to Ben Elias and Sherilyn Schwartz. The author would like to thank N. Christopher Phillips for a number of productive discussions and feedback. He also is grateful to Lawrence G. Brown for answering a question about an appropriate choice of notation, Eusebio Gardella for pointing out the reference [@GKL19], Nasser Golestani for pointing out Lemma 2.3 of [@HO13], Ilan Hirshberg for pointing out Example 5.11 of [@Iz1], Mahdi Moradi for his comment on the proof of Lemma \[Approx\_Commutnat\], Hiroyuki Osaka for pointing out Proposition 3.17 of [@LO19], Gábor Szabó for pointing out the reference [@BSV17], and Hannes Thiel for point out Proposition 2.8 of [@Th20].

Preliminaries  {#Sec_Prelim}
==============

In this preliminary section, we collect some information on the Cuntz semigroup, quasitraces, and the radius of comparison for easy reference and the convenience of the reader.

Throughout, if $A$ is a [C\*-algebra]{}, or if $A = M_{\infty} (B)$ for a C\*-algebra $B$, we write $A_{+}$ for the set of positive elements of $A$ and write ${{\operatorname{U}}}(A)$ for its unitary group. For $a \in A_+$, we denote by $(a -{\varepsilon})_+$ the function $\max(0,t-{\varepsilon})$ on the spectrum of $a$. Also, we denote by ${{\mathcal{K}}}$ the algebra of compact operators on a separable and 