimodule which is compatible with a trace on a unital -algebra , its dimension (square root of Jones index) depends only on its -class. Finally, we show that the dimension map transforms the Kasparov products in to the product of positive real numbers, and determine the subring of generated by the Hilbert -bimodules for a -algebra generated by Jones projections.

     

Planar algebras and the Ocneanu-Szymanski theorem

We give a very simple `planar algebra' proof of the part of the Ocneanu-Szymanski theorem which asserts that for a finite index, depth two, irreducible -subfactor , the relative commutants and admit mutually dual Kac algebra structures. In the hyperfinite case, the same techniques also prove the other part, which asserts that acts on with invariants .

     

Conformal Quantum Field Theory and Subfactors

We survey a recent progress on algebraic quantum field theory in connection with subfactor theory. We mainly concentrate on one-dimensional conformal quantum field theory.     

极大子因子

若N是一个Ⅱ1型因子,G是一个有限群且在N上有一个真外作用α,则当G的阶是素数对,N是Ⅱ1型因子M=N(?)αG的极大子因子．另一方面,假设 N(?) M是Ⅱ1型因子的一个包含,M(?)M1是N(?)M的基本构造,[M:N]= p∈N是素数,N’∩ M=CI,N’∩M1是交换的,N,(?)M深度为2,则N是M的极大子因子．     

Weak Riemannian manifolds from finite index subfactors

Let N ⊂ M be a finite Jones’ index inclusion of II₁ factors and denote by U _N ⊂ U _M their unitary groups. In this article, we study the homogeneous space U _M /U _N , which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit of the Jones projection of the inclusion. We endow with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete); therefore, is a weak Riemannian manifold. We show that enjoys certain properties similar to classic Hilbert–Riemann manifolds. Among them are metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point p ₁ of , there is a ball (of uniform radius r) of the usual norm of M, such that any point p ₂ in the ball is joined to p ₁ by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion , where the last set denotes the Grassmann manifold of the von Neumann algebra generated by M and .