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Continuity of derivations, intertwining maps, and cocycles from Banach algebras. (English) Zbl 1023.46050

For a Banach algebra \(A\) and a Banach \(A\)-bimodule \(E\), a linear map \(S:A\to E\) is called intertwining if the one-dimensional coboundary map \(\delta^1S:(a,b)\mapsto aSb-S(ab)+ (Sa)b\) \((a,b\in A)\) is continuous, it is called left-(right-)intertwining if the map \(b\mapsto (\delta^1S) (a,b)(a\mapsto (\delta^1S) (a,b))\) is continuous for each \(a\in A\) \((b\in A\), resp.), and it is called a derivation if \(\delta^1S=0\). The following implication is shown: If every derivation from \(A\) into each Banach \(A\)-bimodule is continuous, then every left- and every right-intertwining map from \(A\) into each Banach \(A\)-bimodule is continuous. Towards a generalization to higher order cohomology spaces, those unital Banach \(A\)-bimodules are characterized for which each 2-cycle is continuous.
Reviewer: G.Garske (Hagen)

MSC:

46H40 Automatic continuity
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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