Generators of II1 factors

In 2005, Shen introduced a new invariant, G(N), of a di use von Neumann algebra N with a xed faithful trace, and he used this invariant to give a uni ed approach to showing that large classes of II1 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 II1 factor M : if 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 G(M) ≤ n/2. The invariant G(·) is well-behaved under ampli cation, satisfying G(Mt) = t−2G(M) for all t > 0. In particular, if G(LFr) > 0 for any particular r > 1, then the free group factors are pairwise non-isomorphic and are not singly generated for su ciently large values of r. Estimates are given for forming free products and passing to nite index subfactors and the basic construction. We also examine a version of the invariant Gsa(M) de ned only using selfadjoint operators; this is proved to satisfy Gsa(M) = 2G(M). Finally we give inequalities relating a quantity involved in the calculation of G(M) to the free-entropy dimension δ0 of a collection of generators for M . AMS classi cations numbers: 46L10, 46L35 ∗Research supported in part by NSF grant DMS-0600814 †Research supported in part by NSF grant DMS-0401043