Subhomogeneous Operator Systems and Classification of Operator Systems Generated by Λ-Commuting Unitaries

A unital C^* 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 UCP (S) UCP ( S ) which is the matrix state space associated with an operator system S S 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.