# Extended Probabilistic Operator Algebra Seminar (UC Berkeley)

## Seminar Schedule

### Monday Aug. 25

5:00pm-6:00pm, 939 Evans: Steen Thorbjønsen, Infinite Divisibility and Levy Processes in Classical and Free Probability.

### Tuesday Aug. 25

5:00pm-6:00pm, 891 Evans: Narutaka Ozawa, Solid von Neumann Algebras.

### Wednesday Aug. 27

4:00pm-5:00pm, 31 Evans: Dimitri Shlyakhtenko, L2 Betti Numbers for von Neumann Algebras
5:00pm-6:00pm, 31 Evans: Kenley Jung, A hyperfinite inequality for modified microstates free entropy dimension

### Thursday Aug. 28

5:00pm-6:00pm, 939 Evans: Steen Thorbjønsen, A new Application of Random Matrices: Ext(C*red(F2)) is not a group.

### Friday Aug. 29

5:00pm-6:00pm, 939 Evans: Alexandru Nica, Annular non-crossing permutations and second-order asymptotics for random matrices

### Saturday Aug. 30

10:00am-11:00am, 939 Evans: Motohico Mulase, Geometry of Matrix Integrals over Operator Algebras
11:15am-12:15pm, 939 Evans: Dan Voiculescu, Duality transform for the coalgebra of \partial_{X:B}
Lunch
2:00pm-3:00pm, 939 Evans: Roland Speicher, (Random) matrices with classical and non-commutative entries
3:15pm-4:15pm, 939 Evans: Steve Evans, Elementary divisors and determinants of random matrices over local fields
4:30pm-5:30pm, 939 Evans: Hari Bercovici, Partially defined semigroups relative to free multiplicative convolution

### Sunday Aug. 31

10:00am-11:00am, 939 Evans: Persi Diaconis, A probabilistic look at U(infinity)
11:15am-12:15pm, 939 Evans: Dimitri Shlyakhtenko, Some estimates for the non-microstates free entropy dimension
Lunch
2:00pm-3:00pm, 939 Evans: Thierry Cabanal-Duvillard, A matricial representation of the Bercovici-Pata bijection
3:15pm-4:15pm, 939 Evans: Michael Anshelevich, Free Appell polynomials
4:30pm-5:00pm, 939 Evans: Vaughan Jones, Are planar algebras free?

### Monday Sept. 1

10:00am-11:00am, 939 Evans: Marek Bozejko, Models of deformed free probability and Schauder basis for noncommutative L(p)
11:15am-12:15pm, 939 Evans: Alexandru Nica, Non-crossing cumulants of type B
Lunch
2:00pm-3:00pm, 939 Evans: Ken Dykema, Invariant subspaces for DT-operators
3:15pm-4:15pm, 939 Evans: Lars Aagaard, Non-microstates free entropy dimension of DT-operators

## Participants

 Here Not here

 8/22 8/23 8/24 8/25 8/26 8/27 8/28 8/29 8/30 8/31 9/01 9/02 Lars Aagaard Michael Anshelevich Hari Bercovici Marek Bozejko Thierry Cabanal-Duvillard Persi Diaconis Ken Dykema Steve Evans Kenley Jung Motohico Mulase Alexandru Nica Narutaka Ozawa Dimitri Shlyakhtenko Roland Speicher Steen Thorbjensen Dan Voiculescu

## Titles and Abstracts

 Mon 9/1 3:15pm 939 Evans Lars Aagaard: Non-microstates free entropy dimension of DT-operators Abstract: Dykema and Haagerup introduced the class of DT-operators andalso showed that every DT-operator generate L(F2) the von Neumann algebragenerated by the free group on two generators. We will prove that Voiculescu'snon-microstates free entropy dimension is two for all DT-operators.

 Sun 8/31 3:15pm 939 Evans Michael Anshelevich: Free Appell polynomials Abstract: For an algebra with a state, I will introduce a family of multi-linearmaps called free Appell polynomials. They are related to free independenceand especially to processes with free increments. I will describe a numberof properties of these polynomials, explicit combinatorial formulas forthem and, time permitting, some applications.

 Sat 8/30 4:30pm 939 Evans Hari Bercovici: Partially defined semigroups relative to free multiplicative convolution Abstract: It was proved by Nica and Speicher that for any probability measure µon the real line, there exists a continuous family (µt)t>=1 suchthat µ1=µ, and µt+s is the free additive convolution ofµt and µs. We will see that results of this type are true formultiplicative free convolutions as well.

 Mon 9/1 10:00am 939 Evans Marek Bozejko: 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 ( 04 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.

 Sun 8/31 2:00pm 939 Evans Thierry Cabanal-Duvillard: 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.

 Sun 8/31 9:00am 939 Evans Persi Diaconis: 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. 

 Mon 9/1 2:00pm 939 Evans Ken Dykema: Invariant subspaces for DT-operators Abstract: DT-operators arise naturally from free probability theory,and include Voiculescu's circular operator. In joint work with UffeHaagerup, we describe constructions of invariant subspaces relative totheir von Neumann algebras for DT-operators.

 Sat 8/30 3:15pm 939 Evans Steve Evans: Elementary divisors and determinants of random matrices over local fields Abstract: We consider uniformly distributedrandom matrices with entries in the ring of integersof an arbitrary local field. In particular,we show that the sequence ofelementary divisors of such a matrixis in a simple bijective correspondencewith a Markov chain on the nonnegative integers. This chainhas appeared previously in work of Jason Fulman on therational canonical forms of random matrices over finite fields,and it is the key to Fulman's probabilistic proof of theRogers-Ramanujan identities. Our formulae have connectionswith classical identities for $q$-series, particularly the$q$-binomial theorem.

 Wed 8/27 5pm 31 Evans Kenley Jung: 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.

 Sat 8/30 10am 939 Evans Motohico Mulase: 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 muchattention from the probability theoretic point of view in recent years.Although the three ensembles play equally important roles in analysisand probability, only hermitian matrix integrals are mainly considered in geometry and topology. From a purely algebraic point of view, all thesemodels are integrals over a real or complex simple algebra. We can askwhat happens if we consider integrals over non-simple algebras. In thistalk I'll report a general asymptotic expansion formula for matrix-typeintegrals over a finite dimensional von Neumann algebra. As anapplication we obtain a new proof of some results in geometry of Riemannsurfaces, which was previously obtained by Chern-Simons gauge theory.The same technique works for non-orientable surfaces if we consideroperator algebras defined over the reals.

 Fri 8/29 5pm 939 Evans Alexandru Nica: 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 circleand 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 pointout that, as a consequence, annular non-crossing permutations appearin 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.

 Mon 9/1 11:15am 939 Evans Alexandru Nica: 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-crossingcumulants 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.

 Tue 8/26 5pm 891 Evans Narutaka Ozawa: Solid von Neumann Algebras Abstract: We prove that the relative commutant of a diffuse von Neumann subalgebrain a hyperbolic group von Neumann algebra is always injective.

 Sun 8/31 11:15am 939 Evans Dimitri Shlyakhtenko: 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.

 Wed 8/27 4pm 31 Evans Dimitri Shlyakhtenko: 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.

 Sat 8/30 2pm 939 Evans Roland Speicher: (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.

 Mon 8/25 5pm 939 Evans Steen Thorbjønsen: Infinite Divisibility and Levy Processes in Classical and FreeProbability. Abstract: In 1999, Bercovici and Pata introduced a one-to-one correspondence\Lambda between the classes ID(*) and ID(\boxplus) of infinitelydivisible probability measures w.r.t. classical and free convolution,respectively. In the talk, we discuss various properties of \Lambda andhow \Lambda gives rise to a one-to-one correspondence (in law) betweenclassical and free Levy processes. We also introduce anothertransformation \Upsilon:ID(*)\to\ID(*) with the property that for anyµ in ID(*), the free cumulant transform of \Lambda(µ) equalsthe classical cumulant transform of \Upsilon(µ). This mapping turnsput to have some interesting features within classical probability.The talk is on joint work with Ole Barndorff-Nielsen.

 Thu 8/28 5pm 939 Evans Steen Thorbjønsen: A new Application of Random Matrices: Ext(C*red(F2)) is not a group. Abstract: In recent joint work with Uffe Haagerup, we proved thatVoiculescu's random matrix model for a free semi-circular system alsoholds, when one applies the operator norm, rather than the trace. As anapplication, we proved that the ext semi-group of the reducedC*-algebra associated to the free group F2 on two generators is nota group.

 Sat 8/30 11:15am 939 Evans Dan Voiculescu: Duality transform for the coalgebra of \partial_{X:B} Abstract: A duality transform for the coalgebra of the free difference quotientderivation-comultiplication of an operator with respect to a free algebra ofscalars is constructed. The dual object is realized in an algebra of matricialanalytic functions endowed with yet another generalization of the differencequotient derivation.