Geodesic
On the abstract theorem of Picard
Lobachevskii journal of mathematics, Tome 18 (2005), pp. 127-130.

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Let $A$ be a complex Banach algebra with unit. It was shown by Williams [1] that elements $\mathbf a,\mathbf b\in A$ commute if and only if $\sup\limits_{\lambda\in\mathbf C}\|\exp(\lambda\mathbf b)\mathbf a\exp(-\lambda\mathbf b)\|\infty$. This result allows us to obtain an analog of the von Neumann–Fuglede–Putnam theorem in case of normal elements in a complex Banach algebra. In the present paper the results by Williams [1] and Khasbardar et Thakare [2] are refined by using [3, 4, 5]. An abstract version of Picard theorem is obtained in this context. 
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M. I. Karahanyan. On the abstract theorem of Picard. Lobachevskii journal of mathematics, Tome 18 (2005), pp. 127-130. http://geodesic.mathdoc.fr/item/LJM_2005_18_a5/

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[9] I. M. Karahanyan, M. I. Karahanyan, “On the commutativity in Banach algebra”, Izv. Akad. Nauk Armenii, Matematika, 34:4 (1999), 76–81 | MR | Zbl