Dilating covariant representations of the non-commutative disc algebras

TL;DR

This paper proves that contractive covariant representations of non-commutative disc algebra automorphisms can be dilated to unitary representations, establishing the C*-envelope of the associated semicrossed product.

Contribution

It demonstrates the dilation of covariant representations and identifies the C*-envelope for semicrossed products of non-commutative disc algebras under automorphisms.

Findings

Contractive covariant representations dilate to unitary representations.

C*-envelope of the semicrossed product is the crossed product of the Cuntz algebra with the automorphism.

Provides a dilation theory framework for non-commutative disc algebra automorphisms.

Abstract

Let $\phi$ be an isometric automorphism of the non-commutative disc algebra $\fA_n$ for $n \geq 2$. We show that every contractive covariant representation of $(\fA_n, \phi)$ dilates to a unitary covariant representation of $(\O_n, \phi)$. Hence the C*-envelope of the semicrossed product $\fA_n \times_{\phi} \bZ^+$ is $\O_n \times_{\phi} \bZ$.