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Continuity of derivations, intertwining maps, and cocycles from Banach algebras

Research output: Contribution to journalJournal article

Published
<mark>Journal publication date</mark>02/2001
<mark>Journal</mark>Journal of the London Mathematical Society
Issue number1
Volume63
Number of pages11
Pages (from-to)215-225
<mark>State</mark>Published
<mark>Original language</mark>English

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

is continuous, and a linear map D: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.