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Derivations with a hereditary domain. II. (English) Zbl 0918.46048
Let $$A$$ be a complex Banach algebra, $$\text{Rad}_{\mathcal J}(A)$$ its Jacobson radical and $$\text{Rad}_{\mathcal B}(A)$$ its Baer radical, $$B$$ a subalgebra of $$A$$ and $$D: B\rightarrow A$$ a Jordan derivation. Assume $$\dim (\text{Rad}_{\mathcal J}(A)\cap \bigcap_{n=1}^{\infty}B^{n}) < \infty$$ and $$BAB\subset A.$$ The main theorem of this paper asserts that $$B(B\cap {\mathcal S}(D))B\subset \text{Rad}_{\mathcal B}(A),$$ where $${\mathcal S}(D)$$ denotes the separating subspace of $$D.$$ Applications of this result are given.

##### MSC:
 46H40 Automatic continuity
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