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The standard dual of an operator space. (English) Zbl 0726.47030
If X is a linear space of operators on a Hilbert space, then it turns out that there is a natural, enlightening, nontrivial way to regard the dual \(X^*\) also as a linear space of operators on a (different) Hilbert space. The importance of this was pointed out in the work with V. I. Paulsen [‘Tensor products of operator spaces’, J. Functional Anal., to appear], and also independently by Effros and Ruan. The purpose of this paper is to establish some fundamental properties of this construction, and to examine how it interacts with other natural categorical constructions with operator spaces.
In the latter half of the paper we define and study a notion of projectivity (dual to injectivity) for operator spaces, and give a noncommutative version of Grothendieck’s characterization of \(\ell^ 1(I)\) spaces for a discrete set I.

MSC:
47L50 Dual spaces of operator algebras
47L05 Linear spaces of operators
46M10 Projective and injective objects in functional analysis
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