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Primitive ideals and derivations on non-commutative Banach algebras. (English) Zbl 0739.47014
The Singer-Wermer conjecture states that if $$D$$ is a (possibly unbounded) derivation on a commutative Banach algebra then the range of $$D$$ is contained in the (Jacobson) radical of the algebra. This conjecture is now known to be true. However, it is still not currently known whether or not the Singer-Wermer conjecture on derivations extends to non- commutative Banach algebras in the following sense: if $$D$$ is a (possibly unbounded) derivation then is $$D(P)\subseteq P$$ for all primitive ideals $$P$$ of the algebra? This has become known as the non-commutative version of the Singer-Wermer conjecture. We first correct an automatic continuity result in the literature concerning which (and how many) primitive idelas can fail to be invariant. Using this result together with some representation theory we prove a theorem about derivations whose second iteration annihilates some element (specifically, $$D^ 2a=0$$ implies that $$Da$$ is quasinilpotent). This theorem does not require commutativity of the algebra and it is easily seen to imply the Singer-Wermer conjecture. The proof itself is done by contradiction in which the remaining case leads to a new derivation on a commutative subalgebra, and this case can be contradicted by the arguments used in the proof of the Singer-Wermer conjecture.

##### MSC:
 47B47 Commutators, derivations, elementary operators, etc. 46H40 Automatic continuity 46H15 Representations of topological algebras
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