Remarks on commuting exponentials in Banach algebras, II

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*}