Nuclear dimension and sums of commutators

TL;DR

This paper explores expressing certain selfadjoint elements as sums of commutators in C*-algebras, providing bounds based on nuclear dimension and examining cases with infinite nuclear dimension and regularity properties.

Contribution

It establishes bounds on the number of commutators needed in finite nuclear dimension C*-algebras and investigates the problem in non-nuclear settings with regularity assumptions.

Findings

Upper bounds for commutator sums based on nuclear dimension

Existence of simple nuclear C*-algebras with elements not well approximated by finite sums of commutators

Analysis of the problem in non-nuclear C*-algebras with Cuntz semigroup regularity

Abstract

The problem of expressing a selfadjoint 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 C*-algebras of finite nuclear dimension. Upper bounds -- in terms of the nuclear dimension of the C*-algebra -- are given for the number of commutators needed in these sums. An example is given of a simple, nuclear 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, the same problem is investigated on (possibly non-nuclear) simple unital C*-algebras assuming suitable regularity properties in their Cuntz semigroups.