Some notions of transitivity for operator spaces

The famous Mazur Rotation Problem asks whether any separable transitive Banach space (that is, a Banach space where any point on the unit sphere can be mapped into any other point on the unit sphere by a surjective isometry) is necessarily isometric to a Hilbert space. In spite of enormous progress since the 1930's, the problem remains open. In this paper we investigate related non-commutative phenomena. We show that the only completely uniquely maximal (or matrix convex transitive) operator space is a one-dimensional one. Relaxing the conditions somewhat, we show that any matrix-level convex transitive finite dimensional space has to be completely isometric to Pisier's space OH, of corresponding dimension. Finally, we equip l(2) with an operator space structure which is (i) completely almost transitive, and (ii) homogeneous, but not 1-homogeneous.