Commutators greater than a perturbation of the identity

Abstract: Let $a$ and $b$ be elements of an ordered normed algebra $\mathcal A$ with unit $e$. Suppose that the element $a$ is positive and that for some $\varepsilon>0$ there exists an element $x\in \mathcal A$ with $|x|\leq \varepsilon$ such that $$ ab-ba \geq e+x . $$ If the norm on $\mathcal A$ is monotone, then we show $$ |a|\cdot |b|\geq \tfrac{1}{2} \ln \tfrac{1}{\varepsilon} , $$ which can be viewed as an order analog of Popa's quantitative result for commutators of operators on Hilbert spaces. We also give a relevant example of positive operators $A$ and $B$ on the Hilbert lattice $\ell^2$ such that their commutator $A B - B A$ is greater than an arbitrarily small perturbation of the identity operator.