# Are quasitrace extensions unique?

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?

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