I. Mineyev, D. Shlyakhtenko
We show that for any discrete finitely-generated group G and any self-adjoint n-tuple X_1,...,X_n of generators of the group algebra of G, Voiculescu's non-microstates free entropy dimension δ^*(X_1,...,X_n) is exactly equal to β_1 (G)-β_0 (G)+1, where β_i are the L^2 Betti numbers of G.
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We study the microstate free entropy of projections, and establish its basic properties similar to the self-adjoint variable case. Our main contribution is to characterize the pair-block freeness of projections by the additivity of their free entropy (Theorem 4.1), in the proof of which a transportation cost inequality plays an important role. We also briefly discuss the free pressure in relation to the microstate free entropy of projections.
Suppose that μis 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
Dykema and Haagerup introduced the class of DT-operators and also showed that every DT-operator generate 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.
We define and study a relative free entropy quantity, analogous in its properties to Voiculescu's relative free entropy Chi^*(...:B). Our definition uses matricial microstates, unlike his definition, which involves non-commutative Hilbert transform. We prove a change of variable formula and certain maximization results for our quantity. We also exhibit a connection between the free entropy of a matrix with operator entries, relative to the algebra of scalar matrices, with the free entropy of the entries of the matrix.
For certain generating sets of the subfactor pair $M\subset M\rtimes G$ where $G$ is a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Shreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of the form $M\rtimes(\mathbb{Z}/2\mathbb{Z})^{\oplus\infty}$ for a large class of algebras $M$.
In this paper, we extend the notion of microstate free entropy to the bi-free setting. In particular, using the bi-free analogue of random matrices, microstate bi-free entropy is defined. Properties essential to an entropy theory are developed, such as the behaviour of the entropy when transformations on the left variables or on the right variables are performed. In addition, the microstate bi-free entropy is demonstrated to be additive over bi-free collections provided additional regularity assumptions are included and is computed for all bi-free central limit distributions. Moreover, an orbi…
In this paper, we extend the notion of non-microstate free entropy to the bi-free setting. Using a diagrammatic approach involving bi-non-crossing diagrams, bi-free difference quotients are constructed as analogues of the free partial derivations. Adjoints of bi-free difference quotients are discussed and used to define bi-free conjugate variables. Notions of bi-free Fisher information and non-microstate entropy are defined and properties of free entropy are extended to the bi-free setting.
We give an general estimate for the non-microstates free entropy dimension $δ^{*}(X_{1},..., X_{n})$. If $X_{1},..., X_{n}$ generate a diffuse von Neumann algebra, we prove that $δ^{*}(X_{1},..., X_{n})\geq 1$. In the case that $X_{1},..., X_{n}$ are $q$-semicircular variables as introduced by Bozejko and Speicher and $q^{2}n<1$, we show that $δ^{*}(X_{1},..., X_{n})>1$. We also show that for $|q|<\sqrt{2}-1$, the von Neumann algebras generated by a finite family of $q$-Gaussian random variables satisfy a condition of Ozawa and are therefore solid: the relative commutant of any diffuse subalge…
We prove that if $X_{1},...,X_{n} (n >1)$ are selfadjoints in a $W^{*}$-probability space with finite non-microstates free Fisher information, then the von Neumann algebra $W^{*}(X_{1},...,X_{n})$ they generate doesn't have property $Γ$ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy.
Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy $χ_orb$ for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties the relation of $χ_orb$ with the usual free entropy $χ$ is shown. Moreover, the dimension counterpart $δ_{0,orb}$ of $χ_orb$ is discussed, and we obtain the relation of $δ_{0,orb}$ with the original free entropy dimension $δ_0$ with applications to $δ_0$ itself.
We show that in a tracial and finitely generated $W^\ast$-probability space existence of conjugate variables excludes algebraic relations for the generators. Moreover, under the assumption of maximal non-microstates free entropy dimension, we prove that there are no zero divisors in the sense that the product of any non-commutative polynomial in the generators with any element from the von Neumann algebra is zero if and only if at least one of those factors is zero. In particular, this shows that in this case the distribution of any non-constant self-adjoint non-commutative polynomial in the g…
A non-commutative non-selfadjoint random variable z is called R-diagonal, if its *-distribution is invariant under multiplication by free unitaries: if a unitary w is *-free from z, then the *-distribution of z is the same as that of wz. Using Voiculescu's microstates definition of free entropy, we show that the R-diagonal elements are characterized as having the largest free entropy among all variables y with a fixed distribution of y^*y. More generally, let Z be a d*d matrix whose entries are non-commutative random variables X_{ij}. Then the free entropy of the family {X_{ij}} of the entries…
Bi-Free Entropy with Respect to Completely Positive Maps
本文构造了关于完全正映射的非微观态双自由熵概念，通过扩展算子值双自由结构建立共轭变量，并将自由Fisher信息和熵扩展至该设置，证明在双R对角元素处取得极值。
Establishing strong 1-boundedness via non-microstates free entropy techniques
针对满足非素性、含Cartan子代数或Γ性质的迹von Neumann代数，利用非微状态自由熵技术找到特定扩散超有限子代数，给出其强1-有界性的简短证明。
By proving that certain free stochastic differential equations have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain $n$-tuples $X_{1},...,X_{n}$: we show that Abstract. By proving that certain free stochastic differential equations with analytic coefficients have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain n-tuples X_{1},...,X_{n}. In particular, we show that δ_{0}(X_{1},...,X_{n})\geq\dim_{M\bar{\otimes}M^{o}}V where M=W^{*}(X_{1},...,X_{n}) and V=\{(\partial(X_{1}),...,\partial(X_{n}))…
We calculate the microstates free entropy dimension of natural generators in an amalgamated free product of certain von Neumann algebras, with amalgamation over a hyperfinite subalgebra. In particular, some `exotic' Popa algebra generators of free group factors are shown to have the expected free entropy dimension. We also show that microstates and non--microstates free entropy dimension agree for generating sets of many groups. In the appendix by Wolfgang Lueck, the first L^2-Betti number for certain amalgamated free products of groups is calculated.
For a selfadjoint element x in a tracial von Neumann algebra and $α= δ_0(x)$ we compute bounds for $\mathbb H^α(x),$ where $\mathbb H^α(x)$ is the free Hausdorff $α$-entropy of $x.$ The bounds are in terms of $\int \int_{\mathbb R^2 -D} \log |y-z| dμ(y) dμ(z)$ where $μ$ is the Borel measure on the spectrum of x induced by the trace and $D \subset \mathbb R^2$ is the diagonal. We compute similar bounds for the free Hausdorff entropy of a free family of selfadjoints.
We define an analog of Voiculescu's free entropy for n-tuples of unitaries (u_{1},...,u_{n}) in a tracial von Neumann algebra M, normalizing a unital diffuse abelian subalgebra B in M. Using this quantity, we define the free dimension δ_{0}(u_{1},..,u_{n}\btw B). This number depends on (u_{1},... ,u_{n}) only up ``orbit equivalence'' over B. In particular, if R is an measirable equivalence relation on [0,1] generated by n automorphisms α_{1},...,α_{n}, let u_{1},..., u_{n} be the unitaries implementing α_{1},..,α_{n} in the Feldman-Moore crossed product algebra M=W^{*}([0,1],R), and let B be t…
Using Voiculescu's notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebraic invariant. We compute the free Hausdorff dimension in the cases where the set generates a finite dimensional algebra or where the set consists of a single selfadjoint. We show that the free Hausdorff …
We study Voiculescu's microstate free entropy for a single non-selfadjoint random variable. The main result is that certain additional constraints on eigenvalues of microstates do not change the free entropy. Our tool is the method of random regularization of Brown measure which was studied recently by Haagerup and the author. As a simple application we present an upper bound for the free entropy of a single non--selfadjoint operator in terms of its Brown measure and the second moment. We furthermore show that this inequality becomes an equality for a class of DT-operators which was introduced…
We find the microstates free entropy dimension of a large class of $L^{\infty}[0,1]$-circular operators, in the presence of a generator of the diagonal subalgebra.
We continue our study of when topological and measure-theoretic entropy agree for algebraic action of sofic groups. Specifically, we provide a new abstract method to prove that an algebraic action is strongly sofic. The method is based on passing to a "limiting object" for sequences of measures which are asymptotically supported on topological microstates. This limiting object has a natural poset structure that we are able to exploit to prove a max-min principle: if the sofic approximation has ergodic centralizer, then the largest subgroup on which the action is a local weak$^{*}$-limit of mea…
An Elementary Proof of the Inequality $χ\leq χ^*$ for Conditional Free Entropy
本文为自由概率中联系 Voiculescu 两个熵类似物的不等式 $χ\leq χ^*$ 提供了一个不依赖随机微分方程与大偏差理论的初等证明，并将其推广到条件微观态和非微观态自由熵。
We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on $\mathbb{R}^n$. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension c…
We get stationary solutions of a free stochastic partial differential equation. As an application, we prove equality of non-microstate and microstate free entropy dimensions under a Lipschitz like condition on conjugate variables, assuming also R^ω embeddability. This includes an N-tuple of q-Gaussian random variables e.g. for |q|N\leq 0.13.
We introduce a free probabilistic quantity called free Stein irregularity, which is defined in terms of free Stein discrepancies. It turns out that this quantity is related via a simple formula to the Murray--von Neumann dimension of the closure of the domain of the adjoint of the non-commutative Jacobian associated to Voiculescu's free difference quotients. We call this dimension the free Stein dimension, and show that it is a $*$-algebra invariant. We relate these quantities to the free Fisher information, the non-microstates free entropy, and the non-microstates free entropy dimension. In t…
This paper is part of a series aiming at proving that the $\limsup$ and $\liminf$ variants of Voiculescu's free entropy coincide. This is based on a Laplace principle (implying a large deviation principle) for hermitian brownian motion on $[0,1]$. In the current paper, we show that microstates free entropy $χ(X_1,...,X_m)$ and non-microstate free entropy $χ^*(X_1,...,X_m)$ coincide for self-adjoint variables $(X_1,...,X_m)$ satisfying a Schwinger-Dyson equation for subquadratic, bounded below, strictly convex potentials with Lipschitz derivative sufficiently approximable by non-commutative pol…
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