Spaces of Continuous Sesquilinear Forms Associated with Unbounded Operator Algebras

Abstract

Let $\mathcal{A}$ be a closed *-algebra of unbounded operators on a dense invariant domain $\mathcal{D}$ of a Hilbert space, and let ${\mathcal{L}}_{\mathcal{A}}(\mathcal{D},{\mathcal{D}}^{\prime })$ be the vector space of all continuous sequilinear forms on $\mathcal{D}\times \mathcal{D}$ relative to the graph topology of $\mathcal{A}$. We generalize some basic results of the von Neumann algebra theory (von Neumann bicommutant theorem, Kaplansky density theorem) to certain linear subspaces of ${\mathcal{L}}_{\mathcal{A}}(\mathcal{D},{\mathcal{D}}^{\prime })$.