The Rokhlin property and the tracial topological rank

Let $A$ be a unital separable simple \CA with \tr(A)\le 1 and $\alpha$ be an automorphism. We show that if $\alpha$ satisfies the tracially cyclic Rokhlin property then \tr(A\rtimes_{\alpha}\Z)\le 1. We also show that whenever $A$ has a unique tracial state and $\alpha^m$ is uniformly outer for each $m (\not= 0)$ and $\alpha^r$ is approximately inner for some $r>0,$ $\alpha$ 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 $A$ is a unital simple $A{\mathbb T}$-algebra of real rank zero and \alpha\in \Aut(A) which is approximately inner and if $\alpha$ satisfies some Rokhlin property, then the crossed product $A\rtimes_{\alpha}\Z$ is again an $A{\mathbb T}$ -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 $n$ are discussed: (1)$f(0) \leq 0$ and $f$ is $n$-convex in $[0, \alpha)$. (2)For each matrix $a$ with its spectrum in $[0, \alpha)$ and a contraction $c$ in the matrix algebra $M_n$, $f(c^*ac) \leq c^*f(a)c$. (3)The function $f(t)/t$ $(= g(t))$ is $n$-monotone in $(0, \alpha)$. We know that two conditions $(2)$ and $(3)$ are equivalent and if $f$ with $f(0) \leq 0$ is $n$-convex, then $g$ is $(n -1)$-monotone. In this note we consider several extra conditions on $g$ to conclude that the implication from $(3)$ to $(1)$ is true. In particular, we study a class $Q_n([0, \alpha))$ of functions with conditional positive Lowner matrix which contains the class of matrix $n$-monotone functions and show that if $f \in Q_{n+1}([0, \alpha))$ with $f(0) = 0$ and $g$ is $n$-monotone, then $f$ is $n$-convex. We also discuss about the local property of $n$-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 $P \subset A$ with finite index in the sense of Watatani, and prove theorems of the following type. Suppose that $A$ belongs to a class of C*-algebras characterized by some structural property, such as tracial rank zero in the sense of Lin. Then $P$ 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 $\mathcal{Z}$ 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 $\alpha$ from a finite group $G$ on a simple unital C*-algebra $A$ has the tracial Rokhlin property in the sense of Phillips if and only if the canonical conditional expectation $E\colon A \rightarrow A^G$ has the tracial Rokhlin property for an inclusion $A^G \subset A$.Comment: 25 page