It is known that a unital UHF-algebra has a unique tracial state, it is true that it is true that this trace is normal and faithful? I am particularly interested in the universal UHF-algebra, i.e. the one with $K_0$ group isomorphic to $\mathbb{Q} \cap [0,1]$.
I been looking for this in the literature without any success. On the other hand it is clear that if we restrict the trace to any finite factor it is faithful there.
The universal UHF algebra has $K_0$-group $\mathbb Q$ and the unique tracial state is faithful because the algebra is simple.