Complexifications of real operator spaces (Q1419634)

This paper is a continuation of the author's previous work on real operator spaces [Acta Math. Sin. Engl. Ser. 19, 485-496 (2003; Zbl 1058.46037)]. The main result of the present paper is the following interesting theorem on the complexification of real operator spaces. Theorem. Let \(V\) be a real operator space and \(V_c\) its algebraic complexification. Then there is unique complex operator space structure on \(V_c\) which extends the original real operator space structure on \(V\) and such that the natural conjugation on \(V_c\) is completely isometric. This theorem is powerful in the sense that it reduces problems on real operator spaces to the corresponding ones on their complexifications. For instance, \(V\) is injective, nuclear or exact if and only if \(V_c\), equipped with the structure above, is injective, nuclear or exact, respectively.