all $\eta > 0$.

    3.  \[PhiB.Lem\_18\_4\_11.c\] For every $\eta > 0$, there is ${\delta}> 0$ such that $(a - \eta)_{+} \precsim_A (b - {\delta})_{+}$.

The following definition is originally from [@BH82]. It is also Defnition 3.1 of [@Hag14].

\[quasitrace\] A [*quasitrace*]{} on a unital C\*-algebra $A$ is a function $\rho \colon A \to \mathbb{C}$ is if the following hold:

1.  \[quasitrace.a\] $\rho (a^* a) = \rho (a a^*) \geq 0$ for all $a \in A$.

2.  \[quasitrace.b\] $\rho (b + i c ) = \rho (b) + i \rho ( c ) $ for $b, c \in A_{\mathrm{sa}}$.

3.  \[quasitrace.c\] $\rho |_B$ is linear for every commutative C\*-subalgebra $B \subseteq A$.

4.  \[quasitrace.d\] There is a function $\rho_2 \colon M_2 (A) \to \mathbb{C}$ satisfying (\[quasitrace.a\]), (\[quasitrace.b\]), and (\[quasitrace.c\]) with $M_2 (A)$ in place of $A$, and such that, with $(e_{j, k})_{j, k = 1}^{2}$ denoting the standard system of matrix units in $M_2 (\mathbb{C})$, for all $a \in A$ we have $$\rho (a) = \rho_2 (a \otimes e_{1, 1}).$$

A quasitrace $\rho$ on a unital [C\*-algebra]{} is [*normalized*]{} if $\rho (1) = 1$. The set of normalized quasitraces on a unital C\*-algebra $A$ is denoted by ${{\operatorname{QT}}}(A)$.

It was shown 