I'm trying to understand the basics of quasitraces on $C^*$-algebras. Using the terminology of Haagerup, given $n \geq 2$, an $n$-quasitrace $\tau$ on a $C^*$-algebra $A$ is a 1-quasitrace on $A$ which can be extended to a 1-quasitrace on the matrix algebra $M_n(A)$. It was shown by Blackadar and Handelman that any 2-quasitrace is also an $n$-quasitrace for every $n \geq 2$. The space of (normalized) 2-quasitraces on $A$ is denoted $QT(A)$.
As far as I understand, the extension of a quasitrace from $A$ to $M_n(A)$ is not necessarily unique. However, in a paper by Toms, he takes a quasitrace $\tau \in QT(A)$ and then applies it to elements in $M_n(A)$ for arbitrary $n$ (this is on page 4).
Are the quasitrace extensions really unique? If not, why should this be well-defined?