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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 $W^*$-TRO'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 $II_{\lambda, \mu}$ with $\lambda, \mu = 1$ or $\infty$. It is shown that every separable stable $W^*$-TRO (which includes type $I_{\infty,\infty}$, type $II_{\infty, \infty}$ 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 ${\OL}_{1, 1^+}$ space (equivalently, a rectangular
 MSC Classifications: 46L07 - Operator spaces and completely bounded maps [See also 47L25] 46L08 - $C^*$-modules 46L89 - Other noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22]