相关论文: Fractal entropies and dimensions for microstate sp…
For a selfadjoint element x in a tracial von Neumann algebra and $\alpha = \delta_0(x)$ we compute bounds for $\mathbb H^{\alpha}(x),$ where $\mathbb H^{\alpha}(x)$ is the free Hausdorff $\alpha$-entropy of $x.$ The bounds are in terms of…
This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a…
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…
Suppose M is a hyperfinite von Neumann algebra with a tracial state $\phi$ and $\{a_1,...,a_n\}$ is a set of selfadjoint generators for M. We calculate $\delta_0(a_1,...,a_n)$, the modified free entropy dimension of $\{a_1,...,a_n\}$.…
In this paper we introduce the concept of the upper free orbit-dimension of a finite von Neumann algebra, and we derive some of its basic properties. Using this concept, we are able to improve most of the applications of free entropy to…
Suppose $N$ is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in full matrix algebras, we deduce…
We consider the {\it fractal von Neumann entropy} associated with the {\it fractal distribution function} and we obtain for some {\it universal classes h of fractons} their entropies. We obtain also for each of these classes a {\it…
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…
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…
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…
The notion of topological free entropy dimension of $n-$tuples of elements in a unital C$^*$ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological orbit…
Suppose M is a von Neumann algebra with normal, tracial state phi and {a_1,...,a_n} is a set of self-adjoint elements in M. We provide an alternative uniform packing description of delta_0(a_1,...,a_n), the modified free entropy dimension…
If $X, Y,$ and $Z$ are finite sets of selfadjoint elements in a tracial von Neumann algebra and $X$ generates a hyperfinite von Neumann algebra, then $\delta_0(X \cup Y \cup Z) \leq \delta_0(X \cup Y) + \delta_0(X \cup Z) - \delta_0(X).$ We…
Suppose $N \subset M$ is an inclusion of $II_1$-factors of finite index. If $N$ can be generated by a finite set of elements, then there exist finite generating sets $X$ for $N$ and $Y$ for $M$ such that $\delta_0(X) \geq \delta_0(Y)$,…
By juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In…
This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal…
In this work we are interested in the self--affine fractals studied by Gatzouras and Lalley and by the author which generalize the famous general Sierpinski carpets studied by Bedford and McMullen. We give a formula for the Hausdorff…
This paper is an exposition, with some new applications, of our results on the growth of entropy of convolutions. We explain the main result on $\mathbb{R}$, and derive, via a linearization argument, an analogous result for the action of…
In the paper, we introduce a new concept of topological orbit dimension of $n$-tuples of elements in a unital C$^*$ algebra. Using this concept, we conclude that the Voiculescu's topological free entropy dimension of any family of…
We establish several properties of the free Stein dimension, an invariant for finitely generated unital tracial $*$-algebras. We give formulas for its behaviour under direct sums and tensor products with finite dimensional algebras. Among a…