Back to Departmental Colloquium: Fall 2004
Departmental Colloquium
Date: Thursday, Dec 2, 2004
Time: 4:15PM
Location: JWB 335
Special Colloquium
Irreducible Hyperfinite II 1 Subfactors
Title |
Irreducible Hyperfinite II 1 Subfactors |
Abstract |
The goal of this talk is to present a new noncommutative tool in the study of the subfactor theory. The theory of von Neumann algebras was introduced by Murray and von Neumann as the mathematical foundation of quantum mechanics. Every von Neumann algebra can be decomposed as a direct “sum” of factors. Before the `80’s the theory is focused on the classification of factors. Subfactor theory became the mainstream after Vaughan Jones’s work. Subfactor theory studies the position of a subfactor sitting inside the ambient factor. The standard invariant of the inclusion is a complete invariant in “good” cases and can be packed pictorially as the associated planar algebra. The simplest example is the Temperley-Lieb algebra, which gives a representation of braid group. The invariant in this case yields a knot invariant, the celebrated Jones polynomial. Irreducible inclusions are the most benign ones, which allow a classification result in term of planar algebra. A II 1 hyperfinite factor can be approximated by finite dimensional C * algebras and has a unique normalized trace. CAR (canonical anticommutation relations) algebra is a typical example. In this talk, I will present a new technique of constructing irreducible hyperfinite II 1 inclusions. The example is a series of inclusions of II 1 factors inside the Temperley-Lieb algebra R, R + P 1 + P 2 +… + P n +… with the property: The “size” of the subfactor P n compared to R decays exponentially, but each inclusion P n R remains irreducible. One application is an extension of Temperley-Lieb algebra with extremality. When the Jones index is around 4, extremality is equivalent to irreducibility. It is a long standing problem (so-called gap conjecture ) in subfactor theory whether Jones index is “quantized” when it is bigger than 4. The above result represents a progress towards this direction. |
Representation Theory
Commutative Algebra
Topology