On a pattern of reflexive operator spaces

Abstract

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