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CONTINUITY OF DERIVATIONS, INTERTWINING MAPS, AND COCYCLES FROM BANACH ALGEBRAS

Published online by Cambridge University Press:  19 March 2001

H. G. DALES  and
A. R. VILLENA
H. G. DALES
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT
A. R. VILLENA
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Granada, Spain
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Abstract

Let A be a Banach algebra, and let E be a Banach A-bimodule. A linear map S:A→E is intertwining if the bilinear map

formula here

is continuous, and a linear map D[ratio ]A→E is a derivation if δ1D=0, so that a derivation is an intertwining map. Derivations from A to E are not necessarily continuous.

The purpose of the present paper is to prove that the continuity of all intertwining maps from a Banach algebra A into each Banach A-bimodule follows from the fact that all derivations from A into each such bimodule are continuous; this resolves a question left open in [1, p. 36]. Indeed, we prove a somewhat stronger result involving left- (or right-) intertwining maps.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 63 , Issue 1 , February 2001 , pp. 215 - 225
Copyright
The London Mathematical Society 2001

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Footnotes

This paper was written during exchanges between the Department of Pure Mathematics, University of Leeds, and the Departamento de Análisis Matemático, Universidad de Granada. The exchange is supported by Acciones Integradas of the British Council and the Spanish MEC Grant HB97-0007; we acknowledge this support with thanks.