  We initiate the study of matrix convexity for operator spaces. We define the
notion of compact rectangular matrix convex set, and prove the natural analogs
of the Krein-Milman and the bipolar theorems in this context. We deduce a
canonical correspondence between compact rectangular matrix convex sets and
operator spaces. We also introduce the notion of boundary representation for an
operator space, and prove the natural analog of Arveson's conjecture: every
operator space is completely normed by its boundary representations. This
yields a canonical construction of the triple envelope of an operator space.
