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Continuity of derivations on some semiprime Banach algebras. (English) Zbl 0617.46056
Suppose $$B$$ is a commutative Banach algebra with unit. Let $$R$$ and $$N$$ denote respectively, the Jacobson nil radicals of $$B$$. $$N$$ is also called the prime radical of $$B$$ and it consists of all the nilpotent elements of $$B$$. $$B$$ is said to be semiprime if $$N=\{0\}$$. $$B$$ is said to be an integral domain if it doesnot have any non-zero zero divisors. For any linear derivation $$D:B\to B$$, let $$S(D)=\{x\in B:$$ there is $$x_ n\to 0$$ s.t. $$Dx_ n\to x\}$$ be the separating ideal of $$D$$. By the closed graph theorem it is obvious that $$D$$ is continuous if and only if $$S(D)=\{0\}$$. We showed that if derivations are continuous on integral domains, then they are continuous on semiprime Banach algebras. Also we gave some sufficient conditions on $$B$$ and $$R$$ which imply $$S(D)$$ isnilpotent. For example
(i) if every prime ideal is closed in $$B$$ then $$S(D)$$ is contained in N.
(ii) if $$\cap_{n\geq 1}R^ n$$ is contained in every closed prime ideal, then $$S(D)$$ is contained in $$N$$.
Also we proved that if every prime ideal is closed in a commutative semiprime Banach algebra with unit, then every derivation on it is continuous.

##### MSC:
 46J05 General theory of commutative topological algebras 47B47 Commutators, derivations, elementary operators, etc.
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##### References:
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