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Norio Nawata

Publications and source records attributed to Norio Nawata.

16 recordsLinked to original sources
arXiv2024

Strongly outer actions of certain torsion-free amenable groups on the Razak-Jacelon algebra

Let $\mathfrak{C}$ be the smallest class of countable discrete groups with the following properties: (i) $\mathfrak{C}$ contains the trivial group, (ii) $\mathfrak{C}$ is closed under isomorphisms, countable increasing unions and extensions by $\mathbb{Z}$. Note that $\mathfrak{C}$ contains all countable discrete torsion-free abelian groups and poly-$\mathbb{Z}$ groups. Also, $\mathfrak{C}$ is a subclass of the class of countable discrete torsion-free elementary amenable groups. In this paper, we show that if $Γ\in \mathfrak{C}$, then all strongly outer actions of $Γ$ on the Razak-Jacelon algebra $\mathcal{W}$ are cocycle conjugate to each other. This can be regarded as an analogous result of Szabó's result for strongly self-absorbing C$^*$-algebras.

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arXiv2024

$\mathcal{W}$-absorbing actions of finite groups on the Razak-Jacelon algebra

We say that a countable discrete group action $α$ on a C$^*$-algebra $A$ is \textit{$\mathcal{W}$-absorbing} if there exist a C$^*$-algebra $B$ and an action $β$ on $B$ such that $α$ is cocycle conjugate to $β\otimes \mathrm{id}_{\mathcal{W}}$ on $B\otimes \mathcal{W}$ where $\mathcal{W}$ is the Razak-Jacelon algebra. In this paper, we completely classify outer $\mathcal{W}$-absorbing actions of finite groups on $\mathcal{W}$ up to conjugacy and cocycle conjugacy.

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arXiv2024

Approximate representability of finite abelian group actions on the Razak-Jacelon algebra

Let $A$ be a simple separable nuclear monotracial C$^*$-algebra, and let $α$ be an outer action of a finite abelian group $Γ$ on $A$. In this paper, we show that $α\otimes \mathrm{id}_{\mathcal{W}}$ on $A\otimes\mathcal{W}$ is approximately representable if and only if the characteristic invariant of $\tildeα$ is trivial, where $\mathcal{W}$ is the Razak-Jacelon algebra and $\tildeα$ is the induced action on the injective II$_1$ factor $π_{τ_{A}}(A)^{''}$. As an application of this result, we classify such actions up to conjugacy and cocycle conjugacy. In particular, we show the following: Let $A$ and $B$ be simple separable nuclear monotracial C$^*$-algebras, and let $α$ and $β$ be outer actions of a finite abelian group $Γ$ on $A$ and $B$, respectively. Assume that the characteristic invariants of $\tildeα$ and $\tildeβ$ are trivial. Then $α\otimes \mathrm{id}_{\mathcal{W}}$ and $β\otimes \mathrm{id}_{\mathcal{W}}$ are conjugate (resp. cocycle conjugate) if and only if $\tildeα$ on $π_{τ_{A}}(A)^{''}$ and $\tildeβ$ on $π_{τ_{B}}(B)^{''}$ are conjugate (resp. cocycle conjugate). We also construct the model actions.

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arXiv2023

Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra

Let $A$ and $B$ be simple separable nuclear monotracial C$^*$-algebras, and let $α$ and $β$ be strongly outer actions of a countable discrete amenable group $Γ$ on $A$ and $B$, respectively. In this paper, we show that $α\otimes\mathrm{id}_{\mathcal{W}}$ on $A\otimes\mathcal{W}$ and $β\otimes\mathrm{id}_{\mathcal{W}}$ on $B\otimes\mathcal{W}$ are cocycle conjugate where $\mathcal{W}$ is the Razak-Jacelon algebra. Also, we characterize such actions by using the fixed point subalgebras of Kirchberg's central sequence C$^*$-algebras.

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arXiv2022

A characterization of the Razak-Jacelon algebra

Combining Elliott, Gong, Lin and Niu's result and Castillejos and Evington's result, we see that if $A$ is a simple separable nuclear monotracial C$^*$-algebra, then $A\otimes\mathcal{W}$ is isomorphic to $\mathcal{W}$ where $\mathcal{W}$ is the Razak-Jacelon algebra. In this paper, we give another proof of this. In particular, we show that if $\mathcal{D}$ is a simple separable nuclear monotracial $M_{2^{\infty}}$-stable C$^*$-algebra which is $KK$-equivalent to $\{0\}$, then $\mathcal{D}$ is isomorphic to $\mathcal{W}$ without considering tracial approximations of C$^*$-algebras with finite nuclear dimension. Our proof is based on Matui and Sato's technique, Schafhauser's idea in his proof of the Tikuisis-White-Winter theorem and properties of Kirchberg's central sequence C$^*$-algebra $F(\mathcal{D})$ of $\mathcal{D}$. Note that some results for $F(\mathcal{D})$ are based on Elliott-Gong-Lin-Niu's stable uniqueness theorem. Also, we characterize $\mathcal{W}$ by using properties of $F(\mathcal{W})$. Indeed, we show that a simple separable nuclear monotracial C$^*$-algebra $D$ is isomorphic to $\mathcal{W}$ if and only if $D$ satisfies the following properties: (i) for any $θ\in [0,1]$, there exists a projection $p$ in $F(D)$ such that $τ_{D, ω}(p)=θ$, (ii) if $p$ and $q$ are projections in $F(D)$ such that $0<τ_{D, ω}(p)=τ_{D, ω}(q)$, then $p$ is Murray-von Neumann equivalent to $q$, (iii) there exists an injective homomorphism from $D$ to $\mathcal{W}$.

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arXiv2019

Rohlin actions of finite groups on the Razak-Jacelon algebra

Let $A$ be a simple separable nuclear C$^*$-algebra with a unique tracial state and no unbounded traces, and let $α$ be a strongly outer action of a finite group $G$ on $A$. In this paper, we show that $α\otimes \mathrm{id}$ on $A\otimes\mathcal{W}$ has the Rohlin property, where $\mathcal{W}$ is the Razak-Jacelon algebra. Combing this result with the recent classification results and our previous result, we see that such actions are unique up to conjugacy.

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arXiv2018

Trace scaling automorphisms of the stabilized Razak-Jacelon algebra

We classify trace scaling automorphisms of $\mathcal{W}\otimes\mathbb{K}$ up to outer conjugacy, where $\mathcal{W}$ is a certain simple separable nuclear stably projectionless C$^*$-algebra having trivial $K$-groups. Also, we show that all automorphisms of $\mathcal{W}$ with the Rohlin property are outer conjugate to each other. Moreover, we show that the central sequence C$^*$-algebra $F(\mathcal{W})$ of $\mathcal{W}$ is infinitee, which answers a question of Kirchberg.

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arXiv2013

Finite group actions on certain stably projectionless C*-algebras with the Rohlin property

We introduce the Rohlin property and the approximate representability for finite group actions on stably projectionless C*-algebras and study their basic properties. We give some examples of finite group actions on the Razak-Jacelon algebra and show some classification results of these actions. This study is based on the work of Izumi, Robert's classification theorem and Kirchberg's central sequence C*-algebras.

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arXiv2013

Picard groups of certain stably projectionless C*-algebras

We compute Picard groups of several nuclear and non-nuclear simple stably projectionless C*-algebras. In particular, the Picard group of Razak-Jacelon algebra W_2 is isomorphic to a semidirect product of Out(W_2) with R_+^\times. Moreover, for any separable simple nuclear stably projectionless C*-algebra with a finite dimensional lattice of densely defined lower semicontinuous traces, we show that Z-stability and strict comparison are equivalent. (This is essentially based on the result of Matui and Sato, and Kirchberg's central sequence algebras.) This shows if A is a separable simple nuclear stably projectionless C*-algebra with a unique tracial state (and no unbounded trace) and has strict comparison, the following sequence is exact: [{CD} {1} @>>> \mathrm{Out}(A) @>>> \mathrm{Pic}(A) @>>> \mathcal{F}(A) @>>> {1} {CD}] where $\mathcal{F}(A)$ is the fundamental group of A.

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arXiv2011

Fundamental group of uniquely ergodic Cantor minimal systems

We introduce the fundamental group ${\mathcal F}(\mathcal{R}_{G, ϕ})$ of a uniquely ergodic Cantor minimal $G$-system $\mathcal{R}_{G, ϕ}$ where $G$ is a countable discrete group. We compute fundamental groups of several uniquely ergodic Cantor minimal $G$-systems. We show that if $\mathcal{R}_{G, ϕ}$ arises from a free action $ϕ$ of a finitely generated abelian group, then there exists a unital countable subring $R$ of $\mathbb{R}$ such that $\mathcal{F}(\mathcal{R}_{G, ϕ})=R_{+}^\times$. We also consider the relation between fundamental groups of uniquely ergodic Cantor minimal $\mathbb{Z}^n$-systems and fundamental groups of crossed product $C^*$-algebras $C(X)\rtimes_ϕ \mathbb{Z}^n$.

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arXiv2010

Fundamental group of simple $C^*$-algebras with unique trace III

We introduce the fundamental group F(A) of a simple $σ$-unital $C^*$-algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of our previous works. Our definition in this paper makes sense for stably projectionless $C^*$-algebras. We show that there exist separable stably projectionless $C^*$-algebras such that their fundamental groups are equal to $\mathbb{R}_+^\times$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case. This study is motivated by the work of Kishimoto and Kumjian.

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arXiv2010

Fundamental group of simple $C^*$-algebras with unique trace II

We show that any countable subgroup of the multiplicative group $\mathbb{R}_+^{\times}$ of positive real numbers can be realized as the fundamental group $\mathcal{F}(A)$ of a separable simple unital $C^*$-algebra $A$ with unique trace. Furthermore for any fixed countable subgroup $G$ of $\mathbb{R}_+^{\times}$, there exist uncountably many mutually nonisomorphic such algebras $A$ with $G = \mathcal{F}(A)$.

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arXiv2009

Fundamental group of simple $C^*$-algebras with unique trace

We introduce the fundamental group ${\mathcal F}(A)$ of a unital simple $C^*$-algebra $A$ with a unique normalized trace. We compute fundamental groups ${\mathcal F}(A)$ of several nuclear or non-nuclear $C^*$-algebras $A$. K-theoretical obstruction enables us to compute the fundamental group easily. Our study is essentially based on the computation of Picard groups by Kodaka.

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arXiv2009

$C^*$-algebras associated with real multiplication

Noncommutative tori with real multiplication are the irrational rotation algebras that have special equivalence bimodules. Y. Manin proposed the use of noncommutative tori with real multiplication as a geometric framework for the study of abelian class field theory of real quadratic fields. In this paper, we consider the Cuntz-Pimsner algebras constructed by special equivalence bimodules of irrational rotation algebras. We shall show that associated $C^*$-algebras are simple and purely infinite. We compute the K-groups of associated $C^*$-algebras and show that these algebras are related to the solutions of Pell's equation and the unit groups of real quadratic fields. We consider the Morita equivalent classes of associated $C^*$-algebras.

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arXiv2008

Morita equivalent subalgebras of irrational rotation algebras and real quadratic fields

In this paper, we determine the isomorphic classes of Morita equivalent subalgebras of irrational rotation algebras. It is based on the solution of the quadratic Diophantine equations. We determine the irrational rotation algebras that have locally trivial inclusions. We compute the index of the locally trivial inclusions of irrational rotation algebras.

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