Generators of II_1 Factors

TL;DR

This paper studies an invariant of II_1 factors introduced by Shen, exploring its properties, relation to generators, behavior under amplification, and implications for free group factors and their classification.

Contribution

It analyzes the properties of Shen's invariant, relating it to generators, amplification, free group factors, and free entropy dimension, providing new insights into II_1 factor classification.

Findings

$ ext{If } ext{G}(M)<n/2, ext{ then } M ext{ is generated by } n+1 ext{ self-adjoint operators.

$ ext{G}(M)$ is well-behaved under amplification, satisfying } ext{G}(M_t)=t^{-2} ext{G}(M).

Estimates for free products, subfactors, and relations to free entropy dimension.

Abstract

In 2005, Shen introduced a new invariant, $\mathcal G(N)$, of a diffuse von Neumann algebra $N$ with a fixed faithful trace, and he used this invariant to give a unified approach to showing that large classes of ${\mathrm{II}}_1$ factors $M$ are singly generated. This paper focuses on properties of this invariant. We relate $\mathcal G(M)$ to the number of self-adjoint generators of a ${\mathrm{II}}_1$ factor $M$: if $\mathcal G(M)<n/2$, then $M$ is generated by $n+1$ self-adjoint operators, whereas if $M$ is generated by $n+1$ self-adjoint operators, then $\mathcal G(M)\leq n/2$. The invariant $\mathcal G(\cdot)$ is well-behaved under amplification, satisfying $\mathcal G(M_t)=t^{-2}\mathcal G(M)$ for all $t>0$. In particular, if $\mathcal G(\mathcal L\mathbb F_r)>0$ for any particular $r>1$, then the free group factors are pairwise non-isomorphic and are not singly generated for sufficiently large values of $r$. Estimates are given for forming free products and passing to finite index subfactors and the basic construction. We also examine a version of the invariant $\mathcal G_{\text{sa}}(M)$ defined only using self-adjoint operators; this is proved to satisfy $\mathcal G_{\text{sa}}(M)=2\mathcal G(M)$. Finally we give inequalities relating a quantity involved in the calculation of $\mathcal G(M)$ to the free-entropy dimension $\delta_0$ of a collection of generators for $M$.