Singly Generated II_1 Factors

In the paper, we study the generator problem of II$_1$ factors. By defining a new concept related to the number of generators of a von Neumann algebra, we are able to show that a large class of II$_1$ factors are singly generated, i.e., generated by two self-adjoint elements. In particular, this shows that most of II$_1$ factors, whose free entropy dimensions are less than or equal to one, are singly generated

HPS: A C++11 High Performance Serialization Library

Data serialization is a common and crucial component in high performance computing. In this paper, I present a C++11 based serialization library for performance critical systems. It provides an interface similar to Boost but up to 150% faster and beats several popular serialization libraries

On Voiculescu's Semicircular Matrices

Assume $\N$ is a von Neumann algebra of type II$_1$ with a tracial state $\tau_{\N}$, and \M is the von Neumann algebra of the $n\times n$ matrices over $\N$ with the canonical tracial state \tau_{\M}. Let $\mathcal D_n$ be the subalgebra of \M consisting of scalar diagonal matrices in \M. In this article, we study the properties of semicircular elements in \M that are free from $\mathcal D_n$ with respect to \tau_{\M}. Then we define a new concept "matricial distance" of two elements in \M and compute the matricial distance between two free semicircular elements in \M

An analogue of Szego's limit theorem in free probability theory

In the paper, we discuss orthogonal polynomials in free probability theory. Especially, we prove an analogue of of Szego's limit theorem in free probability theory

Embedding Dimensions of Finite von Neumann Algebras

We introduce "embedding dimensions" of a family of generators of a finite von Neumann algebra when the von Neumann algebra can be faithfully embedded into the ultrapower of the hyperfinite II$_1$ factor. These embedding dimensions are von Neumann algebra invariants, i.e., do not depend on the choices of the generators. We also find values of these invariants for some specific von Neumann algebras

A Modified Similarity Degree for C*-algebras

We define variants of Pisier's similarity degree for unital C*-algebras and use direct integral theory to obtain new results. We prove that if every II$_{1}$ factor representation of a separable C*-algebra $\mathcal{A}$ has property $\Gamma$, then the similarity degree of $\mathcal{A}$ is at most 11.Comment: 11 papes. Comments are welcom

Topological Free Entropy Dimension in Unital C^* algebras (II) : Orthogonal Sum of Unital C^*-algebras

In the paper, we obtain a formula for topological free entropy dimension in the orthogonal sum (or direct sum) of unital C^* algebras. As a corollary, we compute the topological free entropy dimension of any family of self-adjoint generators of a finite dimensional C^* algebra

Topological Free Entropy Dimension of in Unital C^*-algebras

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 dimension of one self-adjoint element in a unital C$^*$ algebra. Moreover, we calculate the values of topological free entropy dimensions of families of generators of some unital C$^*$ algebras (for example: irrational rotation C$^*$ algebras or minimal tensor product of two reduced C$^*$ algebras of free groups).Comment: Materials adde

Unital Full Amalgamated Free Products of MF Algebras

In this paper, we consider the question whether a unital full free product of MF algebras with amalgamation over a finite dimensional C*-algebra is an MF algebra. First, we show that, under a natural condition, a unital full free product of two separable residually finite dimensional (RFD) C*-algebras with amalgamation over a finite dimensional C*-algebra is again a separable RFD C*-algebra. Applying this result on MF C*-algebras, we show that, under a natual condition, a unital full free product of two MF algebras is again an MF algebra. As an application, we show that a unital full free product of two AF algebras with amalgamation over an AF algebra is an MF algebra if there are faithful tracial states on each of these two AF algebras such that the restrictions on the common subalgebra agree

Volatilities analysis of first-passage time and first-return time on a small-world scale-free network

In this paper, we study random walks on a small-world scale-free network, also called as pseudofractal scale-free web (PSFW), and analyze the volatilities of first passage time (FPT) and first return time (FRT) by using the variance and the reduced moment as the measures. Note that the FRT and FPT are deeply affected by the starting or target site. We don't intend to enumerate all the possible cases and analyze them. We only study the volatilities of FRT for a given hub (i.e., node with highest degree) and the volatilities of the global FPT (GFPT) to a given hub, which is the average of the FPTs for arriving at a given hub from any possible starting site selected randomly according to the equilibrium distribution of the Markov chain. Firstly, we calculate exactly the probability generating function of the GFPT and FRT based on the self-similar structure of the PSFW. Then, we calculate the probability distribution, the mean, the variance and reduced moment of the GFPT and FRT by using the generating functions as a tool. Results show that: the reduced moment of FRT grows with the increasing of the network order $N$ and tends to infinity while $N\rightarrow\infty$; but for the reduced moments of GFPT, it is almost a constant($\approx1.1605$) for large $N$. Therefore, on the PSFW of large size, the FRT has huge fluctuations and the estimate provided by MFRT is unreliable, whereas the fluctuations of the GFPT is much smaller and the estimate provided by its mean is more reliable. The method we propose can also be used to analyze the volatilities of FPT and FRT on other networks with self-similar structure, such as $(u, v)$ flowers and recursive scale-free trees.Comment: 2 figure, 18 pages, to be appear in JSTA