Existence of nonfaithful tracial states in infinite von Neumann algebra

Question

Can there exist non faithful tracial state in infinite von Neumann algebras ? As faithfulness plays the role to show the vN algebra has to be finite. Further are all hyperfinite $II_{1}$ factors are all classified upto isomorphism? How one can show hyperfinite $II_{1}$ factor is embedded in any $II_{1}$ factor?

Answer 1

Take any von Neumann algebra $N$ with a trace $\tau_0$, let $M=N\oplus N$, and define $$ \tau(x\oplus y)=\tau_0(x). $$

For your second question, there is a single hyperfinite II$_1$-factor, so there is nothing to classify.

For your third question, the answer is here (you should search at least a little before asking!).