A+ CATEGORY SCIENTIFIC UNIT

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image