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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:

  1. How to verify $\Phi$ is a representation modulo the compact? (I think the "faithful" is easy from the isometric.)

  2. 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.

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Since the rank of each $\phi_n$ is finite-dimensional, finite direct sums like $\bigoplus_1^m\phi_n$ are compact (actually, they are finite-rank). So, in the quotient, $\bigoplus_1^\infty\phi_n(a)$ and $\bigoplus_m^\infty\phi_n$ are equal, and so \begin{align} \|\pi\circ\Phi(ab)-[\pi\circ\Phi(a)]\,[\pi\circ\Phi(b)]\|&=\|\pi\left(\bigoplus_1^\infty(\phi_n(ab)-\phi(a)\phi(b)\right)\|\\ &=\|\pi\left(\bigoplus_m^\infty(\phi_n(ab)-\phi(a)\phi(b))\right)\|\\ &\leq\|\bigoplus_m^\infty(\phi_n(ab)-\phi(a)\phi(b))\|\\ &=\sup_m\|\phi_n(ab)-\phi(a)\phi(b)\|\to0\\ \end{align} as $m$ goes to infinity. Thus $\pi\circ\Phi(ab)=[\pi\circ\Phi(a)]\,[\pi\circ\Phi(b)]$.

The "obvious rank-one projections" are $\displaystyle\bigoplus_{n=1}^mI_{k(n)}$, $m=1,2,\ldots$

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  • $\begingroup$ Aha, I got the point. Thanks a lot~ :P $\endgroup$ – Yan kai Nov 2 '14 at 6:50

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