论文标题
Razak-Jacelon代数的特征
A characterization of the Razak-Jacelon algebra
论文作者
论文摘要
结合Elliott，Gong，Lin和Niu的结果以及Castillejos和Evington的结果，我们看到，如果$ a $是一个简单的可分开的可分开的核核单种子c $^*$ - 代数，则$ a \ otimes \ otimies \ otcal {w w} $是$ \ \ nath $ \ nath $ raz iS $ {代数。在本文中，我们给出了另一个证明。特别是，我们表明，如果$ \ Mathcal {d} $是一种简单的可分离核单纹状体$ m_ {2^{\ infty}} $ - 稳定的c $^*$ - algebra，是$ kk $ - 等效于$ \ \ \ \ \ {0 \} $，然后没有$ \ \ rathcal $ \ ring is $ is is i iso {考虑具有有限核维度的C $^*$ - 代数的奇特近似。我们的证明是基于Matui和Sato的技术，Schafhauser的想法证明了Tikuisis-White-Winter定理以及Kirchberg的Central Semecence C $^*$ -Elgebra $ f（\ Mathcal {d}）$ $ \ MATHCAL {D} $。请注意，$ f（\ Mathcal {d}）$的某些结果基于Elliott-Gong-Lin-Niu的稳定唯一性定理。另外，我们通过使用$ f（\ Mathcal {w}）$的属性来表征$ \ Mathcal {W} $。实际上，我们表明，当且仅当$ d $ in [0,1]中的任何$θ\ in [0,1] $中，$ f（d）$ p $ f（d）$ f（d）$ f（d） （ii）如果$ p $和$ q $是$ f（d）$中的预测，以至于$ 0 <τ_{d，ω}（p）=τ_{d，ω}（q）$，那么$ p $是Murray-von neumann等于$ q $，（iii）。
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 $\mathcal{W}$.
