2007/06/30 by Ken Dykema, Allan Sinclair, Roger Smith +1 · 1 citation
Mathematics · #math.OA #msc:46L10 #msc:46L35
published as Oper. Matrices 2, no 4 (2008), 555-582. · 36 Pages, section 8 rewritten
arxiv created 2007/11/02 · arxiv updated 2009/12/01
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 II1 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 II1 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)≤ n/2. The invariant \mathcal G(⋅) is well-behaved under amplification, satisfying \mathcal G(Mt)=t-2\mathcal G(M) for all t>0. In particular, if \mathcal G(\mathcal L\mathbb Fr)>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 Gsa(M) defined only using self-adjoint operators; this is proved to satisfy \mathcal Gsa(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 δ0 of a collection of generators for M.