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Free Probability / Probabilités libres
(Alexandru Nica, Organizer)

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MICHAEL ANSHELEVICH, University of California, Berkeley, California  94720, USA
Free martingale polynomials

We investigate the properties of the free Sheffer systems, which are families of martingale polynomials with respect to the free Levy processes. We classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. We also show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials.

MAREK BOZEJKO, Wroclaw University, Plac Grunwaldzki 2/4, 50384  Wroclaw, Poland
Deformed free probability of Voiculescu

For each real number r in closed interval (0,1) we introduce an r-free product of states on the free product of C^*-algebras and r-free convolution of probability measures on the real line. This make unification of the free (r=1) and Boolean models (r=0) of noncommutative probability. New classes of associative convolutions of measures are considered related to the examples of Muraki-Lu. A new classes of Fock spaces will be also presented and corresponding r-Gaussian random variables. A central limit for the r-convolution and r-cumulants will be also done. This measure is supported on two intervals and is related to the free Poisson (Marcenko-Pastur) measure.

BERNDT BRENKEN, University of Calgary, Calgary, Alberta
Hilbert bimodules and Cuntz-Krieger algebras

We consider some aspects of HIlbert bimodules for arbitrary directed graphs and their associated Cuntz-Pimsner algebras. These algebras may be viewed as Cuntz-Krieger algebras of arbitrary square matrices with non-negative integer or infinite valued entries.

MAN-DUEN CHOI, Department of Mathematics, University of Toronto, Toronto, Ontario  M5S 3G3
The norm estimate for the sum of two matrices

It is often a complicated matter to estimate the the C^*-norm (the usual Hilbert-space operator-norm)of the sum of two complex matrices. Nevertheless, an ultimate answer (without hard computation) can be sought for the best bound of the norm of T = A + B where A and B are (non-commuting ) normal matrices with known eigenvalues. As a sort of manipulation of non-commutativity, the main result can be extended to cover the case of the sum of two non-normal matrices.

KEN DAVIDSON, University of Waterloo and Fields Institute, Ontario
A Perron-Frobenius theorem for completely positive maps

We discuss the spectrum of a completely positive unital map on M_n. In the irreducible case, the intersection of the spectrum with the unit circle is a subgroup; and in general it is the union of cosets of such subgroups. The analysis is related to noncommutative dilation theory and representations of Cuntz algebras.

GEORGE ELLIOTT, University of Toronto, Toronto, Ontario
An amenable properly infinite C^*-algebra with a full finite projection

The recent construction of Rordam of an infinite simple C^*-algebra with a non-zero finite projection can be modified to obtain amenability of the resulting algebra, but so far at the cost of simplicity.

REMUS FLORICEL, Queen's University, Kingston, Ontario
Inner endomorphisms for von Neumann algebras

For a von Neumann algebra M, we introduce a class of unital *-endomorphisms, called k-inner endomorphisms, we clasify them up to cocycle conjugacy and show that an arbitrary unital *-endomorphism of M decomposes as a direct sum of k-inner endomorphisms and a properly outer one.

UWE FRANZ, Ernst-Moritz-Arnd-Universität Greifswald, Institut für Mathematik und Informatik, D-17487  Greifswald, Germany
Free Lévy processes on dual groups

The talk presents several new results on non-commutative stochastic processes with free and stationary increments on dual groups, i.e. free Lévy processes.

First a construction is given that associates an involutive bialgebra to a dual group and a one-to-one correspondence between free Lévy processes on the dual group and a particular class of Lévy processes on the involutive bialgebra is established. Some applications of this construction are presented. This extends a recent result for boolean and monotone Lévy processes to the free case.

Then we give a proof that every Lévy process on a dual group has a natural Markov structure. This generalizes a result by Biane.

FRED GOODMAN, University of Iowa, Iowa, USA
Free probability of type B

The ``usual'' free probability theory of D. Voiculescu may be regarded as a theory of Lie type A, in that the combinatorics underlying the theory is that of the lattice of non-crossing partitions (of type A). Several combinatorialists have studied a type B analogue of the lattice of non-crossing partitions, which is related to the hyperoctahedral groups, just as the lattice of non-crossing partitions is related to the symmetric groups. In joint work with Alexandru Nica and Phillipe Biane, we introduce elements a free probability theory of type B based on the combinatorics of the non-crossing partitions of type B.

MADALIN GUTA, University of Nijmegen, Faculty of Sciences, Mathematics and Informatics, Toernooiveld 1, 6525 ED  Nijmegen, The Netherlands
On a class of generalised Brownian motions

The theory of generalised Brownian motion is briefly reviewed and a new class of examples is presented, associated to the characters of the infinite symmetric group.

MOURAD ISMAIL, University of Southern Florida, Tampa, Florida  33620-5700, USA
Some combinatorial statistics associated with the Rogers-Ramanujan identities

We discuss certain combinatorial statistics which arise from three term recurrence relations associated with special orthogonal polynomials. Combinatorial interpretations for certain properties of these polynomials are given.

DIMA JAKOBSON, Department of Mathematics, McGill University, Montreal, Quebec  H3A 2K6
Spectra of elements in the group ring of SU(2)

We present a new approach for constructing subgroups of SU(2) with the spectral gap. This provides an elementary solution of the Ruziewicz problem, and gives many new examples of subgroups with the spectral gap. We also discuss some other problems related to spectra of elements of the group ring of SU(2).

KENLEY JUNG, University of California, Berkeley, California, USA
Free entropy dimension computations

The talk will deal with the author's results concerning the questions of computation and independence of choice of generators of free entropy dimension for tracial W^*-algebras in the injective case.

MARIUS JUNGE, University of Illinois at Urbana-Champaign, Champaign, Illinois, USA
Martingale inequalities for noncommutative martingales

Due to the pioneering work of Pisier/Xu, martingale inequalities for noncommutative martingales in L_p spaces have evolved considerably. We will discuss the noncommutative analogue of Doob's inequality and recent progress for the order of constants. We apply similar ideas to investigate noncommutative radial limits for in L_p spaces associated to free groups.

This is joint work with Q. Xu.

CLAUS KÖSTLER, Department of Mathematics and Statistics, Queen's University, Kingston, Ontario  K7L 3N6
Inequalities for continuous non-commutative martingales

Analogues of classical Burkholder(-Gundy) inequalities have recently been established for martingales (x_n)_{n Î N} in non-commutative L^p-spaces by Junge, Pisier and Xu. We extend some of this results to martingales (x_t)_{t Î R+} with respect to continuous filtrations. As applications we present L^p-norm estimates of non-commutative Lévy processes which give rise to a theory of operator valued Itô integration in non-commutative L^p-spaces. Especially, this improves results on the construction of stationary Markov processes, as developed by Hellmich, Köstler and Kümmerer.

DAVID KRIBS, University of Iowa, Iowa, USA
Weighted shifts on Fock space

Non-commutative multi-variable versions of weighted shifts arise naturally as `weighted' left creation operators acting on Fock space. We discuss results on various classes of algebras generated by these operators.

LAURENT MARCOUX, Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario  N2L 3G1
Conjugation invariant subspaces of non-selfadjoint operator algebras

We show that a weakly closed subspace S of a nest algebra A is closed under conjugation by invertible elements of A, i.e. that a^-1 S a = S if and only if S is a Lie ideal. A similar result hold for not-necessarily closed subsapces of algebras of infinite multiplicity.

JAMES MINGO, Queen's University
Self-similarity of Hofstadter's butterfly

On l²(Z) let H_q,y be the operator

Hq,y(x)(n) = x(n-1) + x(n+1) + 2 cos(2 pnq+ y)x(n)

for 0 £ q £ 1 and 0 £ y £ 2 p. In 1976 D. Hofstadter showed how to arrange the sets s_q = È_ySp(H_q,y) to form what is now called Hofstadter's butterfly. The main point of Hofstadter's paper was to describe the recursive structure of the picture, however he was unable to give a precise formulation.

We shall give an explicit formulation of the self-similarity and show that on the `rational part' of the butterfly the map is continuous. The main technical tool is the continuous field structure of the rotation algebras and the continuity of the first Chern class.

MAGDALENA MUSAT, Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois  61801, USA
Non-commutative BMO and John-Nirenberg theorem

Non-commutative analogues of classical martingale inequalities, such as the square function inequality, were first proved by G. Pisier and Q. Xu. They also proved the analogue of the classical duality between H¹ and BMO of martingales. Inspired by the commutative results, we will show that:

[BMO, Lp]q=L[(p)/(q)] ,    1 £ p < ¥, 0 < q < 1

holds for the complex interpolation method in the non-commutative setting. As a consequence we obtain a non-commutative version of John-Nirenberg's theorem. We apply the interpolation results to study Schur multipliers on S_p.

This is joint work with Marius Junge.

IAN PUTNAM, University of Victoria, Victoria, British Columbia
Recent results on orbit equivalence for Cantor minimal systems

We consider the dynamical system generated by two commuting homeomorphisms of a Cantor set. We assume the action is free and minimal. We describe a condition which may be described as `the existence of small, positive one-cocycles' and how this can be used to show that the action is orbit equivalent to an AF-relation and also a single transformation. We describe several examples where the condition is known to hold. (Joint work with T. Giordano and C. Skau.)

ZHONG-JIN RUAN, University of Illinois at Urbana-Champaign, Illinois, USA
Approximation properties of non-commutative L_p spaces associated with discrete groups

Let G be a discrete group. We study the approximation properties of L_p(VN(G)) in the category of operator spaces. We show that under certain conditions, L_p(VN(G)) has a very nice local structure, i.e. it is an COL_p space, and has a completely bounded Schauder basis. This is a joint work with Marius Junge.

MARIUS STEFAN, Department of Mathematics, UCLA, Los Angeles, California  90095-1555, USA
Applications of free entropy to certain type II₁-subfactors

We use the free entropy introduced by D. Voiculescu to prove that subfactors of finite index in type II₁-factors generated by finite sets with finite free entropy can not be decomposed as certain cross-products and that their abelian subalgebras must have sufficiently large multiplicities.