Real decomposable maps on operator systems

2025/09/22 by Blecher, David P., Pretorius, Christiaan H.
#47B92 #47L05 #47L07 #47L25 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Primary 46L07 #Secondary: 46M05

paper · doi:10.48550/arxiv.2509.17708

Abstract

We initiate and study the theory of ``real decomposable maps" between real operator systems. (Formally, this is new even in the complex case, which hitherto has restricted itself to the case where the systems are complex C^*-algebras.) We investigate how the definition interacts with the existing theory (which it generalizes) and with the complexification. A surprising term appears in the `Jordan decomposition' of real decomposable maps. This term constitutes a new class of completely bounded maps, a class that also showed up in disguised form in our recent study of real noncommutative (nc) convexity, and whose theory is likely to have applications in that subject. We also check the real case of many important known results related to decomposability, for example about the weak expectation property or injectivity of von Neumann algebras.

Citations

• Exactness and LLP Results via Operator System Methods
• Real Non-Commutative Convexity I
• Real operator systems
• Real structure in operator spaces, injective envelopes and G-spaces
• Similarities and differences between real and complex Banach spaces: an overview and recent developments
• Real operator algebras and real positive maps
• Noncommutative Choquet theory
• Real Operator Algebras and Real Completely Isometric Theory
• The predual of the space of decomposable maps from a C^*-algebra into a von Neumann algebra
• Tensor Products of Operator Systems
• About the QWEP conjecture

Cited by

• Real Non-Commutative Convexity I

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• Completely Order Bounded Maps on Non-Commutative L_p-Spaces