Abstract

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

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