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Derivations with a hereditary domain. (English) Zbl 0922.46047
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 \(BAB\subset B\) and \(\dim [B^k\cap \text{Rad} (A)]< \infty\) hold, for some \(k\in \mathbb{N}\), where \(\text{Rad} (A)\) stands for the Jacobson radical of \(A\). In this situation we show that the separating subspace \({\mathcal S} (D)\) for \(D\) satisfies the property \[ B[ B\cap {\mathcal S} (D) ]B\subset \text{Rad} (A). \] Furthermore, we show several specially relevant situations in which we get a “closability property” nicer than the former one.

MSC:
46H40 Automatic continuity
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