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Hiroyuki Osaka

Publications and source records attributed to Hiroyuki Osaka.

At least 37 records · Page 2Linked to original sources
arXiv2017

On dualities of actions and inclusions

Following the results known in the case of a finite abelian group action on $C\sp*$-algebras we prove the following two theorems; 1. an inclusion $P\subset A$ of (Watatani) index-finite type has the Rokhlin property (is approximately representable) if and only if the dual inclusion is approximately representable (has the Rokhlin property). 2. an inclusion $P\subset A$ of (Watatani) index-finite type has the tracial Rokhlin property (is tracially approximately representable) if and only if the dual inclusion is tracially approximately representable (has the tracial Rokhlin property). Moreover, we provide an alternate proof of Phillips' theorem about the relations between tracial Rokhlin action and tracially approximate representable dual action using a new conceptual framework suggested by authors.

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arXiv2017

Stable rank for crossed products by actions of finite groups on C*-algebras

Let $G$ be a finite group, $A$ a unital separable finite simple nuclear C*-algebra, and $α$ an action of $G$ on $A$. Assume that $A$ absorbs the Jiang-Su algebra $\mathcal{Z}$, the extremal boundary of the trace space of $A$ is compact and finite dimensional and that $α$ fixes any tracial state of $A$. Then tsr$(A \rtimes_αG) = 1$. In particular, when $A$ has a unique tracial state, we conclude it without above conditons on a tracial state space of $A$.

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arXiv2016

Rokhlin Property for Group Actions on Hilbert $C^*$-modules

We introduce Rokhlin properties for certain discrete group actions on $C^*$-correspondences as well as on Hilbert bimodules and analyze them. It turns out that the group actions on any $C^*$-correspondence $E$ with Rokhlin property induces group actions on the associated $C^*$-algebra $\mathcal O_E$ with Rokhlin property and the group actions on any Hilbert bimodule with Rokhlin property induces group actions on the linking algebra with Rokhlin property. Permanence properties of several notions such as nuclear dimension and $\mathcal D$-absorbing property with respect to crossed product of Hilbert $C^*$-modules with groups, where group actions have Rokhlin property, are studied. We also investigate a notion of outerness for Hilbert bimodules.

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arXiv2014

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 $α$ 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 $A^G \subset A$.

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arXiv2012

The linear span of projections in AH algebras and for inclusions of C*-algebras

A $C^*$-algebra is said to have the LP property if the linear span of projections is dense in a given algebra. In the first part of this paper, we show that an AH algebra $A = \underrightarrow{\lim}(A_i,ϕ_i)$ has the LP property if and only if every real-valued continuous function on the spectrum of $A_i$ (as an element of $A_i$ via the non-unital embedding) belongs to the closure of the linear span of projections in $A$. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation. The second contribution of this paper is that for an inclusion of unital $C^*$-algebras $P \subset A$ with a finite Watatani Index, if a faithful conditional expectation $E\colon A \rightarrow P$ has the Rokhlin property in the sense of Osaka and Teruya, then $P$ has the LP property under the condition $A$ has the LP property. As an application, let $A$ be a simple unital $C^*$-algebra with the LP property, $G$ a finite group and $α$ an action of $G$ onto $\mathrm{Aut}(A)$. If $α$ has the Rokhlin property in the sense of Izumi, then the fixed point algebra $A^G$ and the crossed product algebra $A \rtimes_αG$ have the LP property. We also point out that there is a symmetry on CAR algebra, which is constructed by Elliott, such that its fixed point algebra does not have the LP property.

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arXiv2012

Characterization of the monotonicity by the inequality

Let $φ$ be a normal state on the algebra $B(H)$ of all bounded operators on a Hilbert space $H$, $f$ a strictly positive, continuous function on $(0, \infty)$, and let $g$ be a function on $(0, \infty)$ defined by $g(t) = \frac{t}{f(t)}$. We will give characterizations of matrix and operator monotonicity by the following generalized Powers-St\ormer inequality: $$ φ(A + B) - φ(|A - B|) \leq 2φ(f(A)^1/2g(B)f(A)^1/2), $$ whenever $A, B$ are positive invertible operators in $B(H).$

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arXiv2012

On generalized Powers-St$ø$rmer's Inequality

A generalization of Powers-St$ø$rmer's inequality for operator monotone functions on $[0, +\infty)$ and for positive linear functional on general $C^*$-algebras will be proved. It also will be shown that the generalized Powers-St$ø$rmer inequality characterizes the tracial functionals on $C^*$-algebras.

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arXiv2011

Nuclear dimension for an inclusion of unital C*-algebras

Let $P \subset A$ be an inclusion of separable unital C*-algebras with finite Watatani index. Suppose that $E \colon A \rightarrow P$ has the Rokhlin property, that is, there is a projection $e \in A' \cap A^\infty$ such that $E^\infty(e) = ({\rm Index}E)^{-1}1$. We show that if $A$ has nuclear dimension $n$, then $P$ has nuclear dimension less than or equal to $n$. In particular, if an action $α$ of a finite group $G$ on $A$ has the Rokhlin property, then the nuclear dimension of the crossed product algebra $A \rtimes_αG$ is less than or equal to that of $A$.

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arXiv2011

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 $[0, α)$. (2)For each matrix $a$ with its spectrum in $[0, α)$ 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 $(0, α)$. 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 $Q_n([0, α))$ of functions with conditional positive Lowner matrix which contains the class of matrix $n$-monotone functions and show that if $f \in Q_{n+1}([0, α))$ with $f(0) = 0$ and $g$ is $n$-monotone, then $f$ is $n$-convex. We also discuss about the local property of $n$-convexity.

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arXiv2010

Strongly self-absorbing property for inclusions of $C^*$-algebras with a finite Watatani index

Let $P \subset A$ be a inclusion of unital C*-algebras and $E\colon A \to P$ be a conditional expectation of index finite type. We introduce a Rokhlin property for $E$ and discuss about $\mathcal{D}$-absorbing proeprty, where $\mathcal{D}$ is a separable, unital, strongly self-absorbing C*-algebra. In this paper we consider permanent properties for strongly self-absorbing property under inclusions of unital C*-algebras with a finite Watatani index.

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arXiv2010

The Rohlin property for inclusions of $C^*$-algebras with a finite Watatani index

We introduce notions of the Rohlin property and the approximate representability for inclusions of unital $C^*$-algebras. We investigate a dual relation between the Rohlin property and the approximate representability. We prove that a number of classes of unital $C^*$-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. $C^*$-algebras with stable rank one. $C^*$-algebras with real rank zero.

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arXiv2009

Crossed products by finite group actions with the Rokhlin property

We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: (1) AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (2) Simple unital AH algebras with slow dimension growth and real rank zero. (3) C*-algebras with real rank zero or stable rank one. (4) C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. Along the way, we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective.

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arXiv2008

Structure of the spaces of matrix monotone functions and of matrix convex functions and Jensen's type inequality for operators

Let $n \in \N$ and $M_n$ be the algebra of $n \times n$ matrices. We call a function $f$ matrix monotone of order $n$ or $n$-monotone in short whenever the inequality $f(a) \leq f(b)$ holds for every pair of selfadjoint matrices $a, b \in M_n$ such that $a \leq b$ and all eigenvalues of $a$ and $b$ are contained in $I$. Matrix convex (concave) functions on $I$ are similarily defined. The spaces for $n$-monotone functions and $n$-convex functions are written as $P_n(I)$ and $K_n(I)$. In this note we discuss several assertions at each leven $n$ for which we regard themas the problems of double piling structure of those sequences $\{P_n(I)\}_{n\in\N}$ and $\{K_n(I)\}_{n\in\N}$. In order to see clear insight of the aspect of the problems, however, we choose the following three main assertions among them and discuss their mutual dependence: \begin{enumerate} \item[(i)] $f(0)\leq 0$ and $f$ is $n$-convex in $[0,α)$, \item[(ii)] For each matrix $a$ with its spectrum in $[0,α)$ and a contraction $c$ in the matrix algebra $M_n$, \[ f(c^{\star}a c)\leq c^{\star}f(a)c, \] \item[(iii)] The functon $g(t)/t$ is $n$-monotone in $(0,α)$. \end{enumerate} In particular, we show that for any $n \in \N$ two conditions $(ii)$ and $(iii)$ are equivalent.

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arXiv2007

Stable rank for inclusions of C*-algebras

When a unital \ca $A$ has topological stable rank one (write $\tsr(A) = 1$), we know that $\tsr(pAp) \leq 1$ for a non-zero projection $p \in A$. When, however, $\tsr(A) \geq 2$, it is generally faluse. We prove that if a unital C*-algebra $A$ has a simple unital C*-subalgebra $D$ of $A$ with common unit such that $D$ has \PSP and $\sup_{p\in P(D)}\tsr(pAp) < \infty$, then $\tsr(A) \leq 2.$ As an application let $A$ be a simple unital \ca with $\tsr(A) = 1$ and \PSP, $\{G_k\}_{k=1}^n$ finite groups, $\af_k$ actions from $G_k$ to ${\rm Aut}((...((A\times_{\af_1}G_1)\times_{\af_2} G_2)...)\times_{\af_{k-1}}G_{k-1}).$ $(G_0 = \{1\})$ Then $$ \tsr((... ((A\times_{\af_1}G_1)\times_{\af_2} G_2)...)\times_{\af_n}G_n) \leq 2. $$

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arXiv2007

Cancellation for inclusions of C*-algebras of finite depth

Let B be a unital C*-algebra, let A be a unital subalgebra, and let E be a conditional expectation from B to A with index-finite type and a quasi-basis of n elements. Then the topological stable rank satisfies \tsr (B) \leq \tsr (A) + n - 1. As an application, we show that if a unital inclusion A \subset B of C*-algebras has index-finite type and finite depth, and A is simple with stable rank one and Property (SP), then B has cancellation. In particular, if A is a simple unital C*-algebra with stable rank one and Property (SP), and a finite group G acts on A, then the crossed product has cancellation. Separately, if the group is the integers, we obtain cancellation under the additional hypotheses that the group action is outer and is trivial on K_0 (A).

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arXiv2006

Monotone operator functions, gaps and power moment problem

The article is devoted to investigation of the classes of functions belonging to the gaps between classes $P_{n+1}(I)$ and $P_{n}(I)$ of matrix monotone functions for full matrix algebras of successive dimensions. In this paper we address the problem of characterizing polynomials belonging to the gaps $P_{n}(I) \setminus P_{n+1}(I)$ for bounded intervals $I$. We show that solution of this problem is closely linked to solution of truncated moment problems, Hankel matrices and Hankel extensions. Namely, we show that using the solutions to truncated moment problems we can construct continuum many polynomials in the gaps. We also provide via several examples some first insights into the further problem of description of polynomials in the gaps that are not coming from the truncated moment problem. Also, in this article, we deepen further in another way into the structure of the classes of matrix monotone functions and of the gaps between them by considering the problem of position in the gaps of certain interesting subclasses of matrix monotone functions that appeared in connection to interpolation of spaces and in a prove of the L{ö}wner theorem on integral representation of operator monotone functions.

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arXiv2006

Stable rank of inclusion of C*-algebras of depth 2

Let $1 \in A \subset B$ be an inclusion of unital C*-algebras of index-finite type and depth 2. Suppose that $A$ is infinite dimensional simple with $tsr(A) = 1$ and SP-property. Then $tsr(B) \leq 2$. As a corollary when $A$ is a simple C*-algebra with $\tsr(A) = 1$ and SP-property and $α$ an action of a finite group $G$ on $\Aut(A)$, $\tsr(A \rtimes_αG) \leq 2$.

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