On the reflexivity of algebras and linear spaces of operators (Q1076952)

Let A be a weakly closed algebra of operators on Hilbert space H. Assume that for all weakly continuous functionals \(\{\phi_{ij}:\) \(1\leq i\leq 2\), \(1\leq j\leq 3\}\) on A there exist \(x_ 1,x_ 2,y_ 1,y_ 2,y_ 3\in H\) such that \(\phi_{ij}(T)=(Tx_ i,y_ j)\), \(1\leq i\leq 2\), \(i\leq j\leq 3\), \(T\in A\). Assume moreover that the vectors \(x_ 1,x_ 2,y_ 1,y_ 2,y_ 3\) can be chosen not to change very much when the \(\phi_{ij}\) do not change much. It is then proved that A is a reflexive operator algebra. The result is formulated for linear spaces of operators, with a definition of reflexity given by Loginov and Sul'man.