Projective tensor product of proto-quantum spaces

A. Ya. Helemskii¹

¹ Faculty of Mechanics and Mathematics Moscow State (Lomonosov) University Moscow, 111991, Russia

Colloquium Mathematicum, Tome 149 (2017) no. 1, pp. 45-73

Résumé

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.

Détail

DOI : 10.4064/cm6921-11-2016

Keywords: proto quantum space general matricially normed space sense effros ruan presented matrix free language these spaces have special projective tensor product possessing universal property respect completely bounded bilinear operators study general properties tensor product among kind adjoint associativity compute tensor factors notably spaces particular obtain what could called proto quantum version grothendieck theorem about classical projective tensor products spaces finally compare tensor product known projective tensor product operator spaces standard construction latter fit general proto quantum spaces