Remarks on commuting exponentials in Banach algebras, II

Abstract

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