in the discussion after Proposition II.4.6 of [@BH82] that ${{\operatorname{QT}}}( A )\neq \varnothing$ for every stably finite unital C\*-algebra $A$. We refer to [@BH82; @Hag14] for more details about quasitraces.

The following is Definition 6.1 of [@Tom06], except that we allow $r = 0$ in (\[rc\_dfn.a\]). This change makes no difference.

\[rc\_dfn\] Let $A$ be a stably finite unital C\*-algebra.

1.  For $\rho \in {{\operatorname{QT}}}(A)$, define $d_{\rho} \colon {M_{\infty}}(A)_{+} \to [0, \infty)$ by $$d_{\rho} (a) = \lim_{n \to \infty} \rho (a^{1/n}).$$

2.  \[rc\_dfn.a\] Let $r \in [0, {\infty})$. We say that $A$ has [*$r$-comparison*]{} if whenever $a, b \in M_{\infty} (A)_{+}$ satisfy $$d_{\rho} (a) + r < d_{\rho} (b)$$ for all $\rho \in {{\operatorname{QT}}}(A)$, then $a \precsim_A b$.

3.  \[rc\_dfn.b\] The [*radius of comparison*]{} of $A$, denoted ${\operatorname{rc}} (A)$, is $${{\operatorname{rc}}}(A)
     = \inf \big( \big\{ r \in [0, {\infty}) \colon
        {\mbox{$A$ has $r$-comparison}} \big\} \big)$$ if it exists, and $\infty$ otherwise.

It was shown in Proposition 6.3 of [@Tom06] that if $A$ is simple, then $A$ has ${\operatorname{rc}} (A)$-comparison. The following proposition is taken from Proposition 6.2 of [@Tom06].

\[Prp6.2.Tom06\] Let $A$ and $B$ be stably finite unital C\*-algebras. Then:

1.  \[Prp6.2.Tom06.a\] ${{\operatorname{rc}}}(A \oplus B) = \max \big({{\operatorname{rc}}}(A), {{\operatorname{rc}}}(B)\big)$.

2.  