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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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