representations of a non-separable $C^*$-algebra

Question

If $A$ is a non-separable $C^*$-algebra, can we deduce that all the representations of $A$ are infinie dimensional?

Answer 1

No. Take a non-metrizable compact space $X$ so that $C(X)$ is non-separable, but it has lots of one-dimensional representations.

Answer 2

Let $I$ be an uncountable set and fix $j \in I$. Define

$$\pi: \mathbb{C}^{(I)} \to B(\mathbb{C}) = \mathbb{C}: (\lambda_i)_{i \in I} \mapsto \lambda_j$$

Then $\pi$ is a one-dimensional representation of $A$. Moreover, $\mathbb{C}^{(I)}$ is non-separable because the set $\{\delta_i : i \in I\}$ of Dirac-functions satisfies $\| \delta_i - \delta_j\| = 1$ when $i \neq j$, and thus we can build an uncountable set of disjoint open balls, which shows that any dense set must be uncountable. (Credit goes to @Ruy for the above argument).

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If you want something non-abelian simply take the direct sum with a non-abelian $C^*$-algebra.