Operator Algebras Seminar: What is the set of Jones indices of irreducible, hyperfinite subfactors?, Aug. 18

Speaker: Dietmar Bisch, The Vanderbilt UniversitySince Vaughan Jones introduced the theory of subfactors in 1983, it has been an open problem to determine the set of Jones indices of irreducible, hyperfinite subfactors. Jones’ rigidity theorem establishes all indices between 1 and 4, but above 4, there are many open questions. Caceres and I have recently shown that all indices of finite depth subfactors between 4 and 5 are also realized by new hyperfinite subfactors with Temperley-Lieb-Jones standard invariant. They are non-amenable, but have certain nice asymptotic commutativity properties. Our work leads to a conjecture and some results regarding Jones’ index problem. The construction involves new families of commuting squares, a graph planar algebra embedding theorem, and a few tricks that allow us to avoid solving large systems of linear equations to compute invariants of our subfactors. I will explain (somne of) these results, and I will try to make the talk accessible to non-experts in subfactors.