# Characterization of simple C*-algebras via GNS representations

Let $$\mathfrak{A}$$ be a [separable] unital C*-algebra and let $$Q$$ be a dense subset of the state space of $$\mathfrak{A}$$. Suppose that for each $$f\in Q$$ the associated GNS representation is faithful. Is $$\mathfrak{A}$$ simple?

If $$A$$ and $$B$$ are C*-algebras then the state space $$S(A\oplus B)$$ contains a natural copy of $$S(A)$$ and one of $$S(B)$$ such that $$S(A\oplus B)$$ is the convex hull of $$S(A)\cup S(B)$$. Moreover, the convex combinations $$\sigma =\alpha \varphi +\beta \psi ,$$ with $$\varphi \in S(A)$$, $$\psi \in S(B)$$, $$0<\alpha ,\beta <1$$, and $$\alpha +\beta =1$$, form a norm-dense subset of $$S(A\oplus B)$$.
The GNS representation of each such state $$\sigma$$ does not vanish on either $$A$$ or $$B$$ since neither does $$\sigma$$. In case both $$A$$ and $$B$$ are simple, then all of these representations will be faithful, providing a counter-example.
A very concrete example of this situation is simply $$\mathbb C\oplus \mathbb C$$!