The center of $K(H)$ is 0 and $K(H)$ has no nonzero finite dimensional representation. Can we conclude that if the center of a $C^\*$-algebra $A$ is zero, then $A$ has no nonzero finite dimensional representation?


If the above conclusion is not true, does there exist a counterexample?



center of $C^*$-algebra and finite dimensional representation
Let $P$ be a rank $1$ projection in $K(H)$ and let $\mathcal{A}$ be the set of continuous functions $f: [0,1]\to K(H)$ with $f(0) = \alpha P$ for some $\alpha \in \mathbb{C}$. Operations are pointwise. The map $f \mapsto \alpha$ is then a complex homomorphism, i.e., a one dimensional representation, but the center of $\mathcal{A}$ is trivial. (Any nonzero $f \in \mathcal{A}$ must satisfy $f(t) \neq 0$ for some $t > 0$; find $B \in K(H)$ which doesn't commute with $f(t)$ and let $g(s) = sB$. This belongs to $\mathcal{A}$ and doesn't commute with $f$.)


