Vol. 38, No. 1, 1971

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On the range of a derivation

James Patrick Williams

Vol. 38 (1971), No. 1, 273–279

A derivation on an algebra 𝒜 is a linear transformation δ on 𝒜 with the property δ(XY ) = (Y ) + δ(X)Y for all X,Y ∈𝒜. If 𝒜 = () is the Banach algebra of all bounded linear operators on a complex separable infinitedimensional Hilbert space then it is known that every derivation δ on 𝒜 is inner, that is, there is a bounded operator A on such that δ(X) = AX XA = δA(X) for all X ∈ℬ(). (See [8].) In the present note simple necessary and sufficient conditions are obtained that (i) the range (δA) be dense in the weak and ultraweak operator topologies; (ii) the norm closure of the range contain the ideal 𝒦 of compact operators on , (iii) the set of commutalors BX XB where B belongs to the C-algebra generated by A and X is arbitrary be weakly or ultraweakly dense in (). The commutant of the range of a derivation is also computed and it is shown that the ranges of any two nonzero derivations have nonzero intersection.

Mathematical Subject Classification 2000
Primary: 46L99
Received: 22 December 1970
Published: 1 July 1971
James Patrick Williams