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Remarks on commuting exponentials in Banach algebras

Author(s): Christoph Schmoeger
Journal: Proc. Amer. Math. Soc. 127 (1999), 1337-1338.
MSC (1991): Primary 46H99
Posted: January 28, 1999
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Abstract: Suppose that and are elements of a complex unital Banach algebra such that the spectra of and 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.

References:

1.: T.W. Palmer: Banach algebras and the general theory of $^\ast$ -algebra, Vol. I. Cambridge, 1994. MR 95c:46002
2.: E.M.E. Wermuth: A remark on commuting operator exponentials, Proc. Amer. Math. Soc. 125 (1997), 1685-1688. MR 97g:39011


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