The paper continues the
study of differential Banach *-algebras 𝒜S and ℱS of operators associated
with symmetric operators S on Hilbert spaces H. The algebra 𝒜S is the
domain of the largest *-derivation δS of B(H) implemented by S and the
algebra ℱS is the closure of the set of all finite rank operators in 𝒜S with
respect to the norm ∥A∥ = ∥A∥ + ∥δS(A)∥. When S is selfadjoint, ℱS is the
domain of the largest *-derivation of the algebra C(H) implemented by
S. If S is bounded, ℱS= C(H) and 𝒜S= B(H), so 𝒜S is isometrically
isomorphic to the second dual of ℱS . For unbounded selfadjoint operators S the
paper establishes the full analogy with the bounded case: 𝒜S is isometrically
isomorphic to the second dual of ℱS. The paper also classifies the algebras 𝒜S
and ℱS up to isometrical *-isomorphism and obtains some partial results
about bounded but not necessarily isometrical *-isomorphisms of the algebras
ℱS.
