Remarks on commuting exponentials in Banach algebras

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 \[ e^a e^b = e^b e^a \quad \text {implies that} \quad ab = ba. \] In this note we use two elementary facts concerning inner derivations on Banach algebras to give a very short proof of Wermuth’s result.