Free Probability / Probabilités libres
(Alexandru Nica, Organizer)
- 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
- 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)
- 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 Mn.
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
- 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
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
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,
Martingale inequalities for noncommutative martingales
Due to the pioneering work of Pisier/Xu, martingale inequalities for
noncommutative martingales in Lp 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 Lp 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 (xn)n Î N in
non-commutative Lp-spaces by Junge, Pisier and Xu. We extend some of
this results to martingales (xt)t Î R+ with respect
to continuous filtrations. As applications we present Lp-norm
estimates of non-commutative Lévy processes which give rise to a
theory of operator valued Itô integration in non-commutative
Lp-spaces. Especially, this improves results on the construction of
stationary Markov processes, as developed by Hellmich, Köstler and
- 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
- LAURENT MARCOUX, Department of Pure Mathematics, University of Waterloo, Waterloo,
Ontario N2L 3G1
Conjugation invariant subspaces of non-selfadjoint operator
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 Hq,y be the operator
for 0 £ q £ 1 and 0 £ y £ 2 p. In 1976
D. Hofstadter showed how to arrange the sets sq = ÈySp(Hq,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
Hq,y(x)(n) = x(n-1) + x(n+1) + 2 cos(2 pnq+ y)x(n)|
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
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 Sp.
[BMO, Lp]q=L[(p)/(q)] , 1 £ p < ¥, 0 < q < 1|
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 Lp spaces associated
with discrete groups
Let G be a discrete group. We study the approximation properties of
Lp(VN(G)) in the category of operator spaces. We show
that under certain conditions, Lp(VN(G)) has a very nice
local structure, i.e. it is an COLp space, and has a
completely bounded Schauder basis. This is a joint work with Marius
- 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.