Type decomposition and the rectangular AFD property for W*-TRO's

We study the type decomposition and the rectangular AFD property for W*-TRO's. Like von Neumann algebras, every W*-TRO can be uniquely decomposed into the direct sum of W*T'RO's of type I, type II, and type III. We may further consider W*-TRO's of type I_m,n with cardinal numbers m and n, and consider W*-TRO's of type II _λ,μ with λ,μ = 1 or ∞. It is shown that every separable stable W*-TRO (which includes type I_{∞, ∞}, type II_{∞, ∞} and type III) is TRO-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for W* -TRO's. One of our major results is to show that a separable W* -TRO is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular script O sign ℒ_1,1⁺ space (equivalently, a rectangular script O sign ℒ_1,1⁺ space.