Reverse monotone approximation property

A Banach space X is said to have the Reverse Monotone Approximation Property (RMAP) if there exists a net of finite rank operators (T-alpha) on X, converging to the identity point-norm, and such that lim(alpha) parallel to I-X - T-alpha parallel to = 1. We show that the RMAP is rare among "naturally occurring" Banach spaces. For instance, any separable rearrangement invariant function space with the RMAP is isometric to L-2. Similar results are obtained in the non-commutative setting. On the other hand, any separable Banach space with the Commuting Bounded Approximation Property can be renormed to have the RMAP.