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Essentially defined derivations on semisimple Banach algebras. (English) Zbl 1041.46522
From the text: For a semisimple complex Banach algebra \(A\) an ideal \(I\subseteq A\) is said to be essential if \(I\cap J\neq 0\) for every nonzero ideal \(J\subseteq A\). The author proves that every derivation defined on an essential ideal in \(A\) is closable. When \(A\) is ultraprime, the closed graph theorem implies that every such derivation is continuous. Since every prime \(C^*\)-algebra is ultraprime, it follows that every derivation defined on a nonzero ideal of a prime \(C^*\)-algebra is continuous.
MSC:
46H40 Automatic continuity
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References:
[1] Rodriguez, Jordan algebras: Proceedings of a Conference held at Oberwolfach, August 9–15, 1992 pp 97– (1994)
[2] DOI: 10.2307/2373290 · Zbl 0179.18103
[3] DOI: 10.1007/BF01442873 · Zbl 0648.46052
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