AbstractWe 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*- TRO's of type I, type II, and type III. We may further considerW*-TRO's of type Im,n with cardinal numbers m and n, and considerW*-TRO's of type IIλ,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 space (equivalently, a rectangular space).