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A new approach to operator spaces. (English) Zbl 0769.46037
Summary: The authors previously observed that the space of completely bounded maps between two operator spaces can be realized as an operator space. In particular, with the appropriate matricial norms the dual \(V^ \uparrow\) of an operator space \(V\) is completely isometric to a linear space of operators. This approach to duality enables one to formulate new analogues of Banach space concepts and results. In particular, there is an operator space version \(\otimes_ \mu\) of the Banach space projective tensor product \(\widehat {\otimes}\), which satisfies the expected functorial properties. As is the case for Banach spaces, given an operator space \(V\), the functor \(W\mapsto V\otimes_ \mu W\) preserves inclusions if and only if \(V^ \uparrow\) is an injective operator space.

MSC:
46L05 General theory of \(C^*\)-algebras
46M05 Tensor products in functional analysis
46M10 Projective and injective objects in functional analysis
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