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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?

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    $\begingroup$ It's unique. Look up Haagerup's paper that you cited. $\endgroup$ Mar 23 at 1:27
  • $\begingroup$ Ahh, found it. I guess I should have looked at that source a little closer. Thanks! $\endgroup$
    – Sambo
    Mar 23 at 14:27

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