Preprints

A free analogue of Shannon's problem on monotonicity of Entropy.
Abstract: We prove a free probability analog of a result of Artstein-Ball-Barthe-Naor. In particualar we prove that if X_{1},X_{2},... are freely independent identically distributed random variables, then the free entropy chi(X_{1}+...+X_{n}/\sqrt{n}) is monotone increasing for all n. Our proof also leads to a slight simplification of the original argument in the classical case.
Download: http://xxx.lanl.gov/abs/math.OA/0510103

Notes on Free Probability Theory
Abstract: These notes are from a 4-lecture mini-course taught by the author at the conference on von Neumann algebras as part of the ``Geometrie non commutative en mathematiques et physique'' month at CIRM in 2004.
Download: http://xxx.lanl.gov/abs/math.OA/0504063

Remarks on Free Entropy Dimension
Abstract: We prove a technical result, showing that the existence of a closable unbounded dual system in the sense of Voiculescu is equivalent to the finiteness of free Fisher information. This approach allows one to give a purely operator-algebraic proof of the computation of the non-microstates free entropy dimension for generators of groups carried out in an earlier joint work with I. Mineyev. The same technique also works for finite-dimensional algebras. We also show that Voiculescu's question of semi-continuity of free entropy dimension, as stated, admits a counterexample. We state a modified version of the question, which avoids the counterexample, but answering which in the affirmative would still imply the non-isomorphism of free group factors.
Download: http://xxx.lanl.gov/abs/math.OA/0504062

The microstates free entropy dimension of any DT-operator is 2 (with K. Dykema and K. Jung)
Abstract: Suppose that µ is an arbitrary Borel measure on the complex plane with compact support and take c > 0. If Z is a DT(µ,c)-operator as defined by Dykema and Haagerup, then the microstates free entropy dimension of Z is 2.
Download: http://xxx.lanl.gov/abs/math.OA/0412273

Picard groups of topologically stable Poisson structures (with O. Radko)
Abstract: We compute the group of Morita self-equivalences (the Picard group) of a Poisson structure on an orientable surface, under the assumption that the degeneracies of the Poisson tensor are linear. The answer involves mapping class groups of surfaces, i.e., groups of isotopy classes of diffeomorphisms. We also show that the Picard group of these structures coincides with the group of outer Poisson automorphisms.
Download: http://xxx.lanl.gov/abs/math.SG/0408070

Non-microstates free entropy dimension for groups (with I. Mineyev)
Abstract: We show that for any discrete finitely-generated group G and any self-adjoint n-tuple X₁,...,X_n of generators of the group algebra of G, Voiculescu's non-microstates free entropy dimension \delta*(X₁,...,X_n) is exactly equal to \beta₁(G)-\beta₀(G)+1, where \beta_i are the L² Betti numbers of G.
Download: http://xxx.lanl.gov/abs/math.OA/0312242

Non-microstates free entropy dimension for groups (with I. Mineyev)

Abstract: We show that for any discrete finitely-generated group G and any self-adjoint n-tuple X₁,...,X_n of generators of the group algebra of G, Voiculescu's non-microstates free entropy dimension \delta*(X₁,...,X_n) is exactly equal to \beta₁(G)-\beta₀(G)+1, where \beta_i are the L² Betti numbers of G.

Download: http://xxx.lanl.gov/abs/math.OA/0312242

L² Homology for von Neumann algebras (with A. Connes)
Abstract: We define the notion of L² homology and L² Betti numbers for a tracial von Neumann algebra, or, more generally, for any involutive algebra with a trace. The definition of these invariants is obtained from the definition of L² homology for groups, using the ideas from the theory of correspondences. For the group algebra of a discrete group, our Betti numbers coincide with the L² Betti numbers of the group. We find a link between the first L² Betti number and free entropy dimension, which points to the non-vanishing of L² homology for the von Neumann algebra of a free group.
Download: http://xxx.lanl.gov/abs/math.OA/0309343

L² Homology for von Neumann algebras (with A. Connes)

Abstract: We define the notion of L² homology and L² Betti numbers for a tracial von Neumann algebra, or, more generally, for any involutive algebra with a trace. The definition of these invariants is obtained from the definition of L² homology for groups, using the ideas from the theory of correspondences. For the group algebra of a discrete group, our Betti numbers coincide with the L² Betti numbers of the group. We find a link between the first L² Betti number and free entropy dimension, which points to the non-vanishing of L² homology for the von Neumann algebra of a free group.

Download: http://xxx.lanl.gov/abs/math.OA/0309343

On multiplicity and free absorption for free Araki-Woods factors
Abstract: We show that Ozawa's recent results on solid von Neumann algebras imply that there are free Araki-Woods factors, which fail to have free absorption. We also show that a free Araki-Woods factors $\Gamma (\mu, n)$ associated to a measure and a multiplicity function $n$ may non-trivially depend on the multiplicity function.
Download: http://xxx.lanl.gov/abs/math.OA/0302217