Asymptotic reflexivity

We introduce a new version of reflexivity, akin to approximate reflexivity, called Asymptotic Reflexivity. We prove that the unital algebra generated by any operator in B(H), where H is a Hilbert space, is asymptotically reflexive. We also show that a linear subspace S of B(H) is asymptotically reflexive if and only if S∩F(H) is asymptotically, where F(H) is the set of finite rank operators in B(H). This result, in particular, implies that the space F(H) is asymptotically reflexive. An analogous version of Loginov–Shulman Theorem will be also proved for this notion of reflexivity. This result, in particular, implies that any linear subspace of normal operators is asymptotically reflexive. The relation between this notion of reflexivity and completely rank-nonincreasing maps will be studied as well.