## Generators of II_{1<\sub> factors}

*Abstract:* In 2005, Shen introduced a new invariant, 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 II_{1} factors M are singly generated. This paper focuses on properties of this invariant. We relate G(M) to the number of self-adjoint generators of a II_{1} factor M: if G(M)^{-2}G(M) for all t>0. In particular, if G(LF_{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 G_{sa}(M) defined only using self-adjoint operators; this is proved to satisfy G_{sa}(M)=2G(M). Finally we give inequalities relating a quantity involved in the calculation of G(M) to the free-entropy dimension \delta_{0} of a collection of generators for M.

### Publication Details

*Coauthors:* Ken Dykema, Allan Sinclair and Roger Smith

Oper. Matrices, 2 (2008), 555-582.

*Download:* Journal Website, arXiv:0706.1953.

The arXiv version does not include changes made at the suggestion of the referee which can only be found in the final version.

## Seneca Rocks

Climbing with John Roe in West Virgina, 2007.