Illinois Journal of Mathematics

The {OLLP} and $\scr T$-local reflexivity of operator spaces

Z. Dong

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In this paper, we study two `dual' problems in the operator space theory. We first show that if $L$ is a finite-dimensional operator space, then $L$ has the OLLP if and only if for any indexed family of operator spaces $(W_{i})_{i\in I}$ and a free ultrafilter $\mathcal{U}$ on $I$, we have a complete isometry \[ \prod(L\ha W_{i})/\mathcal{U}=L\ha\prod W_{i}/\mathcal{U}. \] Next, we show that if $W$ is an operator space, then $(T_{n}\ck W )^{**}=T_{n}\ck W^{**}$ holds if and only if $W$ is $\mathcal{T}$-locally reflexive, if and only if for any finitely representable operator spaces $V$, we have an isometry $\mathcal{I}(V, W^{*})=(V\ck W)^{*}$.

Article information

Illinois J. Math., Volume 51, Number 4 (2007), 1103-1122.

First available in Project Euclid: 13 November 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L07: Operator spaces and completely bounded maps [See also 47L25]
Secondary: 46B28: Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20]


Dong, Z. The {OLLP} and $\scr T$-local reflexivity of operator spaces. Illinois J. Math. 51 (2007), no. 4, 1103--1122. doi:10.1215/ijm/1258138535.

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