On the entropy and index of the winding endomorphisms of p-adic ring C${}^{\ast }$-algebras

Abstract

For $p\ge 2$, the $p$-adic ring $C^{\ast }$-algebra ${\mathcal{Q}}_p$ is the universal $C^{\ast }$-algebra generated by a unitary $U$ and an isometry $S_p$ such that $S_{p}U=U^p{S}_p$ and ${\sum }_{l=0}^{p-1}{U}^l{S}_p{S}_p^{\ast }{U}^{-l}=1$. For any $k$ coprime with $p$ we define an endomorphism ${\chi }_k\in \mathrm{End}({\mathcal{Q}}_p)$ 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.

References in corpus (2)

• Entropy of shifts on higher-rank graph C*-algebras
• On the cyclic automorphism of the Cuntz algebra and its fixed-point algebra

Cited by in corpus (1)

• On the cyclic automorphism of the Cuntz algebra and its fixed-point algebra