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?



Examples of non-isomorphic $C^\ast$ algebras with isomorphic quasi-state spaces
For any C$^\*$-algebra $A$, we can define its opposite algebra $A^{\mathrm{op}}$, which is the algebra where $ab$ is defined to be $ba$, as calculated in $A$. Let's restrict to unital algebras for simplicity. Then the identity mapping $A \rightarrow A^{\mathrm{op}}$ is linear, positive and unital, and is its own inverse. Therefore it induces a continuous, affine, origin-preserving isomorphism of quasi-state spaces. To get a counterexample, all you need is a C$^\*$-algebra that is not isomorphic to its opposite. As far as I know the first example was [a von Neumann algebra constructed by Connes](https://www.jstor.org/stable/1970940). Since then, there have been several other examples, such as [this separable example](https://arxiv.org/abs/math/0208084).


The missing piece of structure is what is defined and called an *orientation* in [this article](https://projecteuclid.org/euclid.acta/1485890059). The second author of that article is sometimes here on this forum. If you want to apply that article to the non-unital case, just remember that the quasi-state space of a C$^\*$-algebra $A$ is isomorphic to the state space of the unitization $\tilde{A}$ in the obvious manner.


