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

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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 Im,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 ℒ1,1+ space (equivalently, a rectangular script O sign ℒ1,1+ space.

Original languageEnglish (US)
Pages (from-to)843-870
Number of pages28
JournalCanadian Journal of Mathematics
Volume56
Issue number4
DOIs
StatePublished - Aug 2004

ASJC Scopus subject areas

  • Mathematics(all)

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