Commutators greater than a perturbation of the identity

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 $\epsilon >0$ there exists an element $x\in \mathcal{A}$ with $\mathrm{\parallel }x\mathrm{\parallel }\le \epsilon$ such that $$ab-ba\ge e+x\mathrm{.}$$ If the norm on $\mathcal{A}$ is monotone, then we show $$\mathrm{\parallel }a\mathrm{\parallel }\cdot \mathrm{\parallel }b\mathrm{\parallel }\ge \frac{1}{2}\ln \frac{1}{\epsilon },$$ 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 ${\mathrm{\ell }}^2$ such that their commutator $AB-BA$ is greater than an arbitrarily small perturbation of the identity operator.