Unique trace for C*-algebras

Question

Prove that UHF algebra has a unique trace, how about AF C*-algebras?

Does any special meaning for a C*-algebra has a unique trace?

Answer 1

The trace space of even a simple AF algebra can be literally anything (i.e. a metrisable Choquet simplex) as shown by Blackadar

B. Blackadar, Traces on simple AF C*-algebras. J. Funct. Anal. 38 (1980), no. 2, 156–168.

It is also easy to see that for each totally disconnected compact space $K$ (e.g. the Cantor set), the commutative algebra $C(K)$ is AF and it has lots of traces because it has lots of characters!

As for your first question, note that every UHF algebra $A$ arises as a direct limit of full matrix algebras $(M_{n_k}, \varphi_k)$, where the connecting maps $\varphi_k$ are unital. Consequently, every trace on $A$ restricts to a (unique) trace on $M_{n_k}$. Because the union of $M_{n_k}$ is dense in $A$, the trace on $A$ is unique.