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

Volume 262 / 2022

Valeriano Aiello, Stefano Rossi 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.

Authors

  • Valeriano AielloMathematisches Institut
    Universität Bern
    Alpeneggstrasse 22
    3012 Bern, Switzerland
    e-mail
  • Stefano RossiDipartimento di Matematica
    Università degli Studi Aldo Moro di Bari
    Via E. Orabona 4
    70125 Bari, Italy
    e-mail

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