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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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