On the genericity of irreducible subfactors
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918043010990080 |
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| author | Lee, Yoonkyeong Nelson, Brent |
| author_facet | Lee, Yoonkyeong Nelson, Brent |
| contents | We show that finitely generated irreducible $\mathrm{II}_1$ subfactors are generic in the following sense. Given a separable $\mathrm{II}_1$ factor $M$ and an integer $n\geq 2$, equip the set of $n$-tuples of self-adjoint operators in $M$ with norm at most $1$ with the metric $d(x,y) = \max_{1\leq i \leq n} \|x_i - y_i\|_2$. Then the set of $n$-tuples that generate an irreducible subfactor of $M$ forms a dense $G_δ$ set in this metric space. On the way to proving this result, we show that closable derivations vanish on the anticoarse space associated to their kernels, which leads to new applications of conjugate systems in free probability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01838 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the genericity of irreducible subfactors Lee, Yoonkyeong Nelson, Brent Operator Algebras 46L10, 46L54 We show that finitely generated irreducible $\mathrm{II}_1$ subfactors are generic in the following sense. Given a separable $\mathrm{II}_1$ factor $M$ and an integer $n\geq 2$, equip the set of $n$-tuples of self-adjoint operators in $M$ with norm at most $1$ with the metric $d(x,y) = \max_{1\leq i \leq n} \|x_i - y_i\|_2$. Then the set of $n$-tuples that generate an irreducible subfactor of $M$ forms a dense $G_δ$ set in this metric space. On the way to proving this result, we show that closable derivations vanish on the anticoarse space associated to their kernels, which leads to new applications of conjugate systems in free probability. |
| title | On the genericity of irreducible subfactors |
| topic | Operator Algebras 46L10, 46L54 |
| url | https://arxiv.org/abs/2506.01838 |