The primality of subfactors of finite index in the interpolated free group factors

In this paper we prove that any II-subfactor of finite index in the interpolated free group factor is prime for any i.e., it is not isomorphic to tensor products of II-factors.

     

A relation between certain interpolated Cuntz algebras and interpolated free group factors

We investigate von Neumann algebras generated by the real parts of generators of Toeplitz extensions of interpolated Cuntz algebras on sub-Fock spaces. We show that some of them are isomorphic to interpolated free group factors . For example, in case of the golden number the corresponding number is .

     

Free group factors and factors with some decompositions

In this paper, we show that if type von Neumann factors have some decompositions introduced by Liming Ge and Sorin Popa, then these von Neumann factors are not isomorphic to free group factors . Thus we have proved the number defined by Ge and Popa bigger than 3 for all free group factors and we also extend some results of M. Stefan.

     

Maximal injective subalgebras of tensor products of free group factors

In this article, we prove the following results. Let L(F(n_i)) be the free group factor on n_i generators (n_i2) and λ(g_i) be one of standard generators of L(F(n_i)) for 1iN. Let be the abelian von Neumann subalgebra of L(F(n_i)) generated by λ(g_i). Then the abelian von Neumann subalgebra is a maximal injective von Neumann subalgebra of . When N is equal to infinity, we obtain strongly stable II₁ factors (or called McDuff factors) that contain maximal injective abelian von Neumann subalgebras.     

Strongly solid group factors which are not interpolated free group factors

We give examples of non-amenable infinite conjugacy classes groups Γ with the Haagerup property, weakly amenable with constant Λ_cb(Γ) = 1, for which we show that the associated II₁ factors L(Γ) are strongly solid, i.e. the normalizer of any diffuse amenable subalgebra P ì L(G){P \subset L(\Gamma)} generates an amenable von Neumann algebra. Nevertheless, for these examples of groups Γ, L(Γ) is not isomorphic to any interpolated free group factor L(F _t ), for 1 < t ≤  ∞.     

On a free group of transformations defined by an automaton

We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3. Both authors are supported by the NSF grants DMS-0308985 and DMS-0456185. Yaroslav Vorobets is supported by a Clay Research Scholarship.     

Free subgroups of free products and combinatorial hypermaps

We derive a generating series for the number of free subgroups of finite index in ${\Delta }^{+}={\mathbb{Z}}_{p}1{\mathbb{Z}}_q$ by using a connection between free subgroups of ${\Delta }^{+}$ and certain hypermaps (also known as ribbon graphs or “fat” graphs), and show that this generating series is transcendental. We provide non-linear recurrence relations for the above numbers based on differential equations that are part of the Riccati hierarchy.We also study the generating series for conjugacy classes of free subgroups of finite index in ${\Delta }^{+}$, which correspond to isomorphism classes of hypermaps. Asymptotic formulas are provided for the numbers of free subgroups of given finite index, conjugacy classes of such subgroups, or, equivalently, various types of hypermaps and their isomorphism classes.     

An Alternative to Free Entropy for Free Group Factors

In this paper we define a very simple invariant η(V^-) for a k-tuple V^-of unitaries in a finite factor von Neumann algebra, and we show how this invariant can replace free entropy in many of the important applications. We also introduce a notion of metric free entropy and some related concepts.We include proofs, using η, of the theorems of Liming Ge and of D. Voiculescu, respectively, on the primeness of and on the absence of Cartan snbalgebras in the noncommutative free group factors.     

Construction of free subgroups in the group of units of modular group algebras

Let KGbe the group algebra of a p¹ -group Gover a field Kof characteristic p > 0, and let U(KG)be its group of units. If KGcontains a nontrivial bicyclic unit and if Kis not algebraic over its prime field, then we prove that the free product Z_p? Z_p? Z_pcan be embedded in U(KG).     

Two-state free Brownian motions

In a two-state free probability space (A,φ,ψ), we define an algebraic two-state free Brownian motion to be a process with two-state freely independent increments whose two-state free cumulant generating function Rφ,ψ(z) is quadratic. Note that a priori, the distribution of the process with respect to the second state ψ is arbitrary. We show, however, that if A is a von Neumann algebra, the states φ, ψ are normal, and φ is faithful, then there is only a one-parameter family of such processes. Moreover, with the exception of the actual free Brownian motion (corresponding to φ=ψ), these processes only exist for finite time.     

A question of B. Plotkin about the semigroup of endomorphisms of a free group

Let be a free group of finite rank , let be the semigroup of endomorphisms of , and let be the group of automorphisms of .

Theorem. If is an automorphism of , then there is an such that for all .

     

Values of the Pukánszky invariant in free group factors and the hyperfinite factor

Let AMB(L²(M)) be a maximal abelian self-adjoint subalgebra (masa) in a type II₁ factor M in its standard representation. The abelian von Neumann algebra generated by A and JAJ has a type I commutant which contains the projection onto L²(A). Then decomposes into a direct sum of type I_n algebras for n{1,2,…,∞}, and those n's which occur in the direct sum form a set called the Pukánszky invariant, Puk(A), also denoted Puk_M(A) when the containing factor is ambiguous. In this paper we show that this invariant can take on the values S{∞} when M is both a free group factor and the hyperfinite factor, and where S is an arbitrary subset of . The only previously known values for masas in free group factors were {∞} and {1,∞}, and some values of the form S{∞} are new also for the hyperfinite factor.We also consider a more refined invariant (that we will call the measure-multiplicity invariant), which was considered recently by Neshveyev and Størmer and has been known to experts for a long time. We use the measure-multiplicity invariant to distinguish two masas in a free group factor, both having Pukánszky invariant {n,∞}, for arbitrary .     

The non-microstates free entropy dimension of DT-operators

Dykema and Haagerup introduced the class of DT-operators (Amer. J. Math. 126 (2004) 121-189) and also showed that every DT-operator generate (J. Funct. Anal. 209 (2004) 332-366), the von Neumann algebra generated by the free group on two generators. In this paper, we prove that Voiculescu's non-microstates free entropy dimension is 2 for all DT-operators.     

A hydrodynamic exercise in free probability: Setting up free Euler equations

For the free probability analogue of Euclidean space endowed with the Gaussian measure we apply the approach of Arnold to derive Euler equations for a Lie algebra of non-commutative vector fields which preserve a certain trace. We extend the equations to vector fields satisfying non-commutative smoothness requirements. We introduce a cyclic vorticity and show that it satisfies vorticity equations and that it produces a family of conserved quantities.     

A limit theorem for products of free unitary operators

This paper establishes necessary and sufficient conditions for the sequence of products of freely independent unitary operators to converge in distribution to the uniform law on the unit circle. The author would like to express his gratitude to Diana Bloom for her help with editing, and to Professor Raghu Varadhan for useful discussions.     

Free Wishart Processes

Using a matrix approach, we define free Wishart processes of parameter > 0 and prove a free additivity property and invertibility for > 1. For 1, we show that a free Wishart process is a solution of a SDE of square Bessel process type, driven by a free complex Brownian motion. In the case > 1, we establish existence and uniqueness of a strong solution of such a SDE.     

Quantum invariant families of matrices in free probability

We consider (self-adjoint) families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group. For the free orthogonal and hyperoctahedral groups, we obtain complete characterizations of the invariant families in terms of an operator-valued R-cyclicity condition. This is a surprising contrast with the Aldous-Hoover characterization of jointly exchangeable arrays.     

Free moment measures and laws

In [3], it was shown that convex, almost everywhere continuous functions coordinatize a broad class of probability measures on ${\mathbb{R}}^n$ by the map $U?{\left(\mathrm{?}U\right)}_{\mathrm{\#}}{e}^{?U}dx$. We consider whether there is a similar coordinatization of non-commutative probability spaces, with the Gibbs measure $e^{?U}dx$ replaced by the corresponding free Gibbs law. We call laws parameterized in this way free moment laws. We first consider the case of a single (and thus commutative) random variable and then the regime of n non-commutative random variables which are perturbations of freely independent semi-circular variables. We prove that free moment laws exist with little restriction for the one dimensional case, and for small even perturbations of free semi-circle laws in the general case.