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A characterization of the Razak–Jacelon algebra

Norio Nawata

Vol. 16 (2023), No. 8, 1799–1824
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 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 M2-stable C -algebra which is KK-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:

  1. For any 𝜃 [0,1], there exists a projection p in F(D) such that τD,ω(p) = 𝜃.

  2. If p and q are projections in F(D) such that 0 < τD,ω(p) = τD,ω(q), then p is Murray–von Neumann equivalent to q.

  3. 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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