Type Decomposition and the Rectangular AFD Property for W*-TRO’s

Ruan, Zhong-Jin

Canadian journal of mathematics, Tome 56 (2004) no. 4, pp. 843-870

Résumé

We study the type decomposition and the rectangular $\text{AFD}$ property for ${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$ . Like von Neumann algebras, every ${{W}^{*}}-\text{TRO}$ can be uniquely decomposed into the direct sum of ${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$ of $\text{type}\,I,\,\text{type}\,II$ , and $\text{type}\,III$ . We may further consider ${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$ of $\text{type}\,{{I}_{m,n}}$ with cardinal numbers $m$ and $n$ , and consider ${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$ of $type\,I{{I}_{\lambda ,\mu }}\,\text{with}\,\lambda ,\,\mu \,=\,1\,\text{or}\,\infty $ . It is shown that every separable stable ${{W}^{*}}-\text{TRO}$ (which includes $\text{type}\,{{I}_{\infty ,\infty }}$ , $\text{type}\,I{{I}_{\infty ,\infty }}$ and $\text{type}\,III$ ) is $\text{TRO}$ -isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for ${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$ . One of our major results is to show that a separable ${{W}^{*}}-\text{TRO}$ 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 $\mathcal{O}{{\mathcal{L}}_{1,{{1}^{+}}}}$ space (equivalently, a rectangular $\mathcal{O}{{\mathcal{L}}_{1,{{1}^{+}}}}$ space).

Détail

DOI : 10.4153/CJM-2004-038-7

Mots-clés : 46L07, 46L08, 46L89

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