Unbounded derivations, free dilations and indecomposability results for II$_1$ factors

TL;DR

This paper establishes new criteria based on unbounded derivations that determine when II$_1$ factors are prime or have unique Cartan subalgebras, with applications to free probability and group actions.

Contribution

It introduces sufficient conditions involving unbounded derivations for prime factors and uniqueness of Cartan subalgebras, extending understanding of II$_1$ factors.

Findings

Existence of unbounded derivations implies primeness of II$_1$ factors.

Certain derivations ensure the absence of Cartan subalgebras.

Applications to free probability and group actions demonstrate the criteria's utility.

Abstract

We give sufficient conditions, in terms of the existence of unbounded derivations satisfying certain properties, which ensure that a II$_1$ factor $M$ is prime or has at most one Cartan subalgebra. For instance, we prove that if there exists a real closable unbounded densely defined derivation $\delta:M\rightarrow L^2(M)\bar{\otimes}L^2(M)$ whose domain contains a non-amenability set, then $M$ is prime. If $\delta$ is moreover "algebraic" (i.e. its domain $M_0$ is finitely generated, $\delta(M_0)\subset M_0\otimes M_0$ and $\delta^*(1\otimes 1)\in M_0$), then we show that $M$ has no Cartan subalgebra. We also give several applications to examples from free probability. Finally, we provide a class of countable groups $\Gamma$, defined through the existence of an unbounded cocycle $b:\Gamma\rightarrow \mathbb C(\Gamma/\Lambda)$, for some subgroup $\Lambda<\Gamma$, such that the II$_1$ factor $L^{\infty}(X)\rtimes\Gamma$ has a unique Cartan subalgebra, up to unitary conjugacy, for any free ergodic probability measure preserving (pmp) action $\Gamma\curvearrowright (X,\mu)$.