On the entropy and index of the winding endomorphisms of p-adic ring C$^*$-algebras

Normalizers and permutative endomorphisms of the $2$-adic ring $C^*$-algebra

A complete description is provided for the unitary normalizer of the diagonal Cartan subalgebra $\mathcal{D}_2$ in the $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$, which generalizes and unifies analogous results for Cuntz and Bunce-Deddens algebras. Furthermore, the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$ is proved not to be regular. Finally, countably many novel permutative endomorphisms of $\mathcal{Q}_2$ are exhibited with prescribed images of the generator $U$.

2进环$C^*$-代数的自同态及其Weyl群

Endomorphisms of the 2-adic ring $C^*$-algebra and its Weyl group

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。

A look at the inner structure of the $2$-adic ring $C^*$-algebra and its automorphism groups

We undertake a systematic study of the so-called $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$. This is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_2$ such that $S_2U=U^2S_2$ and $S_2S_2^*+US_2S_2^*U^*=1$. Notably, it contains a copy of the Cuntz algebra $\mathcal{O}_2=C^*(S_1, S_2)$ through the injective homomorphism mapping $S_1$ to $US_2$. Among the main results, the relative commutant $C^*(S_2)'\cap \mathcal{Q}_2$ is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$, namely the endomorphis…