# zbMATH — the first resource for mathematics

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
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.