Generators of II₁ factors

Dykema, K., Sinclair, A., Smith, R. and White, S. (2008) Generators of II₁ factors. Operators and Matrices, 2(4), pp. 555-582.

Abstract

In 2005, Junhao 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 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 selfadjoint operators, whereas if M is generated by n+1 self-adjoint operators, then G(M) ≤ n/2. The invariant G(·) is wellbehaved under amplification, satisfying G(Mt)=t−2G(M)for all t>0. In particular, if G(LFr) > 0for any particular r > 1, then the free group factors are pairwise nonisomorphic and are not singly generated for suffciently 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 freeentropy dimension δ₀ of a collection of generators for M.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: White, Professor Stuart
• Authors: Dykema, K., Sinclair, A., Smith, R., and White, S.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Operators and Matrices
• ISSN: 1846-3886