In his article [Group quasi-representations and index theory](https://pdfs.semanticscholar.org/9521/ebb0f8e1bec139e74345d454b60e6be36e82.pdf) (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?



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


