Extended Probabilistic Operator Algebra Seminar (UC Berkeley)

Monday Aug. 25

Tuesday Aug. 25

Wednesday Aug. 27

Thursday Aug. 28

Friday Aug. 29

Saturday Aug. 30

Sunday Aug. 31

Monday Sept. 1

Non-microstates free entropy dimension of DT-operators

Abstract: Dykema and Haagerup introduced the class of DT-operators and
also showed that every DT-operator generate L(F2) the von Neumann algebra
generated by the free group on two generators. We will prove that Voiculescu's
non-microstates free entropy dimension is two for all DT-operators.

Free Appell polynomials

Abstract: For an algebra with a state, I will introduce a family of multi-linear
maps called free Appell polynomials. They are related to free independence
and especially to processes with free increments. I will describe a number
of properties of these polynomials, explicit combinatorial formulas for
them and, time permitting, some applications.

Partially defined semigroups relative to free multiplicative convolution

Abstract: It was proved by Nica and Speicher that for any probability measure µ
on the real line, there exists a continuous family (µt)t>=1 such
that µ1=µ, and µt+s is the free additive convolution of
µt and µs.  We will see that results of this type are true for
multiplicative free convolutions as well.

Models of deformed free probability  and  Schauder basis for noncommutative L(p)

Abstract: We present 2 classes of deformed free probability of Voiculescu:

1.t-free probability, where Kesten measures coming from the random walk on 
the free group with N generators are t-free Gaussian random variables
(parametr t>0, T= 1-1/2N) .

2.r-free probability ( 0<r<1) ,where the symmetrized Marcenko-Pastur measure appear in r-free central limit theorem.  Also Poisson version will be presented
(last results of A.Krystek and H.Yoshida).

Others connections with harmonic analysis on the free group will be done. 
We proved that the natural basis on the free group  F(N) consisting of "trigonometric system"  {delta(x): x is in F(N)} in natural order coming from length function,
Is NOT Schauder basis for non-commutative L(p) for the regular von Neumann algebra of the free group F(N) for p>4 or p<4/3 if   N>1 ( result of G.Fendler and myself).
The similar results hold for the natural basis in the von Neumann algebra generated by q-Gaussian random variables with respect order coming from Wick order ,
if   -1 < q < 1 .
We remember that the classical result of M.Riesz says that trigonometric system is basis for all L(p) for  p>1, if  N=1.

A matricial representation of the Bercovici-Pata bijection

Abstract:  The Bercovici-Pata bijection is a one-to-one relation between classicaly  infinitely divisible laws and freely infinitely divisible laws. We present a quite natural family of 
random matrices which establishes the same relation, as does the Wigner matrix between the gaussian and the semi-circular laws.

A probabilistic look at U(infinity)

Abstract: 
   U(infinity) serves as a kind of limiting object (like Brownian
motion) for
random matrix theory on U(n). I will review the work of Olshanskii and
others giving an extreme point of view of Voiculescu, draw some novel
finite consequences, and indicate some need for other limiting objects.

Invariant subspaces for DT-operators

Abstract: DT-operators arise naturally from free probability theory,
and include Voiculescu's circular operator.  In joint work with Uffe
Haagerup, we describe constructions of invariant subspaces relative to
their von Neumann algebras for DT-operators.

Elementary divisors and determinants of random matrices over local fields

Abstract: We consider uniformly distributed
random matrices with entries in the ring of integers
of an arbitrary local field.  In particular,
we show that the sequence of
elementary divisors of such a matrix
is in a simple bijective correspondence
with a Markov chain on the nonnegative integers.  This chain
has appeared previously in work of Jason Fulman on the
rational canonical forms of random matrices over finite fields,
and it is the key to Fulman's probabilistic proof of the
Rogers-Ramanujan identities.  Our formulae have connections
with classical identities for $q$-series, particularly the
$q$-binomial theorem.

A hyperfinite inequality for modified microstates free entropy dimension

Abstract: Suppose that X, Y and Z are sets of self-adjoint elements in a tracial von Neumann algebra.  Put delta0(Y|X)=delta0(Y,X)-delta0(X).  If X'' is a hyperfinite von Neumann algebra, then delta0(Y,Z|X)<=detal0(Y|X)+delta0(Z|X).  We draw several corollaries from this.

Geometry of Matrix Integrals over Operator Algebras

Abstract: Matrix integrals of real symmetric (GOE), complex hermitian
(GUE) and quaternionic self-adjoint (GSE) matrices have received much
attention from the probability theoretic point of view in recent years.
Although the three ensembles play equally important roles in analysis
and probability, only hermitian matrix integrals are mainly considered 
in geometry and topology. From a purely algebraic point of view, all these
models are integrals over a real or complex simple algebra. We can ask
what happens if we consider integrals over non-simple algebras. In this
talk I'll report a general asymptotic expansion formula for matrix-type
integrals over a finite dimensional von Neumann algebra. As an
application we obtain a new proof of some results in geometry of Riemann
surfaces, which was previously obtained by Chern-Simons gauge theory.
The same technique works for non-orientable surfaces if we consider
operator algebras defined over the reals.

Annular non-crossing permutations and second-order asymptotics  for random matrices

Abstract:We study permutations of {1,..., p+q} which are 
non-crossing in an annulus with p points marked on its external circle
and q points marked on its internal circle. The approach we use goes 
by identifying three possible crossing patterns in an annulus, and by 
defining a permutation to be annular non-crossing when it does not 
display any of these patterns. Starting from this, we prove the 
annular counterpart for a "geodesic condition" observed by 
Biane to characterize non-crossing permutations in a disc. We point
out that, as a consequence, annular non-crossing permutations appear
in the description of the second order asymptotics for the joint 
moments of certain families (Wishart and GUE) of random matrices.
This is joint work with James Mingo.

Non-crossing cumulants of type B

Abstract: The concept of non-crossing cumulants (defined by using 
lattices of non-crossing partitions) is an essential ingredient in 
the combinatorial side of free probability. On the other hand, in a 
development initiated in algebraic combinatorics, V. Reiner has introduced 
a ``type B analogue'' for the lattices of non-crossing partitions. Starting 
from Reiner's combinatorial construction, we propose a concept of non-crossing
cumulants of type B, which in turns leads to a concept of freeness of type B 
(in the appropriate framework of non-commutative probability space). This is 
joint work with Philippe Biane and Fred Goodman.

Solid von Neumann Algebras

Abstract: We prove that the relative commutant of a diffuse von Neumann subalgebra
in a hyperbolic group von Neumann algebra is always injective.

Some estimates for the non-microstates free entropy dimension

Abstract: (joint work with A. Connes). We give lower and upper estimates for the non-microstates free entropy dimension of a self-adjoint n-tuple, which connect it to the dimension of certain L2-homology groups.  Applications include a non-trivial lower bound for the free entropy dimension of an n-tuple  of q-Gaussian random variables, and also an upper bound of 1 for the free entropy dimension (both microstates and micristates-free) of any family of generators of a discrete group with property (T) of Kazhdan.

L2-Betti numbers for von Neumann algebras

Abstract: (joint work with A. Connes).  We extend the notion of L2 Betti numbers defined by Atiyah and Cheeger-Gromov for groups to von Neumann algebras.  We discuss some properties of these invariants.  Attempts to compute them have led, quite unexpectedly, to connections with free entropy dimension and  free probability theory.

(Random) matrices with classical and non-commutative entries

Abstract: I will review some of the questions and answers around classical Gaussian random matrices and will contrast this with corresponding questions around 'non-classical' random matrices.
This is joint work with J. Mingo and A. Nica.

Infinite Divisibility and Levy Processes in Classical and Free
Probability.

Abstract: In 1999, Bercovici and Pata introduced a one-to-one correspondence
\Lambda between the classes ID(*) and ID(\boxplus) of infinitely
divisible probability measures w.r.t. classical and free convolution,
respectively. In the talk, we discuss various properties of \Lambda and
how \Lambda gives rise to a one-to-one correspondence (in law) between
classical and free Levy processes. We also introduce another
transformation \Upsilon:ID(*)\to\ID(*) with the property that for any
µ in ID(*), the free cumulant transform of \Lambda(µ) equals
the classical cumulant transform of \Upsilon(µ). This mapping turns
put to have some interesting features within classical probability.
The talk is on joint work with Ole Barndorff-Nielsen.

A new Application of Random Matrices: Ext(C*red(F2)) is not a group.

Abstract:  In recent joint work with Uffe Haagerup, we proved that
Voiculescu's random matrix model for a free semi-circular system also
holds, when one applies the operator norm, rather than the trace. As an
application, we proved that the ext semi-group of the reduced
C*-algebra associated to the free group F2 on two generators is not
a group.

Duality transform for the coalgebra of \partial_{X:B}

Abstract: A duality transform for the coalgebra of the free difference quotient
derivation-comultiplication of an operator with respect to a free algebra of
scalars is constructed. The dual object is realized in an algebra of matricial
analytic functions endowed with yet another generalization of the difference
quotient derivation.