An introduction to noncommutative measure theory

Abstract

I will talk about the left regular representation of F_2. F_2 is a discrete group of infinite conjugacy classes. Free group factors give rise to D Voiculescu's free probability theory.? I will go through Free Central Limit Theorem. [ Voiculescu, D. V.; Dykema, K. J.; Nica, A. Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. CRM Monograph Series, 1.] ? Alternatively, the inductive limit of permutation group (von Neumann) algebras yields the hyperfinite II_1 factor, which has nontrivial central sequences and thus is not isomorphic to vN(F_2). The infinite tensor product algebra is hyperfinite and so is the Temperley Lieb Algebra. I will talk about the the latter and on top of it the Jones polynomial as a knot invariant, and its annular action on 3-dimensional "box"es. If time allows, I would exploit the Birman–Murakami–Wenzl (BMW) algebra for the pleasure of the audience. [Jones, Vaughan F. R. Subfactors and knots. CBMS Regional Conference Series in Mathematics, 80.]?