A general view of reflexivity

Abstract

Abstract. Various concepts of reflexivity for an algebra or linear space of operators have been studied by operator theorists and algebraists. This paper contains a very general version of reflexivity based on dual pairs of vector spaces over a HausdorfF field. The special cases include topological, algebraic and approximate reflexivity. In addition general versions of hyperreflexivity and direct integrals are introduced. We prove general versions of many known (and some new) theorems, often with simpler proofs.

On the reflexivity and hyperreflexivity of algebras and subspaces

“…Isometries with orthogonal ranges were considered in [18]. In [23] a more abstract definition of reflexivity and hyperreflexivity was presented.…”

Algebraic reflexivity of linear transformations

“…It is well known and easy to prove that if S has a separating vector, then it is algebraically 2-reflexive. The local dimension of S, denoted by k(S), is defined by [1] and [3] are excellent references.) The main result in [6] states that if S is an n-dimensional subspace of B(H), where H is a separable complex Hilbert space, then S is algebraically √ 2n -reflexive.…”

On a pattern of reflexive operator spaces

“…If ref S = S, S is said to be reflexive [2], [8], [11]. In particular, if S is a unital algebra, then ref S = alglatS, where alglatS is the algebra of continuous linear operators that leave invariant all S-invariant subspaces.…”