Generators of II_1 Factors
| dc.creator | Dykema, Ken | |
| dc.creator | Sinclair, Allan | |
| dc.creator | Smith, Roger | |
| dc.creator | White, Stuart | |
| dc.date | 2007-06-13 | |
| dc.date | 2007-11-02 | |
| dc.date.accessioned | 2026-07-07T12:12:14Z | |
| dc.date.available | 2026-07-07T12:12:14Z | |
| dc.description | 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 $δ_0$ of a collection of generators for $M$. | |
| dc.description | 36 Pages, section 8 rewritten | |
| dc.identifier | https://arxiv.org/abs/0706.1953 | |
| dc.identifier | http://arxiv.org/abs/0706.1953 | |
| dc.identifier | Oper. Matrices 2, no 4 (2008), 555-582. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/210479 | |
| dc.subject | Operator Algebras | |
| dc.subject | 46L10; 46L35 | |
| dc.title | Generators of II_1 Factors | |
| dc.type | text |