AI 中文总结
本研究建立自然幺半群与2进环$C^*$-代数$\u27e8\u27e8Q_2$自同态的一一对应，分类指定像的自同态，给出$\u27e8\u27e8O_2$自同态延拓唯一性准则，构造反例解决相关公开问题，刻画保持$C^*(u)$的自同构并描述其Weyl群。
AI 中文摘要
本文建立了一个自然幺半群与2进环$C^*$-代数$\u27e8\u27e8Q_2$的自同态之间的一一对应关系。作为推论，我们对具有指定像的$\u27e8\u27e8Q_2$自同态进行了分类，并推导了若干准则，用于判定$\u27e8\u27e8Q_2$内部Cuntz代数$\u27e8\u27e8O_2$的典范拷贝的自同态延拓为$\u27e8\u27e8Q_2$自同态的唯一性。此外，我们构造了一个可延拓的$\u27e8\u27e8O_2$自同构实例，它无法表示为触发器自同构、规范自同构和内自同构的复合，从而对若干公开问题给出了否定答案。借助这一显式构造，我们还证明了$\u27e8\u27e8Aut(\u27e8\u27e8Q_2,\u27e8\u27e8O_2)$在外自同构群$\u27e8\u27e8Out(\u27e8\u27e8Q_2)$中的典范像是非交换的。最后，我们刻画了整体保持$C^*(u)$的$\u27e8\u27e8Q_2$自同构，并完整描述了Weyl群$\u27e8\u27e8W(\u27e8\u27e8Q_2, C^*(u))$，进而对若干相关公开问题给出了肯定答案。
英文摘要
In this paper, we establish a one-to-one correspondence between a natural monoid and the endomorphisms of the $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$. As a consequence, we classify endomorphisms of $\mathcal{Q}_2$ with prescribed images and derive several criteria for the uniqueness of extensions of endomorphisms of a canonical copy of the Cuntz algebra $\mathcal{O}_2$ inside $\mathcal{Q}_2$ to endomorphisms of $\mathcal{Q}_2$. Moreover, we construct an example of an extendable automorphism of $\mathcal{O}_2$ that is not a composition of the flip-flop automorphism, the gauge automorphisms, and inner automorphisms, thereby providing negative answers to certain open questions. Using this explicit construction, we also show that the canonical image of $\operatorname{Aut}(\mathcal{Q}_2,\mathcal{O}_2)$ in the outer automorphism group $\operatorname{Out}(\mathcal{Q}_2)$ is non-abelian. Finally, we characterize the automorphisms of $\mathcal{Q}_2$ that globally preserve $C^*(u)$, and completely describe the Weyl group $\mathcal{W}(\mathcal{Q}_2, C^*(u))$. Consequently, several related open questions are answered affirmatively.
