A representation $\pi$: $A\rightarrow B(H)$ is said to be irreducible if $\pi(A)$ has no non-trivial invariant subspace. A C*-algebra $A$ is said to be liminal if $\pi(A)=K(H_{\pi})$ for every irreducible representation $\pi$ of $A$. (Here, $K(H_{\pi})$ denotes all the compact operators in $H_{\pi}$)
Then I meet with an exercise:
Every commutative C*-algebra is liminal.
How to prove this question?
As mentioned in the comments, every irreducible representation of an abelian C*-algebra is one-dimensional.
Indeed, if $\pi:A\to B(H)$ is irreducible, then $\pi(A)$ is dense in $B(H)$. As $\pi(A)$ is abelian, we obtain that $B(H)$ is abelian; this implies that $\dim H=1$.