The isomorphism problem for tensor algebras of multivariable dynamical systems

TL;DR

This paper solves the isomorphism problem for tensor algebras of multivariable dynamical systems, establishing a complete invariant based on unitary equivalence, and confirms a longstanding conjecture in the field.

Contribution

It provides a complete classification criterion for tensor algebras of multivariable dynamical systems, resolving a conjecture and extending previous work.

Findings

Unitary equivalence after conjugation is a complete invariant.

Resolved a conjecture of Davidson and Kakariadis.

Extended prior results of Kakariadis and Katsoulis.

Abstract

We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis relating to work of Arveson from the sixties, and extends related work of Kakariadis and Katsoulis.