Hexagon

math.OA — Operator Algebras

The regular pentagonal system and the Smith--Ward problem

Contributed by Shengwei Qiu, Thomas Sinclair

The three-dimensional function system of the regular pentagon fails the lifting property, giving a counterexample to the operator-system formulation of the Smith--Ward problem with C $^*$ -envelope $\mathbb C^5$ . We construct unital $k$ -positive maps from this fixed system into matrix algebras where distance from every ucp map is bounded below independently of $k$ . We also prove that every regular polygon system with at least five vertices detects nuclearity, while its dual characterizes the weak expectation property. These tensor-product characterizations follow from a graph dichotomy: a graph operator system and its dual are C $^*$ -nuclear when the graph is $2$ -chordal; otherwise they detect nuclearity and WEP, respectively.

Primarily AI-generated textHuman understanding: all partsoperator systems; lifting property; nuclearity; weak expectation property

Oct 26, 2026PDF ↗View work →

hexagon:2610.00555v1

math.FA — Functional Analysis

The operator Hilbert space $\mathrm{OH}$ embeds completely isometrically into a noncommutative $\mathrm{L}^1$ -space

Contributed by Cédric ARHANCET

We prove that Pisier's operator Hilbert space $\mathrm{OH}$ embeds completely isometrically into the predual of a $\mathrm{QWEP}$ von Neumann algebra, answering a longstanding question introduced by Pisier and later by Haagerup and Musat. The embedding is obtained as an ultraproduct of free probabilistic models $h_k = \Lambda^{\frac{1}{2}}c_k\Lambda^{\frac{1}{2}}$ realized in the free group factor $\mathrm{VN}(\mathbb{F}_\infty)$ , with the trace and circular family scaled according to the mass of the truncated measure $t^{-\frac{5}{3}} \mathrm{d} t$ .

Primarily human-written textHuman understanding: all partscircular elementscomplete isometriesnoncommutative $\L^1$-spacesoperator Hilbert spaceoperator-valued free probabilitypower-law truncationsultraproducts

Oct 26, 2026PDF ↗View work →

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The regular pentagonal system and the Smith--Ward problem

The operator Hilbert space $\mathrm{OH}$ embeds completely isometrically into a noncommutative $\mathrm{L}^1$ -space

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Matching works

The regular pentagonal system and the Smith--Ward problem

The operator Hilbert space OH\mathrm{OH}OH embeds completely isometrically into a noncommutative L1\mathrm{L}^1L1-space

The operator Hilbert space $\mathrm{OH}$ embeds completely isometrically into a noncommutative $\mathrm{L}^1$ -space