Locally linearly dependent operators and reflexivity of operator spaces*1

Abstract

We obtain two results on the existence of large subspaces of operators of small rank in locally linearly dependent spaces of operators. As a consequence we obtain an upper bound for the rank of operators belonging to a minimal locally linearly dependent space of operators. It has been known that the only obstruction to the reflexivity of a finite-dimensional operator space comes from the operators with small ranks. Our results improve known bounds on the minimal rank that guarantees the reflexivity.