Vol. 38, No. 1, 1971

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 329: 1
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
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