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Journal of the London Mathematical Society 1998 57(2):469-477; doi:10.1112/S0024610798006152
© 1998 by London Mathematical Society
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© The London Mathematical Society

Derivations with a Hereditary Domain

A. R. Villena

Departamento de Analisis Matematico, Facultad de Ciencias, Universidad de Granada 18071 Granada, Spain. E-mail: avillena{at}goliat.ugr.es

Received 7 January 1996. Revision received 19 March 1996.

We investigate the closability of those derivations D defined on a (non necessarily closed) subalgebra B of a complex Banach algebra A for which the conditions

BABsubB and dim[Bk{cap}Rad(A)]<{infty}

hold for some kisinN, where Rad(A) stands for the Jacobson radical of A. In this situation we show that the separating subspace Y(D) for D satisfies the property

B[B{cap}Y(D)]BsubRad(A).

Furthermore, we demonstrate several specially relevant situations in which we get a ‘closability property’ which is more precise than the former one.


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