An operator T on a complex Hilbert space is d-symmetric if i?[6 y J = i?l6yj , where i?(6_J is the uniform closure of the range of the derivation operator S (,X) = TX -XT . It is shown that if the commutator ideal of the inclusion algebra I(T) = {A : R[& A ) C R(6J } for a d-symmetric operator is the ideal of all compact operators then T has countable spectrum and T is a quasidiagonal operator. It is also shown that if for a d-symmetric operator l(T) is the double commutant of T then T is

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... al. Let T be an element of the Banach algebra B(H) of all (bounded linear) operators on a complex Hilbert space H and 6 y the corresponding inner derivation defined by 6 f {X) = TX -XT on B(H) . Let R[6 T ) denote the range of 6_ and i?[6yj its uniform closure. An operator T