Dual Spaces and Isomorphisms of Some Differential Banach *-algebras of Operators

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.