IUMJ

Title: Nuclear dimension and sums of commutators

Authors: Leonel Robert

Issue: Volume 64 (2015), Issue 2, 559-576

Abstract:

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.