Generators of II_1 Factors

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 $\mathcal{G}(M)\le n/2$. 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 $M$.

36 Pages, section 8 rewritten