A characterization of the Razak–Jacelon algebra

Nawata, Norio

doi:10.2140/apde.2023.16.1799

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A characterization of the Razak–Jacelon algebra

Norio Nawata

Vol. 16 (2023), No. 8, 1799–1824

DOI: 10.2140/apde.2023.16.1799

Abstract

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 $A \otimes 𝒲$ is isomorphic to $𝒲$ , where $𝒲$ is the Razak–Jacelon algebra. In this paper, we give another proof of this. In particular, we show that if $𝒟$ is a simple separable nuclear monotracial $M_{2^{\infty}}$ -stable $C^{*}$ -algebra which is $K K$ -equivalent to ${0}$ , then $𝒟$ is isomorphic to $𝒲$ 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 $F (𝒟)$ of $𝒟$ . Note that some results for $F (𝒟)$ are based on Elliott, Gong, Lin and Niu’s stable uniqueness theorem. Also, we characterize $𝒲$ by using properties of $F (𝒲)$ . Indeed, we show that a simple separable nuclear monotracial $C^{*}$ -algebra $D$ is isomorphic to $𝒲$ if and only if $D$ satisfies the following properties:

For any $𝜃 \in [0, 1]$ , there exists a projection $p$ in $F (D)$ such that $τ_{D, ω} (p) = 𝜃$ .
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 $q$ .
There exists an injective homomorphism from $D$ to $𝒲$ .

Keywords

stably projectionless $\mathrm{C}^*$-algebra, Kirchberg's central sequence $\mathrm{C}^*$-algebra, $K\mkern-3mu K$-contractible $\mathrm{C}^*$-algebra, Razak–Jacelon algebra

Mathematical Subject Classification

Primary: 46L35

Secondary: 46L40, 46L80

Milestones

Received: 7 January 2021

Revised: 12 January 2022

Accepted: 14 February 2022

Published: 16 October 2023

Authors

Norio Nawata
Department of Pure and Applied Mathematics Graduate School of Information Science and Technology Osaka University Osaka Japan

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Vol. 16, No. 8, 2023