Freeness of linear and quadratic forms in von Neumann algebras

We characterize the semicircular distribution by freeness of linear and quadratic forms in noncommutative random variables from tracial W^?-probability spaces with relaxed moment conditions.     

A note on noncommutative moment problems

Noncommutative moment problems for C*-algebras are studied. We generalize a result of Hadwin on tracial states to nontracial case. Our results are applied to obtain simple solutions to moment problems on the square and the circle as well as extend the positive unital functionals from a (discrete) complex group algebra to states on the group C*-algebra.     

Crossed products by finite group actions with certain tracial Rokhlin property

We introduce a special tracial Rokhlin property for unital C~*-algebras. Let A be a unital tracial rank zero C~*-algebra(or tracial rank no more than one C~*-algebra). Suppose that α : G → Aut(A) is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C~*-algebra. Then, the crossed product C~*-algebra C~*(G, A, α) has tracia rank zero(or has tracial rank no more than one). In fact,we get a more general results.     

Smale空间上的广群C~*-代数的迹

本文证明由拓扑混合的Smale空间上的渐进等价关系定义的广群C*-代数及其相应的Ruelle代数有唯一的迹态;在拓扑可迁的情形下,证明此C*-代数的迹态构成了一个单形,此单形顶点的个数等于“Smale谱分解”中基本空间的个数,单形的重心是该C*-代数的唯一的αa-不变迹态;此回答了I.Putnam的一个猜测.     

Stability under Small Hilbert–Schmidt Perturbations for <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

This paper studies the tracial stability of C^*-algebras, which is a general property of stability of relations in a Hilbert–Schmidt-type norm defined by a trace on a C^*-algebra. Precise definitions are formulated in terms of tracial ultraproducts. For nuclear C^*-algebras, a characterization of matricial tracial stability in terms of approximation of tracial states by traces of finite-dimensional representations is obtained. For the nonnuclear case, new obstructions and counterexamples are constructed in terms of free entropy theory.     

Tracial Limit of C^＊-algebras

A new limit of C^*-algebras, the tracial limit, is introduced in this paper. We show that a separable simple C^*-algebra A is a tracial limit of C^*-algebras in Z-(k) if and only if A has tracial topological rank no more than k. We present several known results using the notion of tracial limits.     

**-Haagerup Property for C*-Algebras

Extending the notion of Haagerup property for finite von Neumann algebras to the general von Neumann algebras, the authors define and study the $(**)$-Haagerup property for $C^{*}$-algebras in this paper. They first give an answer to Suzuki''s question (2013), and then obtain several results of $(**)$-Haagerup property parallel to those of Haagerup property for $C^{*}$-algebras. It is proved that a nuclear unital $C^{*}$-algebra with a faithful tracial state always has the $(**)$-Haagerup property. Some heredity results concerning the $(**)$-Haagerup property are also proved.     

Tracial algebras and an embedding theorem

We prove that every positive trace on a countably generated ∗-algebra can be approximated by positive traces on algebras of generic matrices. This implies that every countably generated tracial ∗-algebra can be embedded into a metric ultraproduct of generic matrix algebras. As a particular consequence, every finite von Neumann algebra with separable pre-dual can be embedded into an ultraproduct of tracial ∗-algebras, which as ∗-algebras embed into a matrix-ring over a commutative algebra.     

Plancherel–Rotach Asymptotic Expansion for Some Polynomials from Indeterminate Moment Problems

We study the Plancherel–Rotach asymptotics of four families of orthogonal polynomials: the Chen–Ismail polynomials, the Berg–Letessier–Valent polynomials, and the Conrad–Flajolet polynomials I and II. All these polynomials arise in indeterminate moment problems, and three of them are birth and death process polynomials with cubic or quartic rates. We employ a difference equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to a conjecture about large degree behavior of polynomials orthogonal with respect to solutions of indeterminate moment problems.     

On topologically finite-dimensional simple <Emphasis Type="Italic">C</Emphasis>*-algebras

We show that, if a simple C*-algebra A is topologically finite-dimensional in a suitable sense, then not only K₀(A) has certain good properties, but A is even accessible to Elliott’s classification program. More precisely, we prove the following results:If A is simple, separable and unital with finite decomposition rank and real rank zero, then K₀(A) is weakly unperforated.If A has finite decomposition rank, real rank zero and the space of extremal tracial states is compact and zero-dimensional, then A has stable rank one and tracial rank zero. As a consequence, if B is another such algebra, and if A and B have isomorphic Elliott invariants and satisfy the Universal coefficients theorem, then they are isomorphic.In the case where A has finite decomposition rank and the space of extremal tracial states is compact and zero-dimensional, we also give a criterion (in terms of the ordered K₀-group) for A to have real rank zero. As a byproduct, we show that there are examples of simple, stably finite and quasidiagonal C*-algebras with infinite decomposition rank.Supported by: EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280) and Deutsche Forschungsgemeinschaft (SFB 478).     

The boundary of the Krein space tracial numerical range, an algebraic approach and a numerical algorithm

In this article, tracial numerical ranges associated with matrices in an indefinite inner product space are investigated. The boundary equations of these sets are obtained, and the case of the boundary being a polygon is studied. As an application, a numerical algorithm for plotting the tracial numerical range of an arbitrary complex matrix is presented. Our approach uses the elementary idea that the boundary may be traced by computing the supporting lines.     

The construction of numerical integration rules of degree three for product regions

Numerical integration formulas in n-dimensional Euclidean space of degree three are discussed. In this paper, for the product regions a method is presented to construct numerical integration formulas of degree three with 2n real points and positive weights. The presented problem is a little different from those dealt with by other authors. All the corresponding one-dimensional integrals can be different from each other and they are also nonsymmetrical. In this paper an n-dimensional numerical integration problem is turned into n one-dimensional moment problems, which simplifies the construction process. Some explicit numerical formulas are given. Furthermore, a more generalized numerical integration problem is considered, which will shed light on the final solution to the third degree numerical integration problem.     

On-line control model for network construction projects

An activity-on-arc network project of the PERT type with random activity durations is considered. For each activity, its accomplishment is measured in percentages of the total project. When operated, each activity utilizes resources of a pregiven capacity and no resource reallocation is undertaken in the course of the project's realization. Each activity can be operated at several possible speeds that are subject to random disturbances and correspond to one and the same resource capacity; that is, these speeds depend only on the degree of intensity of the project's realization. For example, in construction projects partial accomplishments are usually measured in percentages of the total project, while different speeds correspond to different hours a day per worker. The number of possible speeds is common to all activities. For each activity, speeds are sorted in ascending order of their average values—namely speeds are indexed. It is assumed that at any moment t>0 activities, in operation at that moment, have to apply speeds of one and the same index that actually determines the project's speed. The progress of the project can be evaluated only via inspection at control points that have to be determined. The project's due date and the chance constraint to meet the deadline are pregiven. An on-line control model is suggested that, at each control point, faces a stochastic optimization problem. Two conflicting objectives are imbedded in the model:(1) to minimize the number of control points, and(2) to minimize the average index of the project's speeds which can be changed only at a control point.At each routine control point, decision-making centers on determining the next control point and the new index of the speeds (for all activities to be operated) up to that point. The model's performance is verified via simulation.The developed on-line control algorithm can be used for various PERT network projects which can be realized with different speeds, including construction projects and R&D projects.     

Stable Rank One and Real Rank Zero for Crossed Products by Finite Group Actions with the Tracial Rokhlin Property

The authors prove that the crossed product of an infinite dimensional simple separable unital C＊-algebra with stable rank one by an action of a finite group with the tracial Rokhlin property has again stable rank one. It is also proved that the crossed product of an infinite dimensional simple separable unital C＊-algebra with real rank zero by an action of a finite group with the tracial Rokhlin property has again real rank zero.     

Purely infinite corona algebras of simple C*-algebras

In this paper, we study the problem of when the corona algebra of a non-unital C*-algebra is purely infinite. A complete answer is obtained for stabilisations of simple and unital algebras that have enough comparison of positive elements. Our result relates the pure infiniteness condition (from its strongest to weakest forms) to the geometry of the tracial simplex of the algebra, and to the behaviour of corona projections, despite the fact that there is no real rank zero condition.     

Efficient algorithms for wavelength assignment on trees of rings

A fundamental problem in communication networks is wavelength assignment (WA): given a set of routing paths on a network, assign a wavelength to each path such that the paths with the same wavelength are edge-disjoint, using the minimum number of wavelengths. The WA problem is NP-hard for a tree of rings network which is well used in practice. In this paper, we give an efficient algorithm which solves the WA problem on a tree of rings with an arbitrary (node) degree using at most 3L wavelengths and achieves an approximation ratio of 2.75 asymptotically, where L is the maximum number of paths on any link in the network. The 3L upper bound is tight since there are instances of the WA problem that require 3L wavelengths even on a tree of rings with degree four. We also give a 3L and 2-approximation (resp. 2.5-approximation) algorithm for the WA problem on a tree of rings with degree at most six (resp. eight). Previous results include: 4L (resp. 3L) wavelengths for trees of rings with arbitrary degrees (resp. degree at most eight), and 2-approximation (resp. 2.5-approximation) algorithm for trees of rings with degree four (resp. six).     

An upper triangular decomposition theorem for some unbounded operators affiliated to II<Subscript>1</Subscript>-factors

Results of Haagerup and Schultz [17] about existence of invariant subspaces that decompose the Brown measure are extended to a large class of unbounded operators affiliated to a tracial von Neumann algebra. These subspaces are used to decompose an arbitrary operator in this class into the sum of a normal operator and a spectrally negligible operator. This latter result is used to prove that, on a bimodule over a tracial von Neumann algebra that is closed with respect to logarithmic submajorization, every trace is spectral, in the sense that the trace value on an operator depends only on the Brown measure of the operator.     

A tracial quantum central limit theorem

We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be considered a generalization of Johansson's theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity.