Subhomogeneous Operator Systems and Classification of Operator Systems Generated by $\Lambda$-Commuting Unitaries

TL;DR

This paper extends the concept of subhomogeneity from $C^*$-algebras to operator systems, providing classification results for operator systems generated by $ ext{Lambda}$-commuting unitaries and related non-commutative structures.

Contribution

It introduces a new framework for subhomogeneity in operator systems and applies it to classify systems generated by $ ext{Lambda}$-commuting unitaries and non-commutative tori.

Findings

Two $N$-subhomogeneous operator systems are completely order equivalent iff they are $N$-order equivalent.

Unital $N$-positive maps into finite dimensional $N$-subhomogeneous systems are completely positive.

Classification of operator systems generated by $q$-commuting unitaries and higher-dimensional non-commutative tori.

Abstract

A unital $C^*$-algebra is called $N$-subhomogeneous if its irreducible representations are finite dimensional with dimension at most $N$. We extend this notion to operator systems, replacing irreducible representations by boundary representations. This is done by considering the matrix state space associated with an operator system and identifying the boundary representations as absolute matrix extreme points. We show that two $N$-subhomogeneous operator systems are completely order equivalent if and only if they are $N$-order equivalent. Moreover, we show that a unital $N$-positive map into a finite dimensional $N$-subhomogeneous operator system is completely positive. We apply these tools to classify pairs of $q$-commuting unitaries up to $*$-isomorphism. Similar results are obtained for operator systems related to higher dimensional non-commutative tori.