Geodesic
On the abstract theorem of Picard
Lobachevskii journal of mathematics, Tome 18 (2005), pp. 127-130 
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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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] J. P. Williams, “On commutativity and numerical range in Banach algebras”, J. Functional Analysis, 10 (1972), 326–329 | DOI | MR | Zbl

[2] S. K. Khasbardar, N. K. Thakare, “Commutativity in a Banach Algebra”, Bolletino U.M.I. (5), 15-A (1978), 581–584 | MR | Zbl

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[4] M. I. Karahanyan, “Asymptotic properties of commutators of Banach algebras elements”, Izv. Akad. Nauk Armenii, Matematika, 19:6 (1978), 405–421

[5] M. I. Karahanyan, “Asymptotic properties of commutators”, Izv. Akad. Nauk Armenii, Matematika, 29:1 (1994), 43–49 | MR

[6] J. F. Bonsall and J. Dunkan, Complete Normed Algebras, Springer-Verlag, 1973 | MR | Zbl

[7] E. A. Gorin, “Estimation of a partial involution in a Banach algebra”, Russian Journal of Mathematical Physics, 5:1 (1997), 117–118 | MR | Zbl

[8] P. R. Halmosh, A Hilbert Space Problem Book, Princeton, New-Jersey, 1967 | MR

[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