Remarks on commuting exponentials in Banach algebras

Abstract

Suppose that a a and b b are elements of a complex unital Banach algebra such that the spectra of a a and b b are 2 π i 2\pi i -congruence-free. E.M.E. Wermuth has shown that then \[ e a e b = e b e a implies that a b = b a . e^a e^b = e^b e^a \quad \text {implies that} \quad ab = ba. \] In this note we use two elementary facts concerning inner derivations on Banach algebras to give a very short proof of Wermuth’s result.