Let $A$ (resp. $B$) be a *unital* $C^\ast$-algebra, $\mathcal{Q}(A)$ (resp. $\mathcal{Q}(B)$) the compact convex subset of $A^\ast$ equipped with the $\sigma(A^\ast, A)$ (resp. $\sigma(B^\ast, B)$) topology. Suppose $\mathcal{Q}(A)$ and $\mathcal{Q}(B)$ are isomorphic in the sense that there exists a bijective affine homeomorphism $\varphi$ from $\mathcal{Q}(A)$ onto $\mathcal{Q}(B)$, does it follow that $A$ and $B$ are isomorphic as $C^\ast$ algebras?

In the abelian case (i.e. both $A$ and $B$ are commutative), the answer is affirmative, as we can simply look at the compact subspace of pure states and conclude by the Gelfand duality. But in the non-abelian case, the argument above fails. In fact, I strongly suspect that there exists non-isomorphic unital $C^\ast$-algebras $A$ and $B$ with $\mathcal{Q}(A)$ and $\mathcal{Q}(B)$ being isomorphic. Can anyone give a concrete example of this?