Representation is a compact perturbation of a direct sum of one-dimensional representations

Question

In his article Group quasi-representations and index theory (p. 11), Dadarlat claims that a *-representation $\phi_1\colon C(Y)\to L(H)$ of a commutative C*-algebra "is a compact perturbation of a *-representation of $C(Y)$ which is a direct sum of one-dimensional representations" by Voiculescu's theorem. How does this work?

The version of Voiculescu's theorem that I know is the one given in the book by Higson and Roe, analytic K-homology on page 63, and it states that if $\rho\colon E\to L(H)$ is a non-degenerate representation of a unital separable C*-algebra, and $\sigma\colon E\to L(H')$ is a completely positive map which vanishes on the pre-image of the compact operators under $\rho$, then $\sigma$ is, up to compact perturbation, equal to $V^*\rho V$, for $V\colon H'\to H$ an isometry.

Is this the right theorem to use here, and if so, how does the proof go?

Answer 1

In practice, there are several non-obvious corollaries of Voiculescu's theorem that are collectively called "by Voiculescu's Theorem." In this case it is a consequence of Voiculescu's theorem (the one you mention) that every representation of a separable C*-algebra is approximately unitarily equivalent modulo the compact operators to a direct sum of irreducible representations. You can find a good discussion of many of the equivalences and corollaries to Voiculescu's theorem in Voiculescu's original paper (see Corollary 1.6 specifically) or in Davidson's "C*-algebras by Example" (see Corollary II.5.9 specifically).