Here is a proposition in a book "C*-algebra and Finite-Dimensional Approximations" P239
Proposition 7.1.8 Every type I C*-algebra with a faithful tracial state is RFD.
Proof Let $\tau$ be a faithful trace on type I algebra $A$ and let $\pi_{\tau}: A\rightarrow B(L^{2}(A, \tau))$ be the associated GNS representation. Since $\tau$ is faithful, so is $\pi_{\tau}$.
By the structure theory of type I von Neumann algebras we have $$\pi_{\tau}(A)''\cong\prod\limits_{n}L^{\infty}(X_{n}, \mu_{n})\otimes B(H_{n}).$$
But $\pi_{\tau}(A)''$ has a faithful trace (the vector state in the GNS representation is tracial and faithful) and hence the dimensions of all the $H_{n}$'s must be finite. Since it is simple to show that $C(X)\otimes M_{k}(\mathbb{C})$ is RFD, the remainder of the proof is straightforward.
My question
What is the faithful trace of $\pi_{\tau}(A)''$ in the proof? Is it $T \longmapsto <Tv_{\tau}, v_{\tau}>$? ($v_{\tau}$ is the cyclic vector of GNS.) Is it tracial?
How to obtain that $H_{n}$ is finite dimensional?
How to show that $C(X)\otimes M_{k}(\mathbb{C})$ is RFD.
Here is the definition of RFD:
Definition 7.1.6. A C*-algebra $A$ is called residually finite-dimensional (RFD) if there exist finite-dimensional *-homomorphisms $\pi_{n}: A\rightarrow M_{k(n)}(\mathbb{C})$ s.t. $\oplus \pi_{n}: A\rightarrow \prod M_{k(n)}(\mathbb{C})$ is faithful.