Here is a question from the proof of the "QD C*-algebra's representation theorem" in P245 of book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka.
For a separable unital C*-algebra $A$, let $\phi_{n}: A\rightarrow M_{k(n)}(\mathbb{C})$ be u.c.p maps which are asymptotically multiplicative (i.e., $||\phi_{n}(ab)-\phi_{n}(a)\phi_{n}(b)||\rightarrow 0$ for all $a, b\in A$) and isometric (i.e., $||a||=\lim_{n\rightarrow\infty}||\phi_{n}(a)||$ for all $a\in A$). Consider the u.c.p. map $$\Phi: A\rightarrow \prod_{n=1}^{\infty}M_{k(n)}(\mathbb{C})\subset B(\bigoplus_{n=1}^{\infty}\ell_{k(n)}^{2}),~~\Phi(a)=\bigoplus_{n=1}^{\infty}\phi_{n}(a).$$
Then, $\Phi$ is a faithful representation modulo the compact.
My question are:
How to verify $\Phi$ is a representation modulo the compact? (I think the "faithful" is easy from the isometric.)
In the proof, the author says $\Phi$ obviously has finite-rank projections which commute with its image and tend strongly to one. What are the "obvious" finite-rank projections?
Now, I attach the definition of representation modulo the compact in P19 of this book.
If $\pi: B(H)\rightarrow Q(H)$ is the canonical mapping onto the Calkin algebra, $A$ is a unital C*-algebra and $\phi: A\rightarrow B(H)$ is a unital completely positive map, then we say that $\phi$ is a representation modulo the compacts if $\pi\circ\phi: A\rightarrow Q(H)$ is a *-homomorphism.