On Equivalence for Representations of Toeplitz Algebras

TL;DR

This paper introduces new notions of equivalence for representations of Toeplitz algebras, with implications for $C^*$-dynamics and endomorphism conjugacy, generalizing previous results by Laca and Enomoto-Watatani.

Contribution

It defines novel equivalence concepts for Toeplitz algebra representations and connects them to multiplicity, extending known results in the field.

Findings

New equivalence notions for Toeplitz algebra representations

Application to $C^*$-dynamics and endomorphism conjugacy

Recovery of classical results as special cases

Abstract

Two new notions of equivalence for representations of a Toeplitz algebra $\mathcal{E}_n$, $n<\infty$, on a common Hilbert space are defined. Our main results apply to $C^*$-dynamics and the conjugacy of certain $*$-endomorphisms. One particular case of the relations is shown to coincide with the multiplicity of a representation. Previously known results due to Laca and Enomoto-Watatani are recovered as special cases.