6,055 research outputs found
The Rokhlin property and the tracial topological rank
Let be a unital separable simple \CA with \tr(A)\le 1 and be
an automorphism. We show that if satisfies the tracially cyclic
Rokhlin property then \tr(A\rtimes_{\alpha}\Z)\le 1. We also show that
whenever has a unique tracial state and is uniformly outer for
each and is approximately inner for some
satisfies the tracial cyclic Rokhlin property. By applying the
classification theory of nuclear \CA s, we use the above result to prove a
conjecture of Kishimoto: if is a unital simple -algebra of
real rank zero and \alpha\in \Aut(A) which is approximately inner and if
satisfies some Rokhlin property, then the crossed product
is again an -algebra of real rank zero. As
a by-product, we find that one can construct a large class of simple \CA s with
tracial rank one (and zero) from crossed products.Comment: 21 page
Double piling structure of matrix monotone functions and of matrix convex functions II
We continue the analysis in [H. Osaka and J. Tomiyama, Double piling
structure of matrix monotone functions and of matrix convex functions, Linear
and its Applications 431(2009), 1825 - 1832] in which the followings three
assertions at each label are discussed: (1) and is
-convex in . (2)For each matrix with its spectrum in and a contraction in the matrix algebra , . (3)The function is -monotone in . We know that two conditions and are equivalent and if
with is -convex, then is -monotone. In this note
we consider several extra conditions on to conclude that the implication
from to is true. In particular, we study a class
of functions with conditional positive Lowner matrix which contains the class
of matrix -monotone functions and show that if
with and is -monotone, then is -convex. We also
discuss about the local property of -convexity.Comment: 13page
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
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