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

TL;DR

This paper explores the convex geometric structure of positive definite functions on compact groups and their connection to quantum channels as Fourier multipliers, revealing algebraic characterizations via convex set homeomorphisms.

Contribution

It establishes a correspondence between the affine homeomorphisms of positive definite function sets and Jordan automorphisms of group von Neumann algebras, linking convex geometry with operator algebra structure.

Findings

Two compact groups have isomorphic von Neumann algebras iff their positive definite function sets are affinely homeomorphic.

The group of affine homeomorphisms of the convex set is characterized by Jordan *-automorphisms.

The study connects convex geometric properties with algebraic isomorphisms of quantum channels.

Abstract

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 $\mathrm{VN}(G)$. 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 $\mathrm{VN}(G)$. We show that the von Neumann algebras of two compact groups $G$ and $H$ are $*$-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 $*$-automorphisms of $\mathrm{VN}(G)$.