arXiv 1709.005542017-09math.OA

Cuntz-Krieger代数的超刚性子集及零点刚性性质

Hyperrigid subsets of Cuntz-Krieger algebras and the property of rigidity at zero

Guy Salomon

备注：: 22 pages. To appear in J. Operator Theory

摘要：

生成一个$C^*$-代数$A$的子集$\mathcal{G}$称为超刚性的，如果对每个忠实的非退化$*$-表示$A\subseteq B(H)$和一个酉完全正映射序列$ϕ_n:B(H) \to B(H)$，我们有其中所有收敛在范数内的\[ \lim_{n\to\infty}ϕ_n(g)= g~~\text{for all } g\in \mathcal{G} ~~ \implies ~~ \lim_{n\to\infty}ϕ_n(a)= a~~\text{for all } a\in A \].本文证明了与无孤立点的行有限有向图$G$相关联的Cuntz-Krieger代数$\mathcal{O}(G)$的部分等距集$\mathcal{E}=\{S_e:e\in E\}$是超刚性的.此外，我们定义和检查一个密切相关的概念：在$0$刚性的属性。一个$C^*$-代数$A$的生成子集$\mathcal{G}$称为在$0$上是刚性的，如果对于每个对每个$g\in \mathcal{G}$满足$\lim_{n\to \infty}φ_n(g)=0$的压缩正映射序列$φ_n:A\to \mathbb C$，我们对每个$a\in A$都有那个$\lim_{n\to \infty}φ_n(a)=0$。我们表明，当结合，超刚性和刚性$0$相当于一个更强的概念超刚性，我们连接到唯一的扩展属性。然而，对于发电机组$\mathcal{E}$，情况并非如此。更精确地说，我们证明了，对于任何图$G$，生成$\mathcal{O}(G)$的Cuntz-Krieger家族的子集在$0$是刚性的当且仅当它们包含每个顶点投影。

英文摘要：

A subset $\mathcal{G}$ generating a $C^*$-algebra $A$ is said to be hyperrigid if for every faithful nondegenerate $*$-representation $A\subseteq B(H)$ and a sequence $ϕ_n:B(H) \to B(H)$ of unital completely positive maps, we have that \[ \lim_{n\to\infty}ϕ_n(g)= g~~\text{for all } g\in \mathcal{G} ~~ \implies ~~ \lim_{n\to\infty}ϕ_n(a)= a~~\text{for all } a\in A \] where all convergence are in norm. In this paper, we show that for the Cuntz-Krieger algebra $\mathcal{O}(G)$ associated to a row-finite directed graph $G$ with no isolated vertices, the set of partial isometries $\mathcal{E}=\{S_e:e\in E\}$ is hyperrigid. In addition, we define and examine a closely related notion: the property of rigidity at $0$. A generating subset $\mathcal{G}$ of a $C^*$-algebra $A$ is said to be rigid at $0$ if for every sequence of contractive positive maps $φ_n:A\to \mathbb C$ satisfying $\lim_{n\to \infty}φ_n(g)=0$ for every $g\in \mathcal{G}$, we have that $\lim_{n\to \infty}φ_n(a)=0$ for every $a\in A$. We show that, when combined, hyperrigidity and rigidity at $0$ are equivalent to a somewhat stronger notion of hyperrigidity, and we connect this to the unique extension property. This, however, is not the case for the generating set $\mathcal{E}$. More precisely, we show that for any graph $G$, subsets of the Cuntz-Krieger family generating $\mathcal{O}(G)$ are rigid at $0$ if and only if they contain every vertex projection.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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