Generators of II_1 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 ${\mathrm{I}\mathrm{I}}_1$ 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 ${\mathrm{I}\mathrm{I}}_1$ factor $M$: if $\mathcal{G}(M)<n\mathrm{/}2$, then $M$ is generated by $n+1$ self-adjoint operators, whereas if $M$ is generated by $n+1$ self-adjoint operators, then $\mathcal{G}(M)\le n\mathrm{/}2$. The invariant $\mathcal G(\cdot)$iswell-behavedunderamplification,satisfying$\mathcal G(M_t)=t^{-2}\mathcal G(M)$forall$t>0$\mathrm{.}Inparticular,if$\mathcal G(\mathcal L\mathbb F_r)>0$foranyparticular$r>1$, 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 $M$.