The Birman-Schwinger principle in von Neumann algebras of finite type

We introduce a relative index for a pair of dissipative operators in a von Neumann algebra of finite type and prove an analog of the Birman-Schwinger principle in this setting. As an application of this result, revisiting the Birman-Krein formula in the abstract scattering theory, we represent the de la Harpe-Skandalis determinant of the characteristic function of dissipative operators in the algebra in terms of the relative index.     

Constrained von Neumann Inequalities

An equivalent formulation of the von Neumann inequality states that the backward shift S* on ?₂ is extremal, in the sense that if T is a Hilbert space contraction, then ‖p(T)‖?‖p(S*)‖ for each polynomial p. We discuss several results of the following type: if T is a Hilbert space contraction satisfying some constraints, then S* restricted to a suitable invariant subspace is an extremal operator. Several operator radii are used instead of the operator norm. Applications to inequalities of coefficients of rational functions positive on the torus are given.     

Free orbit dimension of finite von Neumann algebras

We introduce a new free entropy invariant, which yields improvements of most of the applications of free entropy to finite von Neumann algebras, including those with Cartan subalgebras, simple masas, property T, property Γ, nonprime factors, and thin factors.     

Reversibility of linear cellular automata

In this paper the reversibility problem for linear cellular automata defined by a characteristic matrix of the form of a pentadiagonal matrix is tackled. Specifically, a criterion for the reversibility in terms of the number of cells is stated and, in these cases, the inverse cellular automata are explicitly computed.     

On the notion of relative property (T) for inclusions of von Neumann algebras

We prove that the notion of rigidity (or relative property (T)) for inclusions of finite von Neumann algebras recently defined by the second author is equivalent to a weaker property, in which no “continuity constants” are required. The proof is by contradiction and uses infinite products of completely positive maps, regarded as correspondences.     

A secret sharing scheme based on cellular automata

A new secret sharing scheme based on a particular type of discrete delay dynamical systems: memory cellular automata, is proposed. Specifically, such scheme consists of a (k, n)-threshold scheme where the text to be shared is considered as one of the k initial conditions of the memory cellular automata and the n shares to be distributed are n consecutive configurations of the evolution of such cellular automata. It is also proved to be perfect and ideal.     

Modelling nonlinear sequence generators in terms of linear cellular automata

In this work, a wide family of LFSR-based sequence generators, the so-called clock-controlled shrinking generators (CCSGs), has been analyzed and identified with a subset of linear cellular automata (CA). In fact, a pair of linear models describing the behavior of the CCSGs can be derived. The algorithm that converts a given CCSG into a CA-based linear model is very simple and can be applied to CCSGs in a range of practical interest. The linearity of these cellular models can be advantageously used in two different ways: (a) for the analysis and/or cryptanalysis of the CCSGs and (b) for the reconstruction of the output sequence obtained from this kind of generators.     

Density of periodic points, invariant measures and almost equicontinuous points of cellular automata

Revisiting the notion of μ-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure μ by iterations of such automata converges in Cesàro mean to an invariant measure μ_c. Moreover the dynamical system (cellular automaton F, invariant measure μ_c) has still the μ_c-almost equicontinuity property and the set of periodic points is dense in the topological support of the measure μ_c. We also show that the density of periodic points in the topological support of a measure μ occurs for each μ-almost equicontinuous cellular automaton when μ is an invariant and shift ergodic measure. Finally using most of these results we give a non-trivial example of a couple (μ-equicontinuous cellular automaton F, shift and F-invariant measure μ) such that the restriction of F to the topological support of μ has no equicontinuous points.     

Synthesis of cryptographic interleaved sequences by means of linear cellular automata

This work shows that a class of pseudorandom binary sequences, the so-called interleaved sequences, can be generated by means of linear multiplicative polynomial cellular automata. In fact, these linear automata generate all the solutions of a type of linear difference equations with binary coefficients. Interleaved sequences are just particular solutions of such equations. In this way, popular nonlinear sequence generators with cryptographic application can be linearized in terms of simple cellular automata.     

On maximal injective subalgebras of tensor products of von Neumann algebras

Let Mi be a von Neumann algebra, and Bi be a maximal injective von Neumann subalgebra of Mi, i=1,2. If M₁ has separable predual and the center of B₁ is atomic, e.g., B₁ is a factor, then is a maximal injective von Neumann subalgebra of . This partly answers a question of Popa.     

Multiplication operators defined by covering maps on the Bergman space: The connection between operator theory and von Neumann algebras

In this paper, we combine methods of complex analysis, operator theory and conformal geometry to construct a class of Type II factors in the theory of von Neumann algebras, which arise essentially from holomorphic coverings of bounded planar domains. One will see how types of such von Neumann algebras are related to algebraic topology of planar domains. As a result, the paper establishes a fascinating connections to one of the long-standing problems in free group factors. An interplay of analytical, geometrical, operator and group theoretical techniques is intrinsic to the paper.     

Persistence of sums of correlated increments and clustering in cellular automata

Let $T$ be the first return time to $(?\infty ,0]$ of sums of increments given by a functional of a stationary Markov chain. We determine the asymptotic behavior of the survival probability, $\mathbb{P}(T\ge t)～C{t}^{?1∕2}$ for an explicit constant $C$. Our analysis is based on a connection between the survival probability and the running maximum of the time-reversed process, and relies on a functional central limit theorem for Markov chains. As applications, we recover known clustering results for the 3-color cyclic cellular automaton and the Greenberg–Hastings model, and we prove a new clustering result for the 3-color firefly cellular automaton.     

A note on cocycle-conjugate endomorphisms of von Neumann algebras

We show that two cocycle-conjugate endomorphisms of an arbitrary von Neumann algebra that satisfy certain stability conditions are conjugate endomorphisms, when restricted to some specific von Neumann subalgebras. As a consequence of this result, we obtain a new criterion for conjugacy of Powers shift endomorphisms acting on factors of type

     

Dilations of semigroups on von Neumann algebras and noncommutative Lp-spaces

We prove that any weak* continuous semigroup ${(T_t)}_{t?0}$ of factorizable Markov maps acting on a von Neumann algebra M equipped with a normal faithful state can be dilated by a group of Markov ?-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative ${\mathrm{L}}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh's ${\mathrm{H}}^{\infty }$ functional calculus of the generators of these semigroups on the associated noncommutative ${\mathrm{L}}^p$-spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of ${\mathbb{R}}^n$.     

The range of operators on von Neumann algebras

We prove that for every bounded linear operator , where is a non-reflexive quotient of a von Neumann algebra, the point spectrum of is non-empty (i.e., for some the operator fails to have dense range). In particular, and as an application, we obtain that such a space cannot support a topologically transitive operator.

     

On a property ofL p spaces on semifinite von Neumann algebras

A characterization of the traces in a broad class of weights on von Neumann algebras is obtained. A new property of the domain ideals of these traces is proved. In the semifinite case, a relation for a faithful normal trace is established. This result is new even for the algebra of all bounded operators on a Hilbert space. Applications of the main result to the structure theory of von Neumann algebras and to the Köthe duality theory for ideal spaces of Segal measurable operators are given.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 185–190, August, 1998.The author wishes to express his deep gratitude to Professor A. N. Sherstnev for setting the problem and for fruitful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00025.     

Asymptotic behavior of Markov semigroups on preduals of von Neumann algebras

We develop a new approach for investigation of asymptotic behavior of Markov semigroup on preduals of von Neumann algebras. With using of our technique we establish several results about mean ergodicity, statistical stability, and constrictiviness of Markov semigroups.     

基于元胞自动机的植物病虫害传播的计算机仿真

利用元胞自动机方法建立植物病虫害传播的数学模型。在此基础上,分别对两种不同病虫害来源的情况进行仿真。仿真结果表明,在参数给定的情况下,无论病虫来源于自身还是外界,植物病虫害的传播均在一定时间后达到稳定状态,不同状态元胞占有率相近;相同参数下,同病虫来源于自身相比,植物病虫从外界入侵时,植物被感染的变化率较低,病虫害传播路径较有规律,有利于病虫害源的确定和病虫害的治理。