Generators of II₁ Factors

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 $M$.