Continuity of Derivations, Intertwining Maps, and Cocycles from Banach Algebras

Let A be a Banach algebra, and let E be a Banach A-bimodule. A linear map S :AMNE is intertwining if the bilinear map (a, b)PN (δ"S ) (a, b)B a[Sb®S(ab)­Sa[b, A¬AMNE, is continuous, and a linear map D :AMNE is a deriŠation if δ"D ̄ 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. A much-studied question asks when every derivation from A to E is automatically continuous, that is, when Z "(A,E ) ̄:"(A,E ) in the notation to be given below. There are many strong classical results about this question, but a full characterization of A and E such that Z "(A,E ) ̄:"(A,E ) is not expected in the near future. An analogous question asks when all 2-cocycles from A into E are automatically continuous, that is, when Z #(A,E ) ̄:#(A,E ). Perhaps surprisingly, we can give a full solution to this question. We now give a fuller description of our results. Let E and F be linear spaces (over #). Then the linear space of all linear maps from E to F is denoted by ,(E,F ), and, for n `., the linear space of all n-linear maps from E¬...¬E to F is denoted by ,n(E,F ). Now suppose that A is an algebra and that E is an A-bimodule. Then the map δ" :,(A,E )MN,#(A,E ) is defined by setting (δ"S ) (a, b) ̄ a[Sb®S(ab)­Sa[b (a, b `A) for S `,(A,E ). A map S `,(A,E ) is a deriŠation if δ"S ̄ 0; such maps form the subspace Z "(A,E ) of ,(A,E ). The map δ# :,#(A,E )MN,$(A,E ) is defined by setting