Commutators greater than a perturbation of the identity

Abstract

Let a and b be elements of an ordered normed algebra A with unit e. Suppose that the element a is positive and that for some ε>0 there exists an element x∈A with ‖x‖≤ε such thatab−ba≥e+x. If the norm on A is monotone, then we show‖a‖⋅‖b‖≥12ln⁡1ε, 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 ℓ2 such that their commutator AB−BA is greater than an arbitrarily small perturbation of the identity operator.