## Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 843-870 |

Number of pages | 28 |

Journal | Canadian Journal of Mathematics |

Volume | 56 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2004 |

## ASJC Scopus subject areas

- Mathematics(all)