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Kazunori Kodaka

Publications and source records attributed to Kazunori Kodaka.

18 recordsLinked to original sources
arXiv2021

Strong Morita equivalences for completely positive linear maps and GNS-C*-correspondences

We will consider the set of all completely positive linear maps from a unital $C^*$-algebra to the $C^*$-algebra of all (bounded) adjointable right Hilbert $C^*$-module maps, which are automatically bounded, on a right Hilbert $C^*$-module and we will introduce strong Morita equivalence for elements in this set. In this paper, we will give the following result: If two classes of two unital $C^*$-algebras are strongly Morita equivalent, respectively, then we can construct a bijective correspondence between two sets of all strong Morita equivalence classes of completely positive linear maps given as above. Furthermore, we will discuss the relation between strong Morita equivalence for completely positive linear maps and strong Morita equivalence for GNS-$C^*$-correspondences.

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arXiv2021

Free coactions of a finite dimensional $C^*$-Hopf algebra and strong Morita equivalence

We shall introduce a notion of free coactions of a finite dimensional $C^*$-Hopf algebra on a $C^*$-algebra modifying a notion of free actions of a discrete group on a $C^*$-algebra and we shall study several properties on coactions of a finite dimensional $C^*$-Hopf algebra on $C^*$-algebras, which are relating to strong Morita equivalence for inclusions of $C^*$-algebras.

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arXiv2021

Fell bundles over a countable discrete group and strong Morita equivalence for inclusions of $C^*$-algebras

We consider two saturated Fell bundles over a countable discrete group, whose unit fibers are $σ$-unital $C^*$-algebras. Then by taking the reduced cross-sectional $C^*$-algebras, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the unit fiber algebras, which is a $σ$-unital $C^*$-algebra, in the multiplier $C^*$-algebra of the reduced cross-sectional $C^*$-algebra is trivial. We show that the two saturated Fell bundles are then equivalent up to some automorphism of the group.

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arXiv2021

Strong Morita equivalence for conditional expectations

We consider two inclusions of $C^*$-algebras whose small $C^*$-algebras have approximate units of the large $C^*$-algebras and their two spaces of all bounded bimodule linear maps. We suppose that the two inclusions of $C^*$-algebras are strongly Morita equivalent. In this paper, we shall show that there exists an isometric isomorphism from one of the spaces of all bounded bimodule linear maps to the other space and we shall study on basic properties about the isometric isomorphism. And, using this isometric isomorphism, we define the Picard group for a bimodule linear map and discuss on the Picard group for a bimodule linear map.

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arXiv2021

Strong Morita equivalence for completely positive linear maps on $C^*$-algebras

We will introduce the notion of strong Morita equivalence for completely positive linear maps and study its basic properties. Also, we will discuss the relation between strong Morita equivalence for bounded $C^*$-bimodule linear maps and strong Morita equivalence for completely positive linear maps. Furthermore, we will show that if two unital $C^*$-algebras are strongly Morita equivalent, then there is a $1-1$ correspondence between the two sets of all strong Morita equivalence classes of completely positive linear maps on the two unital $C^*$-algebras and we will show that the corresponding two classes of the completely positive linear maps are also strongly Morita equivalent.

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arXiv2020

Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group

We consider two twisted actions of a countable discrete group on $σ$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions is irreducible, that is, the relative commutant of one of the $σ$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.

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arXiv2020

Bimodule maps from a unital $C^*$-algebra to its $C^*$-subalgebra and strong Morita equivalence

Let $A \subset C$ and $B \subset D$ be unital inclusions of unital $C^*$-algebras. Let ${}_A \mathbf{B}_A (C, A)$ (resp. ${}_B \mathbf{B}_B (D, B)$) be the space of all bounded $A$-bimodule (resp. $B$-bimodule) linear maps from $C$ (resp. $D$) to $A$ (resp. $B$). We suppose that $A \subset C$ and $B \subset D$ are strongly Morita equivalent. We shall show that there is an isometric isomorphism $f$ of ${}_A \mathbf{B}_A (C, A)$ onto ${}_B \mathbf{B}_B (D, B)$ and we shall study on basic properties about $f$.

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arXiv2019

The Picard groups of inclusions of $C^*$-algebras induced by equivalence bimodules

Let $A$ and $B$ be $σ$-unital $C^*$-algebras and $X$ and $Y$ an $A-A$-equivalence bimodule and a $B-B$-equivalence bimodule, respectively. Also, let $A\rtimes_X \mathbb{Z}$ and $B\rtimes_Y \mathbb{Z}$ be the crossed products of $A$ and $B$ by $X$ and $Y$, respectively. Furthermore, let $A\subset A\rtimes_X \mathbb{Z}$ and $B\subset B\rtimes_Y \mathbb{Z}$ be the inclusions of $C^*$-algebras induced by $X$ and $Y$, respectively. We suppose that $A' \cap M(A\rtimes_X \mathbb{Z})=\mathbb{C} 1$. In this paper we shall show that the inclusions $A\subset A\rtimes_X \mathbb{Z}$ and $B\subset B\rtimes_Y \mathbb{Z}$ are strongly Morita equivalent if and only if there is an $A-B$-equivalence bimodule $M$ such that $Y\cong \widetilde{M}\otimes_A X \otimes_A M$ or $\widetilde{Y}\cong \widetilde{M}\otimes_A X \otimes_A M$ as $B-B$-equivalence bimodules, where $\widetilde{M}$ and $\widetilde{Y}$ are the dual $B-A$-equivalence bimodule and the dual $B-B$-equivalence bimodule of $M$ and $Y$, respectively. Applying this result, we shall compute the Picard group of the inclusion $A\subset A\rtimes_X \mathbb{Z}$ under the assumption that $A' \cap M(A\rtimes_X \mathbb{Z})=\mathbb{C} 1$.

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arXiv2019

Equivalence bundles over a finite group and strong Morita equivalence for unital inclusions of unital $C^*$-algebras

Let $\mathcal{A}= \{A_t \}_{t \in G}$ and $\mathcal{B}= \{B_t \}_{t\in G}$ be $C^*$-algebraic bundles over a finite group $G$. Let $C=\oplus_{t \in G}A_t$ and $D=\oplus_{t\in G}B_t$. Also, let $A=A_e$ and $B=B_e$, where $e$ is the unit element in $G$. We suppose that $C$ and $D$ are unital and $A$ and $B$ have the unit elements in $C$ and $D$, respectively. In this paper, we shall show that if there is an equivalence $\mathcal{A}-\mathcal{B}$-bundle over $G$ with some properties, then the unital inclusions of unital $C^*$-algebras $A \subset C$ and $B \subset D$ induced by $\mathcal{A}$ and $\mathcal{B}$ are strongly Morita equivalent. Also, we suppose that $\mathcal{A}$ and $\mathcal{B}$ are saturated and that $A' \cap C= \mathbf{C} 1$. We shall show that if $A \subset C$ and $B \subset D$ are strongly Morita equivalent, then there are an automorphism $f$ of $G$ and an equivalence bundle $\mathcal{A}-\mathcal{B}^f $-bundle over $G$ with the some properties, where $\mathcal{B}^f$ is the $C^*$-algebraic bundle induced by $\mathcal{B}$ and $f$, which is defined by $\mathcal{B}^f = \{B_{f(t)} \}_{t \in G}$. Furthermore, we shall give an application.

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arXiv2019

The Picard groups for unital inclusions of unital $C^*$-algebras

We shall introduce the notion of the Picard group for an inclusion of $C^*$-algebras. We shall also study its basic properties and the relation between the Picard group for an inclusion of $C^*$-algebras and the ordinary Picard group. Furthermore, we shall give some examples of the Picard groups for unital inclusions of unital $C^*$-algebras.

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arXiv2018

The generalized Picard groups for finite dimensional $C^*$-Hopf algebra coactions on unital $C^*$-algebras

We shall generalize the notion of the strong Morita equivalence for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra and define the Picard groups with respect to the generalized strong Morita equivalence. We call them the generalized Picard groups for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algeba. We shall investigate basic properties of the generalized Picard groups and clarify the relation between the generalized Picard groups and the Picard groups for unital inclusions of unital $C^*$-algebras.

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arXiv2017

Coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras, unital inclusions of unital $C^*$-algebras and the strong Morita equivalence

Let $A$ and $B$ be unital $C^*$-algebras and let $H$ be a finite dimensional $C^*$-Hopf algebra. Let $H^0$ be its dual $C^*$-Hopf algebra. Let $(ρ, u)$ and $(σ, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. In this paper, we shall show the following theorem: We suppose that the unital inclusions $A\subset A\rtimes_{ρ, u}H$ and $B\subset B\rtimes_{σ, v}H$ are strongly Morita equivalent. If $A'\cap (A\rtimes_{ρ, u}H)=\BC1$, then there is a $C^*$-Hopf algebra automorphism $λ^0$ of $H^0$ such that the twisted coaction $(ρ, u)$ is strongly Morita equivalent to the twisted coaction $((\id_B \otimesλ^0 )\circσ\, , \, (\id_B \otimesλ^0 \otimesλ^0 )(v))$ induced by $(σ, v)$ and $λ^0$.

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arXiv2015

Equivariant Picard groups of $C^*$-algebras with finite dimensional $C^*$-Hopf algebra coactions

Let $A$ be a $C^*$-algebra and $H$ a finite dimensional $C^*$-Hopf algebra with its dual $C^*$-Hopf algebra $H^0$. Let $(ρ, u)$ be a twisted coaction of $H^0$ on $A$. We shall define the $(ρ, u, H)$-equivariant Picard group of $A$, which is denoted by $\Pic_H^{ρ, u}(A)$, and discuss basic properties of $\Pic_H^{ρ, u}(A)$. Also, we suppose that $(ρ, u)$ is the coaction of $H^0$ on the unital $C^*$-algebra $A$, that is, $u=1\otimes 1^0$. We investigate the relation between $\Pic(A^s )$, the ordinary Picard group of $A^s$ and $\Pic_H^{ρ^s}(A^s )$ where $A^s$ is the stable $C^*$-algebra of $A$ and $ρ^s$ is the coaction of $H^0$ on $A^s$ induced by $ρ$. Furthermore, we shall show that $\Pic_{H^0}^{\widehatρ}(A\rtimes_{ρ, u}H)$ is isomorphic to $\Pic_H^{ρ, u}(A)$, where $\widehatρ$ is the dual coaction of $H$ on the twisted crossed product $A\rtimes_{ρ, u}H$ of $A$ by the twisted coaction $(ρ, u)$ of $H^0$ on $A$.

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arXiv2015

The strong Morita equivalence for coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras

Following Jansen and Waldmann, and Kajiwara and Watatani, we shall introduce notions of coactions of a finite dimensional $C^*$-Hopf algebra on a Hilbert $C^*$-bimodule of finite type in the sense of Kajiwara and Watatani and define their crossed product. We shall investigate their basic properties and show that the strong Morita equivalence for coactions preserves the Rohlin property for coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras.

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arXiv2012

The Rohlin property for coactions of finite dimensional $C^*$-Hopf algebras on unital $C^*$-algebras

We shall introduce the approximate representability and the Rohlin property for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra and discuss some basic properties of approximately representable coactions and coactions with the Rohlin property of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra. Also, we shall give an example of an approximately representable coaction of a finite dimensional $C^*$-Hopf algebra on a simple unital $C^*$-algebra which has also the Rohlin property and we shall give the 1-cohomology vanishing theorem for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra and the 2-cohomology vanishing theorem for twisted coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra. Furthermore, we shall introduce the notion of the approximately unitary equivalence of coactions of a finite dimensional $C^*$-Hopf algebra $H$ on a unital $C^*$-algebra $A$ and show that if $ρ$ and $σ$, coactions of $H$ on a separable unital $C^*$-algebra $A$, which have the Rohlin property, are approximately unitarily equivalent, then there is an approximately inner automorphism $α$ on $A$ such that $σ=(α\otimes\id)\circρ\circα^{-1}$.

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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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