On the entropy and index of the winding endomorphisms of $p$-adic ring $C^*$-algebras
Volume 262 / 2022
Studia Mathematica 262 (2022), 305-326
MSC: Primary 46L55, 46L40, 28D20; Secondary 37A25.
DOI: 10.4064/sm201125-9-2
Published online: 18 October 2021
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 to $p$ 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.
