Nuclear dimension and sums of commutators Leonel Robert 46L05Sums of commutatorsnuclear dimensionstrict comparison The problem of expressing a self-adjoint element that is zero on every bounded trace as a finite sum (or a limit of sums) of commutators is investigated in the setting of $\mathrm{C^*}$-algebras of finite nuclear dimension. Upper bounds ---in terms of the nuclear dimension of the $\mathrm{C^*}$-algebra---are given for the number of commutators needed in these sums. An example is given of a simple, nuclear $\mathrm{C^*}$-algebra (of infinite nuclear dimension) with a unique tracial state and with elements that vanish on all bounded traces and yet are ``badly" approximated by finite sums of commutators. Finally, we investigate the same problem on (possibly non-nuclear) simple unital $\mathrm{C^*}$-algebras, assuming suitable regularity properties in their Cuntz semigroups. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5472 10.1512/iumj.2015.64.5472 en Indiana Univ. Math. J. 64 (2015) 559 - 576 state-of-the-art mathematics http://iumj.org/access/