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 l² with an operator space structure which is (i) completely almost transitive, and (ii) homogeneous, but not 1-homogeneous.