On the genericity of irreducible subfactors

TL;DR

This paper demonstrates that finitely generated irreducible subfactors are dense and generic within the space of self-adjoint operator tuples in a separable II_1 factor, using techniques from free probability.

Contribution

It establishes the genericity of finitely generated irreducible subfactors in separable II_1 factors and introduces new applications of conjugate systems in free probability.

Findings

Irreducible subfactors form a dense G_delta set in the operator tuple space.

Closable derivations vanish on anticoarse spaces associated with their kernels.

New applications of conjugate systems in free probability are developed.

Abstract

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 $G_\delta$ 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.