Conjugate Dynamical Systems on C*-algebras

TL;DR

This paper characterizes when semicrossed products of C*-dynamical systems are isometrically isomorphic, establishing a connection to outer conjugacy under certain conditions on the *-endomorphisms.

Contribution

It proves that semicrossed products are isometrically isomorphic if and only if the systems are outer conjugate, extending previous results to cases with injective or surjective endomorphisms.

Findings

Semicrossed products are isometrically isomorphic iff systems are outer conjugate.

The result holds when the endomorphism is injective or surjective.

The conclusion applies to various other cases as well.

Abstract

Let $(A, \alpha)$ and $(B, \beta)$ be C*-dynamical systems where $\alpha$ and $\beta$ are arbitrary *-endomorphisms. When $\alpha$ is injective or surjective, we show that the semicrossed products $A \times_\alpha \mathbb{Z}$ and $B \times_\beta \mathbb{Z}$ are isometrically isomorphic if and only if $(A, \alpha)$ and $(B, \beta)$ are outer conjugate. This conclusion also holds in various other cases as well.