Suppose $\mathcal A_1$ and $\mathcal A_2$ are unital C$^*$-algebras with faithful states $\varphi_1,\psi_1$ of $\mathcal A_1$ and $\varphi_2, \psi_2$ of $\mathcal A_2$.

Are the reduced free products $(\mathcal A_1, \varphi_1) * (\mathcal A_2, \varphi_2)$ and $(\mathcal A_1,\psi_1) * (\mathcal A_2, \psi_2)$ isomorphic as C$^*$-algebras?

Because of the faithfulness of the states both of these algebras are C$^*$-norm completions of the algebraic free product $\mathcal A_1 \circledast_\mathbb C \mathcal A_2$.

Suppose $\mathcal A_1$ and $\mathcal A_2$ are unital C$^*$-algebras with faithful states $\varphi_1,\psi_1$ of $\mathcal A_1$ and $\varphi_2, \psi_2$ of $\mathcal A_2$.

Are the reduced free products $(\mathcal A_1, \varphi_1) * (\mathcal A_2, \varphi_2)$ and $(\mathcal A_1,\psi_1) * (\mathcal A_2, \psi_2)$ isomorphic as C$^*$-algebras?

Because of the faithfulness of the states both of these algebras are C$^*$-norm completions of the algebraic free product $\mathcal A_1 \circledast_\mathbb C \mathcal A_2$. The question then is whether this free product can be thought of as the unique spatial free product akin to the spatial tensor product being unique.