Higher reflections and entropy of canonical shifts for inclusions of $C^*$-algebras with finite Watatani index

TL;DR

This paper investigates the structure of inclusions of simple $C^*$-algebras with finite Watatani index, introducing reflection operators and connecting the entropy of canonical shifts to the Watatani index.

Contribution

It introduces reflection operators on relative commutants and links the entropy of canonical shifts to the Watatani index, refining Fourier inequalities in the process.

Findings

Reflection operators are unital, involutive, $*$-preserving anti-homomorphisms.

Fourier inequalities are established on higher relative commutants.

The entropy of the canonical shift is connected to the minimal Watatani index.

Abstract

Given a unital inclusion of simple $C^*$-algebras equipped with a conditional expectation of index-finite type, we study Fourier transforms and rotation operators and introduce the reflection operators on the relative commutants. We prove that the reflections are unital, involutive, $*$-preserving anti-homomorphisms that preserve certain Markov-type traces. As an application, we prove Fourier theoretic inequalities on the higher relative commutants and refine the existing constant in Young's inequality as presented in the current literature. By employing the reflection operators, we define a canonical shift on the von Neumann algebra generated by the relative commutants. We establish a connection between the Connes-St{\o}rmer entropy of the canonical shift and the minimal Watatani index.