The notion of a dual of
an operator space which is again an operator space has been introduced
independently by Vern Paulsen and the author, and by Effros and Ruan. Its
significance in the theory of tensor products of operator spaces has already
been partially explored by the aforementioned. Here we establish some other
fundamental properties of this dual construction, and examine how it interacts with
other natural categorical constructs for operator spaces. We define and study
a notion of projectivity for operator spaces, and give a noncommutative
version of Grothendieck’s characterization of l1(I) spaces for a discrete set
I.
