Operator Algebra Seminar
Fall 2003

<!--
//-->

Schedule for Winter 2004

Monday, December 2, 2003, 4:30 - 5:30, Jeffery 319
Speaker: Claus Koestler, Queen's University

Monday, November 24, 2003, 4:30 - 5:30, Jeffery 319

Jamie Mingo, Queen's University

Orthogonal Polynomials and non-crossing annular diagrams

If X_n is a n W n self-adjoint Gaussian random matrix then a
theorem of Johansson (1998) shows that the random variables
{Tr(T_k(X_n))}_k converge, as n tends to infinity, to
independent Gaussian random variables, where {T_k} are the
Chebyshev polynomials of the second kind (suitably
centred). Cabanal-Duvillard (2001) extended this result to
the case of a pair of independent self-adjoint Gaussian
random matrices X_n and Y_n, in that he showed that the
random
variables

{Tr(T_j(X_n)), Tr(T_k(Y_n)), Tr(S_{l,m}(X_n,
Y_n))}_{j,k,l,m}

converge to independent Gaussian random variables, where
S_{m,n} is a family of Chebyshev polynomials of the first
kind
in two non-commuting variables.

I shall give a diagrammatic interpretation of the result of
Cabanal-Duvillard motivated by a theorem of Andu Nica and me
which shows that the correlation of certain ensembles of
random matrices is given by non-crossing annular diagrams.
Some extensions to the case of Wishart ensembles will be
given.

This is joint work with Tim Kusalik and Roland Speicher
(Queen's).

Monday, November 17, 4:00 - 5:30, Jeffery 319
Speaker: Roland Speicher, Queen's University

TITLE: q-Levy processes (after M. Anshelevich)

Abstract: I will present some of the main results of a
recent paper of Anshelevich on q-Levy processes.

Monday, November 10, 4:00 - 5:30, Jeffery 319
Speaker: Matthias Neufang, Carleton University

TITLE: Representations of multiplier algebras

ABSTRACT:
I shall present a common representation theoretical
framework for various multiplier algebras arising in
Abstract Harmonic Analysis, such as the measure algebra and
the completely bounded multipliers of the Fourier algebra on
a locally compact group G.  The image algebras are
completely characterized as certain normal completely
bounded bimodule maps on B(L_2(G)). The study of our
representations reveals various intriguing properties of the
algebras. Furthermore, it provides a simple description of
their Kac algebraic duality. I shall finally discuss several
extensions of our program, especially to non-normal maps.

Part of this work is joint with Zhong-Jin Ruan and Nico Spronk.

Monday, November 3, 4:30 - 6:00, Jeffery 319
Speaker: Roland Speicher , Queen's University

Title: Some new results about the q-deformed von Neumann
algebras

Abstract: I will present a recent work of Shlyakhtenko on
estimates for non-microstates free entropy, with
applications to q-semicircular families.

Monday, October 20, 4:30- 6:00, Jeffery 319
Speaker: Michael Roth, Queen's University

Title:  Random Matrices and Moduli Spaces II

Abstract: This is the continuation of the talk from
Oct. 6th. We will review the setup from last time, and
outline Kontsevich's argument for his basic identity.

We will then look at Kontsevich's matrix model and see how
essentially the same formula arises, allowing a proof of
Witten's conjecture.

Finally we will conclude with some other relations between
tautological numbers on moduli spaces and numbers arising in
the "large N limit" for the standard matrix model.

Tuesday, October 14, 4:30 - 6:00 pm, Jeffery 422
Speaker: Benoit Collins, Kyoto University

Title: Jacobi matrix ensembles, free probability and
universality results

Abstract: The Jacobi unitary matrix ensembles have already
been widely studied in the physical and mathematical
literature. Most "universality" properties that have been
obtained for the local scaling of eigenvalues of this
ensemble arise from the case of the dimension tending to
infinity and the parameters $(a,b)$ being fixed.  We present
a result stating that for $\pi,\pi'$ two independent random
selfadjoint projectors of fixed rank, $\pi\pi'\pi$ is a
Jacobi ensemble.  Therefore the Jacobi unitary ensemble with
suitably varying parameters is a matrix model for the
semigroup of free additive convolution for Bernoulli
measures. We establish universality results for these
models.

Monday, October 6, 4:30 - 6:00, Jeffery 319
Speaker: Michael Roth, Queen's University

There are (according to physicists) two models for 2d
quantum gravity.  One coming from the moduli space of
curves, and one coming from random matrix theory.

There are certain numbers coming from the moduli space of
curves (tautological intersection numbers) which are of
interest to geometers.  Based on an the generating function
of these numbers as the partition function of the theory,
and by analogy with the Matrix model, Witten proposed in
1991 that this generating function should satisfy the KdV
hierarchy of differential equations.

This conjecture was proved by Kontsevich in 1992, partly by
comparison with a particular matrix model.

The talk is intended as a brief survey of developments
linking tautological intersection numbers and numbers coming
in the "large N limit" from matrix models.

I will (briefly) define the relevant moduli spaces and
intersection numbers, explain Witten's conjecture, and
sketch Kontsevich's proof.

I also hope to have time to explain some more recent
developments, including a second proof of Kontsevich's
theorem, and further links between the tautological
intersection numbers, Hurwitz numbers, and large N limits.
These last results are largely due to Okounkov and
Pandharipande.

Monday, September 29, 4:30 - 6:00, Jeffery 319
SPEAKER: Andrew Toms, University of Copenhagen

Title:  Simple nuclear C*-algebras:  a growing menagerie

Abstract: The recent advent of 'fast dimension growth' AH
(approximately homogeneous) C*-algebras has led to ever more
unusual examples of simple nuclear C*-algebras.  We present
several such examples, and discuss their implications for
the classification and structure theory of C*-algebras.

Monday, September 22, Jeffery 319, 4:30-6 pm
Speaker: Roland Speicher, Queen's

Title: Random matrices with classical and non-commutative
entries

matrices, in particular, in connection with fluctuations
around the semi-circular limit.  Most of the talk is on
classical random matrices, but I will also contrast some
points with the corresponding situation for matrices with
non-commutative entries.

Thursday, September 11, 2:30 - 4 pm, Jeffery 115
SPEAKER: Professor Wlodek Bryc, University of Cincinnati

TITLE: Spectral measure of large random Hankel, Markov and
Toeplitz matrices.

Abstract: For Hankel and Toeplitz matrices generated by
i.i.d. random variables $\{X_k\}$ of zero mean and unit
variance, and for symmetric Markov matrices generated by
i.i.d. random variables $\{X_{i,j}\}_{j>i}$ of zero mean and
unit variance, scaling the eigenvalues by $\sqrt{n}$ we
prove the almost sure, weak convergence of the spectral
measures to universal, non-random, symmetric distributions
$\gamma_T$ and $\gamma_H$ , $\gamma_M$ of unbounded support.
We show that $\gamma_H$ is not unimodal and that $\gamma_T$
is not normal.  The moments of $\gamma_T$ and $\gamma_H$ are
the sum of volumes of solids related to Eulerian
numbers. The distribution $\gamma_M$ has a bounded smooth
density given by the free convolution of the semi-circle and
normal densities.

Thursday, September 11, 4:30 - 6 pm, Jeffery 115
SPEAKER: Professor Marek Bozejko, University of Wroclaw

TITLE: Radial functions on the free groups and von Neumann
algebras generated by q-gaussian random variables and the
Schauder basis problem

Abstract: We consider radial functions on the free group and
on the von Neumann algebras generated by free and by
q-Gaussian random variables.  We show that these algebras
are maximal Abelian subalgebras.  Next we deal with the
Schauder basis problem in L(p) spaces for the von Neumann
algebras generated by the free generators and by free and
q-Gaussian random variables. We show that the natural basis
corresponding to trigonometric system is not a Schauder
basis for p > 3 and p< 3/2.  Relations with integral
singular operators similar to the Hilbert transform will
also be presented.

Monday, September 8, 3:00 - 4:30, Jeffery 422
SPEAKER: Claus Koestler, Queen's University
TITLE: The type of a white noise

ABSTRACT:  In non-commutative  probability theory emerged an
operator   algebraic  notion of  white  noises.  It captures
classical   examples  of white   noises   as  well   as many
non-commutative examples.  Among  the classical examples are
Gaussian  and Poisson white  noise, the  more familiar among
the non-commutative examples are fermionic, free and bosonic
white noises.

In my talk I will prove the general result that white noises
can't  be of infinite type,  referring to the classification
scheme of von Neumann algebras.