Abstract view
Type Decomposition and the Rectangular AFD Property for $W^*$TRO's


Published:20040801
Printed: Aug 2004
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 $W^*$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 TROisomorphic 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