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

Cédric Arhancet, Lei Li

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 $\mathrm{VN}(G)$.

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

Abstract

If 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 . We study the convex geometry of the convex set of normalized positive definite functions, equipped with the topology induced by the norm topology of the Fourier algebra , and its relation with the structure of . We show that the von Neumann algebras of two compact groups and are -isomorphic if and only if the convex sets and are affinely homeomorphic. We also describe the group of affine homeomorphisms of in terms of Jordan -automorphisms of .
Paper Structure (18 sections, 13 theorems, 49 equations)

This paper contains 18 sections, 13 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Preliminaries
  4. Geometry
  5. States of operator algebras
  6. Projections
  7. Harmonic analysis
  8. A description of the normal state space of a group von Neumann algebra
  9. Convex structure of $\mathrm{P}_1(G)$ and structure of the group von Neumann algebra
  10. Topology on $\mathrm{P}_1(G)$.
  11. Facial structure of $\mathrm{P}_1(G)$ and von Neumann algebra $\mathrm{VN}(G)$
  12. Affine homeomorphisms of the set $\mathrm{P}_1(G)$ of normalized positive definite functions
  13. Jordan $*$-homomorphisms
  14. The group of affine homeomorphisms of $\mathrm{P}_1(G)$
  15. Declaration of interest
  16. ...and 3 more sections

Key Result

Proposition 2.1

Let $\mathcal{M}$ be a von Neumann algebra with normal state space $\mathcal{K}$. We denote by $\mathcal{F}$ the set of all norm closed faces of $\mathcal{K}$ and by $\mathcal{P}$ the set of all orthogonal projections in $\mathcal{M}$, each equipped with its natural ordering. Then there exists a bij Moreover, this correspondence preserves the order, i.e. and the disjointness, i.e. Furthermore, $

Theorems & Definitions (16)

  • Proposition 2.1
  • Proposition 2.2
  • Example 2.3
  • Proposition 2.4
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 4.1
  • Proposition 4.2
  • Lemma 4.3
  • ...and 6 more