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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Remarks on commuting exponentials
in Banach algebras


Author: Christoph Schmoeger
Journal: Proc. Amer. Math. Soc. 127 (1999), 1337-1338
MSC (1991): Primary 46H99
Published electronically: January 28, 1999
MathSciNet review: 1476391
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Abstract: Suppose that $a$ and $b$ are elements of a complex unital Banach algebra such that the spectra of $a$ and $b$ are $2\pi i$-congruence-free. E.M.E. Wermuth has shown that then

\begin{displaymath}e^a e^b = e^b e^a \quad \text{implies that} \quad ab = ba. \end{displaymath}

In this note we use two elementary facts concerning inner derivations on Banach algebras to give a very short proof of Wermuth's result.


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Additional Information

Christoph Schmoeger
Affiliation: Mathematisches Institut I, Universität Karlsruhe, D-76128 Karlsruhe, Germany
Email: christoph.schmoeger@math.uni-karlsruhe.de

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04701-2
PII: S 0002-9939(99)04701-2
Keywords: Commuting exponentials
Received by editor(s): August 5, 1997
Published electronically: January 28, 1999
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1999 American Mathematical Society