Vol. 53, No. 2, 1974

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Derivations of C-algebras have semi-continuous generators

Dorte Olesen and Gert Kjærgaard Pedersen

Vol. 53 (1974), No. 2, 563–572

For each derivation δ of a C-algebra A with δ(x) = δ(x) there exists a minimal positive element h in the enveloping von Neumann algebra A′′ such that δ(x) = hxxh. It is shown that the generator h belongs to the class of lower semi-continuous elements in A′′. From this it follows that if the function π →∥π δis continuous on the spectrum of A then h multiplies A. This immediately implies that each derivation of a simple C-algebra is given by a multiplier of the algebra. Another application shows that each derivation of a countably generated monotone sequentially closed C. algebra is inner.

Mathematical Subject Classification 2000
Primary: 46L10
Received: 17 May 1973
Published: 1 August 1974
Dorte Olesen
Gert Kjærgaard Pedersen