Operator Algebras Seminar, Winter 2001

• Date: 1/17/01 (Wednesday) Room: SH 6635, Time: 1-3 pm (Note: Special date and time!)
  • Speaker: Yasuyuki Kawahigashi, University of Tokyo
  • Title: Subfactors and extensions of endomorphisms
  • Abstract: Extensions of endomorphisms of a subfactor to the ambient factor have turned out to be a quite useful and interesing method in the theory of subfactors in connection to conformal field theory, quantum doubles, and also within subfactor theory itself. We will present applications of this idea.
    

• Date: 1/18/01 (Thursday) (Note: Mathematics Colloquium!)
  • Speaker: Yasuyuki Kawahigashi, University of Tokyo
  • Title: Operator algebras and topological quantum field theory
  • Abstract: Since the discovery of the Jones polynomial for links, the theory of operator algebras has provided several interesting tools to study topological structures in 3-dimensions. We will present the current status of such applications and explain how useful operator algebraic approaches are.
  
  

• Date: 1/19/01
  • Speaker: Keiko Kawamuro, University of Tokyo
  • Title: An extension of completely positive maps compatible with Jones basic construction

• Date: 1/26/01
  • no meeting due to the MSRI workshop on free probability theory

• Date: 2/2/01
  • Speaker: Imre Tuba, UCSB
  • Title: Low-dimensional braid representations
  • Abstract: We have classified all simple representations of the braid group $B_3$ with dime nsion $d \leq 5$ over any algebraically closed field. In particular, we proved t hat a simple $d$-dimensional representation is determined up to isomorphism by t he eigenvalues of the image of the two braid generators and a choice of a root o f their product for $d=4$ and $5$. We also showed that such representations exis t whenever the eigenvalues are not zeros of certain explicitly given polynomials and constructed the matrices via which the generators act.
    We have found a necessary and sufficient condition for the unitarizability of si mple representations of $B_3$ of dimension $d \leq 5$. In particular, we showed that a simple representation is unitarizable if and only if the eigenvalues are distinct of norm $1$, and satisf y certain inequalities, which can be computed explicitly.
    We have used these results to compute categorical dimensions of objects in braid ed tensor categories. Kazhdan and Wenzl characterized tensor categories of Lie t ype A, using Hecke algebras. We expect to use these results in classifying braid ed tensor categories whose Grothendieck semiring is isomorphic to the one of the representation category of a classical Lie group or one of its associated fusio n categories.

• Date: 2/9/01
  • Speaker: Davide Castelvecchi, UCSB
  • Title: Noncommutative invariants of foliations
  • Abstract: I will describe certain measure-theoretic objects on a foliation that can be seen as tracial weights on suitable von Neumann algebras. I apply the theory to proving foliated Morse inequalities which involve Connes' L² Betti numbers.

• Date: 2/16/01
  • no meeting due to job talk by Timofeyev

• Date: 2/23/01
  • Speaker: Huaxin Lin, University of Oregon (visiting UCSB)
  • Title: Classification of simple C*-algebras of tracial rank zero
  • Abstract: We use $K$-theoretical invariant to give a classification result for simple nuclear C*-algebras of tracial rank zero.

• Date: 3/2/01
  • Time:3:00-4:00 pm
  • Speaker: Mihai Pimsner, University of Pennsylvania
  • Title: Hilbert C*-bimodules and subfactors
    
    
  • Time:4:15 pm
  • Speaker: Sorin Popa, UCLA
  • Title: A new characterization of the standard invariant of a subfactor

• Date: 3/9/01
  • Speaker: Mikael Rordam, University of Copenhagen (visiting UCSB)
  • Title: On Elliott's classification program

• Date: 3/16/01
  • Speaker: Uffe Haagerup, Odense University
  • Title: The Invariant Subspace Problem Relative to a von Neumann Factor of type II₁
  • Abstract: The invariant subspace problem relative to a von Neumann algebra M (on a Hilbert space H) can be formulated in the following way: Does every operator T in M have a non-trivial closed invariant subspace E affiliated with M (i.e. of the form E = P(H) for a projection P in M).
    Recently we have shown, that under the assumption that M is embedable in an ultrapower R-omega of the hyperfinite II₁ factor (which might well be true for all II₁ factors) one can to EVERY operator T in M and EVERY Borelset in the complex plane associate a closed T-invariant subspace E(T,B) of H affiliated with M.
    In comparison with the classical Apostol-Foias decomposition theory, the key new idea is to replace the spectrum of an operator with L.G.Brown's spectral distribution measure mu(T), and the space E(T,B) we construct is the unique closed T-invariant subspace affiliated with M, which "cut" the Brown measure mu(T) into a part concentrated on B and another part concentrated on the complement C\B.
    This shows in particular, that unless mu(T) is concentrated in a single point, T has a non-trivial closed invariant subspace affiliated with M.