On a pattern of reflexive operator spaces

A linear subspace $M$ is a separating subspace for an operator space $S$ if the only member of $S$ annihilating $M$ is 0. It is proved in this paper that if $S$ has a strictly separating vector $x$ and a separating subspace $M$ satisfying $Sx \cap [SM] = \{0\}$, then $S$ is reflexive. Applying this to finite dimensional $S$ leads to more results on reflexivity. For example, if dim $S = n$, and every nonzero operator in $S$ has rank $> n^{2}$, then $S$ is reflexive.