Relative annihilators and relative commutants in non-selfadjoint operator algebras

We extend von Neumann’s Double Commutant Theorem to the setting of nonselfadjoint operator algebras A, while restricting the notion of commutants of a subset S of A to those operators in A which commute with every operator in S. If A is a completely distributive commutative subspace lattice algebra acting on a Hilbert space H, we obtain an alternate characterization (to those of Erdos–Power and of Deguang) of the weak operator closed ideals of A. In the case of nest algebras, we use this characterization to formulate an explicit characterization of the relative (double) commutants and relative (double) annihilators of these ideals. We also describe a property of subspaces of the algebra for which the relative commutants can be expressed as an extension of the relative annihilator by the scalar operators.