On the convex structure of the space of quantum channels which act as Fourier multipliers

If $G$ is a compact group, continuous normalized positive definite functions are in one-to-one correspondence with unital quantum channels acting as Fourier multipliers on the group von Neumann algebra $\mathrm{V}\mathrm{N}(G)$. We study the convex geometry of the convex set ${\mathrm{P}}_1(G)$ of normalized positive definite functions, equipped with the topology induced by the norm topology of the Fourier algebra $\mathrm{A}(G)$, and its relation with the structure of $\mathrm{V}\mathrm{N}(G)$. We show that the von Neumann algebras of two compact groups $G$ and $H$ are $\ast$-isomorphic if and only if the convex sets ${\mathrm{P}}_1(G)$ and ${\mathrm{P}}_1(H)$ are affinely homeomorphic. We also describe the group of affine homeomorphisms of ${\mathrm{P}}_1(G)$ in terms of Jordan $\ast$-automorphisms of $\mathrm{V}\mathrm{N}(G)$.