Automorphisms of free products and their application to multivariable dynamics

TL;DR

This paper characterizes automorphisms of free product algebras and applies these results to classify multivariable dynamical systems through their semicrossed product algebras, revealing new insights into their structure and symmetries.

Contribution

It establishes a complete description of automorphisms of free product algebras and introduces a new conjugacy concept for multivariable dynamical systems.

Findings

Automorphisms are permutations and component automorphisms of free products.

A new dynamical conjugacy characterizes semicrossed product algebras.

Conditions for isometric isomorphism between semicrossed and tensor algebras.

Abstract

We examine the completely isometric automorphisms of a free product of noncommutative disc algebras. It will be established that such an automorphism is given simply by a completely isometric automorphism of each component of the free product and a permutation of the components. This mirrors a similar fact in topology concerning biholomorphic automorphisms of product spaces with nice boundaries due to Rudin, Ligocka and Tsyganov. This paper is also a study of multivariable dynamical systems by their semicrossed product algebras. A new form of dynamical system conjugacy is introduced and is shown to completely characterize the semicrossed product algebra. This is proven by using the rigidity of free product automorphisms established in the first part of the paper. Lastly, a representation theory is developed to determine when the semicrossed product algebra and the tensor algebra of a dynamical system are completely isometrically isomorphic.