Real operator systems
David P. Blecher, Travis B. Russell
TL;DR
This work develops a comprehensive theory of real operator systems, establishing a real variant of the Choi–Effros–Ozawa characterization and analyzing how complexification interacts with core constructions. It develops the real C*-covers, archimedeanization, quotients, duality, and the real MIN/MAX framework, revealing fundamental differences from the complex theory (notably the nonexistence of MIN/MAX functors in general) while identifying when real theories align with their complex counterparts. It then builds a full tensor product landscape for real operator systems, including minimal, maximal, commuting, and injective tensor products, and proves stability results showing that CPFP, WEP, DCEP, SLLP, and exactness are preserved under complexification. The paper culminates with real versions of the Kirchberg conjectures and Tsirelson’s problem, demonstrating equivalences with the complex case and providing intrinsic real-system perspectives on these central questions. Overall, it provides a robust, real-grounded framework for operator system theory with practical implications for real quantum information and the study of real C*-algebras.
Abstract
Operator systems are the unital self-adjoint subspaces of the bounded operators on a Hilbert space. Complex operator systems are an important category containing the C*-algebras and von Neumann algebras, which is increasingly of interest in modern analysis and also in modern quantum physics (such as quantum information theory). They have an extensive theory, and have very important applications in all of these subjects. We present here the real case of the theory of (complex) operator systems, and also the real case of their remarkable tensor product theory, due in the complex case to Paulsen and his coauthors and students (such as Kavruk), building on pioneering earlier work of Kirchberg and others. We uncover several notable differences between the real and complex theory, including the absence of minimal and maximal functors in the category of real operator systems. We also develop very many foundational structural results for real operator systems, and elucidate how the complexification interacts with the basic constructions in the subject. In the final two sections of our paper we study real analogues of the Kirchberg conjectures (and of several important related problems that have attracted much interest recently), and study the deep relationships between them.