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A Kleinecke-Shirokov type condition with Jordan automorphisms. (English) Zbl 1058.46506

Summary: Let \(\varphi\) be a Jordan automorphism of an algebra \({\mathcal A}\). The situation when an element \(a \in {\mathcal A}\) satisfies \(\frac{1}{2}(\varphi(a) + \varphi^{-1}(a)) = a\) is considered. The result which we obtain implies the Kleinecke-Shirokov theorem and Jacobson’s lemma.

MSC:

46H05 General theory of topological algebras
47B47 Commutators, derivations, elementary operators, etc.
47B48 Linear operators on Banach algebras
16W20 Automorphisms and endomorphisms
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