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Projective tensor product of proto-quantum spaces

Volume 149 / 2017

A. Ya. Helemskii Colloquium Mathematicum 149 (2017), 45-73 MSC: 46L07, 46M05. DOI: 10.4064/cm6921-11-2016 Published online: 18 April 2017

Abstract

A proto-quantum space is a (general) matricially normed space in the sense of Effros and Ruan presented in a ‘matrix-free’ language. We show that these spaces have a special (projective) tensor product possessing the universal property with respect to completely bounded bilinear operators. We study some general properties of this tensor product (among them a kind of adjoint associativity), and compute it for some tensor factors, notably for $L_1$-spaces. In particular, we obtain what could be called the proto-quantum version of the Grothendieck theorem about classical projective tensor products by $L_1$-spaces. Finally, we compare the new tensor product with the known projective tensor product of operator spaces, and show that the standard construction of the latter is not fit for general proto-quantum spaces.

Authors

  • A. Ya. HelemskiiFaculty of Mechanics and Mathematics
    Moscow State (Lomonosov) University
    Moscow, 111991, Russia
    e-mail

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