33 research outputs found
Normal intermediate subfactors
Let be an irreducible inclusion of type type II factors
with finite Jones index. We shall introduce the notion of normality for
intermediate subfactors of the inclusion . If the depth of is 2, then an intermediate subfactor for is normal
in if and only if the depths of and
are both 2. In particular, if is the crossed product of a
finite group , then is normal in if and only
if is a normal subgroup of .Comment: 25 pages, amslatex, to appear in J. Math. Soc. Japa
The Jiang-Su absorption for inclusions of unital C*-algebras
In this paper we will introduce the tracial Rokhlin property for an inclusion
of separable simple unital C*-algebras with finite index in the
sense of Watatani, and prove theorems of the following type. Suppose that
belongs to a class of C*-algebras characterized by some structural property,
such as tracial rank zero in the sense of Lin. Then belongs to the same
class. The classes we consider include:(1) Simple C*-algebras with real rank
zero or stable rank one, (2) Simple C*-algebras with tracial rank zero or
tracial rank less than or equal to one, (3) Simple C*-algebras with the
Jiang-Su algebra absorption, (4) Simple C*-algebras for which the
order on projections is determined by traces, (5) Simple C*-algebras with the
strict comparison property for the Cuntz semigroup. The conditions (3) and (5)
are important properties related to Toms and Winter's conjecture, that is, the
properties of strict comparison, finite nuclear dimension, and Z-absorption are
equivalent for separable simple infinite-dimensional nuclear unital
C*-algebras. We show that an action from a finite group on a
simple unital C*-algebra has the tracial Rokhlin property in the sense of
Phillips if and only if the canonical conditional expectation has the tracial Rokhlin property for an inclusion .Comment: 25 page
The Rohlin property for inclusions of -algebras with a finite Watatani index
We introduce notions of the Rohlin property and the approximate
representability for inclusions of unital -algebras. We investigate a dual
relation between the Rohlin property and the approximate representability. We
prove that a number of classes of unital -algebras are closed under
inclusions with the Rohlin property, including:
AF algebras, AI algebras, AT algebras, and related classes characterized by
direct limit decomposition using semiprojective building blocks. -algebras
with stable rank one. -algebras with real rank zero.Comment: We revised the section 4 and its correspondent part in Introduction
in the original pape