On the convex structure of the space of quantum channels which act as Fourier multipliers
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911480985681920 |
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| author | Arhancet, Cédric Li, Lei |
| author_facet | Arhancet, Cédric Li, Lei |
| contents | 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 $\mathrm{VN}(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00276 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the convex structure of the space of quantum channels which act as Fourier multipliers Arhancet, Cédric Li, Lei Operator Algebras Functional Analysis Group Theory 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 $\mathrm{VN}(G)$. |
| title | On the convex structure of the space of quantum channels which act as Fourier multipliers |
| topic | Operator Algebras Functional Analysis Group Theory |
| url | https://arxiv.org/abs/2603.00276 |