Classification of Finitely Generated Operator Systems

Abstract

For the past few decades operator systems and their C*-envelopes have provided an invaluable tool for studying the theory of C*-algebras and positive maps. They provide the natural context in which to study the theory of completely positive maps. Furthermore, many of the important open problems in quantum information theory have found equivalent formulations in terms of operator systems. The question of the classification of operator systems and computing their C*-envelopes have been the center of much interest. Borrowing from the theory of representations of commutative C*-algebras by affine maps, we construct a new tool for classifying certain types of finitely generated operator systems. Using this tool, we show that all the information regarding such operator systems is usually encoded in the joint spectra of their generating operators. Using this tool we completely classify operator systems generated by finitely many normal operators. We also provide a different proof for the classification theorem of operator systems generated by a unitary with spectrum size different that 4. Furthermore, we settle the classification problem for operator systems generated by a single unitary with four points in its spectrum. In addition, we compute the C*-envelopes of such operator systems. Furthermore, we apply this tool to the classification problem of those operator systems generated by a unilateral shift with arbitrary multiplicity or by an isometry and we compute their C*-envelopes.