An asymptotic variant of the Fuglede–Putnam theorem on commutators for elements of Banach algebras

The Fuglede–Putnam theorem (in Moore's asymptotic form) on the commutators of normal operators of a Hilbert space is generalized, in particular, in the following form. Let $a_1,a_2,b_1$ and $b_2$ be the elements of a complex Banach algebra such that $[a_1,b_1]=[a_2,b_2]=0$ and $\|e^{\overline\lambda a_1-\lambda b_1}\|=o(|\lambda|^{1/2})$, $\|e^{\overline\lambda a_2-\lambda b_2}\|=o(|\lambda|^{1/2})$ as $\lambda\to\infty$. Then the inequality $\|b_1x-xb_2\|\le\varphi(\|a_1-xa_2\|)$, where $\varphi(\varepsilon)\to0$ as $\varepsilon\to0$, holds uniformly in every ball $\|x\|\le R<\infty$.