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

Abstract

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 $\mathcal{I}(V, W^{*})=(V\ck W)^{*}$.