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

IF 0.7 3区 数学 Q2 MATHEMATICS
Valeriano Aiello, S. Rossi
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引用次数: 1

Abstract

For $p\geq 2$, the $p$-adic ring $C^*$-algebra $\mathcal{Q}_p$ is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_p$ such that $S_pU=U^pS_p$ and $\sum_{l=0}^{p-1}U^lS_pS_p^*U^{-l}=1$. For any $k$ coprime with $p$ we define an endomorphism $\chi_k\in{\rm End}(\mathcal{Q}_p)$ by setting $\chi_k(U):=U^k$ and $\chi_k(S_p):=S_p$. We then compute the entropy of $\chi_k$, which turns out to be $\log |k|$. Finally, for selected values of $k$ we also compute the Watatani index of $\chi_k$ showing that the entropy is the natural logarithm of the index.
$p$-进环$C^*$-代数的缠绕自同态的熵和指标
对于$p\geq 2$, $p$ -adic环$C^*$ -algebra $\mathcal{Q}_p$是由一个幺正$U$和一个等距$S_p$生成的普数$C^*$ -代数,使得$S_pU=U^pS_p$和$\sum_{l=0}^{p-1}U^lS_pS_p^*U^{-l}=1$。对于任何具有$p$的$k$协素数,我们通过设置$\chi_k(U):=U^k$和$\chi_k(S_p):=S_p$来定义一个自同态$\chi_k\in{\rm End}(\mathcal{Q}_p)$。然后我们计算$\chi_k$的熵,结果是$\log |k|$。最后,对于$k$的选定值,我们还计算了$\chi_k$的Watatani指数,表明熵是该指数的自然对数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Studia Mathematica
Studia Mathematica 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
72
审稿时长
5 months
期刊介绍: The journal publishes original papers in English, French, German and Russian, mainly in functional analysis, abstract methods of mathematical analysis and probability theory.
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