Given a *-representation $$ \Psi\colon C(K) \to B(H), $$ for $C(K)$ the continuous functions on a compact Hausdorff space and $B(H)$ the bounded operators on a Hilbert space $H$. We can always decompose $\Psi$ as a direct sum of cyclic representations $$ \Psi_i \colon C(K) \to B(H_i), $$ cyclic meaning that there exists $\xi_i \in H_i$ with $\overline{\big\{\Psi(f)\xi_i \colon f \in C(K)\big\}} = H_i$.
Apparently a *-representation is non-degenerate iff it can be decomposed as a direct sum of cyclic representations (e.g. Arveson, "Invitation to C*-algebras, Ex. 1.3F).
But wouldn't this mean that every *-representation of a unital commutative C*-algebra is degenerate (since every unital commutative C*-algebra is isometrically isomorphic to some $C(K)$)?
I am quite skeptical that this would be true since I have not come across the statement as in the title of my post. What am I missing?
EDIT: Turns out that I was missing the fact that all *-representations I considered were unital.
That's not what Arveson says. He says that a representation is non-degenerate if and only if $H$ can be decomposed as a direct sum of cyclic subspaces.
Any C$^*$-algebra has an arbitrary number of degenerate representations: given $\pi:A\to B(H)$, consider $\tilde\pi:A\to B(H\oplus H)$, where $$ \tilde\pi(a)(\xi\oplus\eta)=\pi(a)\xi\oplus 0. $$
But no one cares about them, because you can always change a representation $\pi:A\to B(H)$ into a non-degenerate one by changing $H$ by $\overline{\pi(A)H}$.