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

2026/03/31 by Cédric Arhancet, Lei Li
Mathematics · #math.OA #math.FA #math.GR

paper · pdf

14 pages, minor improvements

arxiv created 2026/07/30 · arxiv updated 2026/07/31

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 VN(G). We study the convex geometry of the convex set P₁(G) of normalized positive definite functions, equipped with the topology induced by the norm topology of the Fourier algebra A(G), and its relation with the structure of 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 P₁(G) and P₁(H) are affinely homeomorphic. We also describe the group of affine homeomorphisms of P₁(G) in terms of Jordan *-automorphisms of VN(G).

Citations

• Quantum information theory and Fourier multipliers on quantum groups
• Multipliers of the Fourier Algebras of Some Simple Lie Groups and Their Discrete Subgroups

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