Remarks on commuting exponentials in Banach algebras, II

Schmoeger, Christoph

doi:10.1090/S0002-9939-00-05465-4

Remarks on commuting exponentials in Banach algebras, II
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Proc. Amer. Math. Soc. 128 (2000), 3405-3409 Request permission

Abstract:

Suppose that $a$ and $b$ are elements of a complex unital Banach algebra such that the spectrum of $a$ is $2\pi i$-congruence-free and $e^ae^b = e^be^a$. We show that then $ab-ba$ is the sum of nilpotent elements. If $r(b)$ denotes the spectral radius of $b$, then we show that the additional assumption $r(b)<2 \pi$ implies that \begin{equation*} b (ab-ba)^2 = (ab-ba)^2 b. \end{equation*}

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