Topologized Hilbert spaces

Let HPHP be a Hausdorff topological vector space with the underlying vector space HH being a Hilbert space such that PP is coarser than the norm topology. A densely defined PP-PP-continuous operator on HH is called PP-maximal if it has no non-trivial PP-PP-continuous extension, and it is said to be PP-adjointable if its adjoint is also PP-PP-continuous.We show that if PP is locally convex, the collection MP⋆(H) of all densely defined PP-maximal PP-adjointable operators is a ⁎⁎-algebra under the multiplication given by the PP-maximal extension of the composition and the involution ⋄⋄ given by the PP-maximal extension of the adjoint. Examples include rigged Hilbert spaces and O⁎O⁎-algebras.In the general (not necessarily locally convex) case, we associate with HPHP a ⁎⁎-algebra Lb⋆(HP ˜) which is a ⁎⁎-subalgebra of MP⋆(H) when PP is locally convex. If PP is the measure topology on HH corresponding to a tracial von Neumann algebra M⊆L(H)M⊆L(H), then the image of the representation of the measurable operator algebra on the completion HP ˜ of HH with respect to PP, can be regarded as a ⁎⁎-subalgebra of Lb⋆(HP ˜).In the case when PP is normable, it is shown that Lb⋆(HP ˜) is a Banach ⁎⁎-algebra. Examples of such Banach ⁎⁎-algebras include LL∞[0,1]⋆(L2[0,1]):={Ψ∈B(L2[0,1]):Ψ(L∞[0,1])⊆L∞[0,1];Ψ⁎(L∞[0,1])⊆L∞[0,1]} (under a suitable norm) as well as LT(ℓ2)⋆(S(ℓ2)):={Φ∈B(S(ℓ2)):Φ(T(ℓ2))⊆T(ℓ2);Φ⁎(T(ℓ2))⊆T(ℓ2)}, where S(ℓ2)S(ℓ2) and T(ℓ2)T(ℓ2) are the spaces of Hilbert–Schmidt operators and of trace-class operators respectively, on ℓ2ℓ2.