Topologized Hilbert spaces

Let H P be a Hausdorff topological vector space with the underlying vector space H being a Hilbert space such that P is coarser than the norm topology. A densely defined P - P -continuous operator on H is called P -maximal if it has no non-trivial P - P -continuous extension, and it is said to be P -adjointable if its adjoint is also P - P -continuous. w that if P is locally convex, the collection M P ⋆ ( H ) of all densely defined P -maximal P -adjointable operators is a ⁎ -algebra under the multiplication given by the P -maximal extension of the composition and the involution ⋄ given by the P -maximal extension of the adjoint. Examples include rigged Hilbert spaces and O ⁎ -algebras. general (not necessarily locally convex) case, we associate with H P a ⁎ -algebra L b ⋆ ( H P   ˜ ) which is a ⁎ -subalgebra of M P ⋆ ( H ) when P is locally convex. If P is the measure topology on H corresponding to a tracial von Neumann algebra M ⊆ L ( H ) , then the image of the representation of the measurable operator algebra on the completion H P   ˜ of H with respect to P , can be regarded as a ⁎ -subalgebra of L b ⋆ ( H P   ˜ ) . case when P is normable, it is shown that L b ⋆ ( H P   ˜ ) is a Banach ⁎ -algebra. Examples of such Banach ⁎ -algebras include L L ∞ [ 0 , 1 ] ⋆ ( L 2 [ 0 , 1 ] ) : = { Ψ ∈ B ( L 2 [ 0 , 1 ] ) : Ψ ( L ∞ [ 0 , 1 ] ) ⊆ L ∞ [ 0 , 1 ] ; Ψ ⁎ ( L ∞ [ 0 , 1 ] ) ⊆ L ∞ [ 0 , 1 ] } (under a suitable norm) as well as L T ( ℓ 2 ) ⋆ ( S ( ℓ 2 ) ) : = { Φ ∈ B ( S ( ℓ 2 ) ) : Φ ( T ( ℓ 2 ) ) ⊆ T ( ℓ 2 ) ; Φ ⁎ ( T ( ℓ 2 ) ) ⊆ T ( ℓ 2 ) } , where S ( ℓ 2 ) and T ( ℓ 2 ) are the spaces of Hilbert–Schmidt operators and of trace-class operators respectively, on ℓ 2 .