Biholomorphisms of the unit ball of C^n and semicrossed products

TL;DR

This paper characterizes when semicrossed products of non-commutative disc algebras are isomorphic, showing they correspond precisely to conjugate automorphisms, thus linking algebraic isomorphism to automorphism conjugacy.

Contribution

It establishes a complete classification of semicrossed products of certain operator algebras based on automorphism conjugacy, extending known results to non-commutative settings.

Findings

Semicrossed products are isomorphic iff automorphisms are conjugate.

Results apply to both non-commutative disc algebra and d-shift algebra.

Provides a classification linking algebra isomorphisms to automorphism conjugacy.

Abstract

Assume that $\phi_1$ and $\phi_2$ are automorphisms of the non-commutative disc algebra $\fA_n$, $n \geq 2$. We show that the semicrossed products $\fA_n \times_{\phi_1} \bZ^+$ and $\fA_n \times_{\phi_2} \bZ^+$ are isomorphic as algebras if and only if $\phi_1$ and $\phi_2$ are conjugate via an automorphism of $\fA_n$. A similar result holds for semicrossed products of the d-shift algebra $\A_d$, $d \geq 2$.