identify GNS construction as asubalgebra of $R^{\omega}$

Question

I have two questions In lemma6.5.5.

1.why can we identify the GNS representation of $\prod M_{k(n)}(\Bbb C)/\bigoplus M_{k(n)}(\Bbb C)$ with respect to $\tau _{\omega}$ with a subalgebra of $R^{\omega}$. 2.why is the Gns construction unique?

Answer 1

For the first point:

Just assume $k(n) > 1$ for infinitely many $n$ (otherwise the claim is trivial). Note that $$ R = \overline{\bigotimes}_n M_{k(n)} $$ Any sequence $(x_n)_n \in \prod M_{k(n)}$ gives a sequence in $\prod R$ by $x_n \mapsto 1^{\otimes (n - 1)} \otimes x_n$. This gives a map $\prod M_k(n) \to \prod R$. Compose that with the quotient map to get $\prod M_k(n) \to R^\omega$. Recall that $R^\omega = \prod R / J_\omega$. Now check that the $\bigoplus M_{k(n)}$ lays inside the tracial ideal $J_\omega$ and that the traces agree.