A characterization of the Razak-Jacelon algebra

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${}^{\ast }$-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${}^{\ast }$-algebra which is $KK$-equivalent to $\{0\}$, then $\mathcal{D}$ is isomorphic to $\mathcal{W}$ without considering tracial approximations of C${}^{\ast }$-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${}^{\ast }$-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${}^{\ast }$-algebra $D$ is isomorphic to $\mathcal{W}$ if and only if $D$ satisfies the following properties: (i) for any $\theta \in [0,1]$, there exists a projection $p$ in $F(D)$ such that ${\tau }_{D,\omega }(p)=\theta$, (ii) if $p$ and $q$ are projections in $F(D)$ such that $0<{\tau }_{D,\omega }(p)={\tau }_{D,\omega }(q)$, then $p$ is Murray-von Neumann equivalent to $q$, (iii) there exists an injective homomorphism from $D$ to $\mathcal{W}$.

23 pages, added references, fixed typos(v2), fixed typos(v3) to appear in Analysis and PDE

References in corpus (4)

• Quasidiagonality of nuclear C*-algebras
• Nuclear dimension of simple C*-algebras
• Nuclear dimension of simple stably projectionless C*-algebras
• Trace scaling automorphisms of the stabilized Razak-Jacelon algebra