The classification problem for finitely generated operator systems and spaces

Abstract

The classification of separable operator spaces and systems is commonly believed to be intractable. We analyze this belief from the point of view of Borel complexity theory. On one hand we confirm that the classification problems for arbitrary separable operator systems and spaces are intractable. On the other hand we show that the finitely generated operator systems and spaces are completely classifiable (or smooth); in fact a finitely generated operator system is classified by its complete theory when regarded as a structure in continuous logic. In the particular case of operator systems generated by a single unitary, a complete invariant is given by the spectrum of the unitary up to a rigid motion of the circle, provided that the spectrum contains at least 5 points. As a consequence of these results we show that the relation on compact subsets of ${\mathbb{C}}^n$, given by homeomorphism via a degree 1 polynomial, is smooth.

v2: 32 pages. Minor corrections in Section 4

References in corpus (5)

• Characterisations of the weak expectation property
• Minimal and Maximal Operator Spaces and Operator Systems in Entanglement Theory
• The noncommutative Choquet boundary III: Operator systems in matrix algebras
• Completeness of the isomorphism problem for separable C*-algebras
• The complexity of the homeomorphism relation between compact metric spaces

Cited by in corpus (5)

• Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization
• Crossed products of operator systems
• Complexity of distances: Theory of generalized analytic equivalence relations
• Complexity of distances: Reductions of distances between metric and Banach spaces
• Embeddings and $C^{\ast }$-envelopes of exact operator systems