遗传本质$C^*$-包含的唯一伪期望
Unique Pseudo-Expectations for Hereditarily Essential $C^*$-Inclusions
AI总结：
本文研究遗传本质$C^*$-包含的伪期望唯一性公开问题，对扭$C^*$-动力系统对应的几类包含给出肯定回答，同时构造反例否定一般情形，并指出正则情形仍可能成立。
摘要：
若对每个中间$C^*$-代数$\u27eb\ud835\u2c9c \u2286 \ud835\u2c9e \u2286 \ud835\u2c9f\u27ec$以及每个非零理想$\u27eb\{0\} \u2260 \ud835\u2c9b \u22b4 \ud835\u2c9e\u27ec$，都有$\ud835\u2c9b \u2229 \ud835\u2c9c \u2260 \{0\}$，则称$C^*$-包含$\ud835\u2c9c \u2286 \ud835\u2c9f$是遗传本质的。也就是说，$\ud835\u2c9c$可检测每个中间$C^*$-代数$\ud835\u2c9c \u2286 \ud835\u2c9e \u2286 \ud835\u2c9f$中的理想。根据Pitts和Zarikian的结果，含单位元的$C^*$-包含$\ud835\u2c9c \u2286 \ud835\u2c9f$是遗传本质的，当且仅当$\ud835\u2c9c \u2286 \ud835\u2c9f$的每个伪期望$\u03b8:\ud835\u2c9f \u2192 I(\ud835\u2c9c)$都是忠实的。一个存在十年的公开问题是：遗传本质$C^*$-包含是否必定具有唯一伪期望？在本注记中，我们对几类重要的$C^*$-包含给出了肯定回答，尤其是形如$\ud835\u2c9c \u2286 \ud835\u2c9c \u22ca_{\u03b1,r}^\u03c3 G$的包含，其中$(\ud835\u2c9c,G,\u03b1,\u03c3)$是扭$C^*$-动力系统。另一方面，我们通过构造具有多个条件期望的形如$C_r^*(G) \u2286 C(X) \u22ca_{\u03b1,r} G$的$C^*$-不可约包含，对一般问题给出了否定解答。我们的结果留下了如下可能性：该问题对正则遗传本质$C^*$-包含可能有肯定答案。
英文摘要：
The $C^*$-inclusion $\mathcal{A} \subseteq \mathcal{B}$ is said to be hereditarily essential if for every intermediate $C^*$-algebra $\mathcal{A} \subseteq \mathcal{C} \subseteq \mathcal{B}$ and every non-zero ideal $\{0\} \neq \mathcal{J} \unlhd \mathcal{C}$, we have that $\mathcal{J} \cap \mathcal{A} \neq \{0\}$. That is, $\mathcal{A}$ detects ideals in every intermediate $C^*$-algebra $\mathcal{A} \subseteq \mathcal{C} \subseteq \mathcal{B}$. By a result of Pitts and Zarikian, a unital $C^*$-inclusion $\mathcal{A} \subseteq \mathcal{B}$ is hereditarily essential if and only if every pseudo-expectation $θ:\mathcal{B} \to I(\mathcal{A})$ for $\mathcal{A} \subseteq \mathcal{B}$ is faithful. A decade-old open question asks whether hereditarily essential $C^*$-inclusions must have unique pseudo-expectations? In this note, we answer the question affirmatively for some important classes of $C^*$-inclusions, in particular those of the form $\mathcal{A} \subseteq \mathcal{A} \rtimes_{α,r}^σG$, for a twisted $C^*$-dynamical system $(\mathcal{A},G,α,σ)$. On the other hand, we settle the general question negatively by exhibiting $C^*$-irreducible inclusions of the form $C_r^*(G) \subseteq C(X) \rtimes_{α,r} G$ with multiple conditional expectations. Our results leave open the possibility that the question might have a positive answer for regular hereditarily essential $C^*$-inclusions.
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。
