A Semigroup Crossed Product Arising in Number Theory

Recently Bost and Connes have studied an interesting C*-algebraicHecke algebra arising in number theory. Here it is shown thatthis algebra can be realised as a semigroup crossed product,and be profitably studied using methods developed by the authorsfor analysing Toeplitz algebras. One main result is a characterisationof faithful representations of the Hecke algebra.     

Jones Index of a Quantum Dynamical Semigroup

In this paper we consider a semigroup of completely positive maps τ=(τ _t ,t≥0) with a faithful normal invariant state φ on a type-II ₁ factor A₀\mathcal{A}_{0} and propose an index theory. We achieve this via a Kolmogorov’s type of construction for stationary Markov processes which naturally associate a nested family of isomorphic von-Neumann algebras. In particular this construction generalises well known Jones construction associated with a sub-factor of a type-II₁ factor.     

Monotonic Solutions of a Functional Equation Arising from Simultaneous Utility Representations

The functional equation ?(h(x)) ? ?(h(y)) + ?(y) = ?(h(x ? y) + y) was solved by Aczél, Luce and Marley on the assumption that the functions are different iable. They posed the question of its strictly monotonic continuous solutions. The problem is solved using a uniqueness theorem. The continuity assumption on the functions is removed and the equation is also solved on a restricted domain.     

A Functional Equation Arising from Simultaneous Utility Representations

Suppose that two classes of utility representations of preferences, one additive and one increasing increments, hold simultaneously over uncertain binary alternatives (gambles). This assumption leads to the functional equation $$ f[h(x-y)+y]=f[h(x)]-f[h(y)]+f(y)\qquad (\kappa >x\geq y\geq 0), $$ and to the inequality h(z) ≤ z (z ∈ [0, κ[), where the functions ? and h are strictly increasing maps of the real interval [0, κ[ onto the real intervals [0, λ[ and [0, μ[, respectively, κ, λ, μ ∈]0, ∞]. We present all solutions under the additional assumption of (first-order) differentiability.     

Ideals and Representations of Certain Semigroup Algebras

We characterize various types of ideals in ¹-Munn algebras with a bounded approximate identity, in terms of the ideals of their underlying algebras. Moreover, we show the correspondence between different types of *-representations of ¹-Munn algebras and *-representations of their underlying algebras. In the context of semigroup algebras, .rst we prove the conjecture which was proposed in [8]. Then we apply the results of the first part to build various representations of unital semigroup algebras of completely 0-simple inverse semigroups. This generalizes some of the Barnes results [1] with different and much shorter proofs.     

A Parametrization of the Irreducible Representations of a Compact Inverse Semigroup

The aim in this article is to provide a parametrization of the finite dimensional irreducible representations of a compact inverse semigroup in terms of the irreducible representations of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new, and more conceptual, proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Moreover, we also prove that every norm continuous irreducible *-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.     

Covariant Representations for Coactions of Hopf C*-Algebras

Given a coaction of a Hopf C*-algebra A on a C*-algebra B withan -invariant C*-subalgebra C, and a conditional expectationE: B C commuting with , it is shown that if (, u) is a covariantrepresentation of the system (C, A, |_C), then there is an associatedcovariant representation of the system (B, A, ), where is the representation induced from up to B via E, and is a unitary corepresentation of Anaturally associated with u. Some applications are also discussed,including a lifting of ergodic coactions to von Neumann algebras,and a characterization of the amenability of multiplicativeunitary operators via infinite tensor product covariant representations.2000 Mathematics Subject Classification 46L55, 46L60, 46L87,46L89, 81R15, 81R50, 81T05.     

Conditions Sufficient for the Conservativity of a Minimal Quantum Dynamical Semigroup

Conditions sufficient for a minimal quantum dynamical semigroup (QDS) to be conservative are proved for the class of problems in quantum optics under the assumption that the self-adjoint Hamiltonian of the QDS is a finite degree polynomial in the creation and annihilation operators. The degree of the Hamiltonian may be greater than the degree of the completely positive part of the generator of the QDS. The conservativity (or the unital property) of a minimal QDS implies the uniqueness of the solution of the corresponding master Markov equation, i.e., in the unital case, the formal generator determines the QDS uniquely; moreover, in the Heisenberg representation, the QDS preserves the unit observable, and in the Schrödinger representation, it preserves the trace of the initial state. The analogs of the conservativity condition for classical Markov evolution equations (such as the heat and the Kolmogorov--Feller equations) are known as nonexplosion conditions or conditions excluding the escape of trajectories to infinity.     

On the Mechanism for Nonlinear Representations of the Lorentz Group Arising in Quantum Field Theory

A mechanism is identified that leads to the correct law for the relativistic Wigner function transformation with respect to the Lorentz group as long as the corresponding relativistic wave functions have special transformation properties.     

非太阳自反Banach空间控制系统的半群理论

本文利用C0 半群的对偶理论研究了非太阳自反Banach空间上的一类新控制系统 ,得到近似可控的一些新结果 .     

Representations of Hecke Algebras and Dilations of Semigroup Crossed Products

A family of Hecke C^*-algebras can be realised as crossed productsby semigroups of endomorphisms. It is shown by dilating representationsof the semigroup crossed product that the category of representationsof the Hecke algebra is equivalent to the category of continuousunitary representations of a totally disconnected locally compactgroup.     

一类退化半导体方程弱解存在性的研究

文章研究了当■(s)=sm(m>1),b(s)=s2和初值为u0,v0∈L 2(Ω)时一类退化半导体方程弱解的存在性．文章首先将原问题正则化,然后对正则化问题在L 2(Ω)空间上做出了有界估计,最后利用收敛性得到了问题的结论．     

Number of Representations of p-One-Dimensional Forms by a Genus

We study branching of representations of a locally p-one-dimensional form by a genus of positive definite integral quadratic forms. We give a complete list of minimal representations by a genus for forms of square level. GaussMinkowski formulas are obtained for heights of representations over the ring of integers. As an application, we obtain formulas for heights of primitive representations by genera for specific forms constructed by the method of orthogonal complement. Bibliography: 6 titles.     

A Nonlinear Elliptic System Arising in Electron Density Theory

Of concern is the nonlinear system

in R³ Here λ_i,C_j are given positive constants. Such a system arises in spin polarized ground state electron density theory. The parametres λ_:1:, λ_:2 should be such that

where N_1>,N₂ are given positive numbers satisfying where Z is some give positive number. (Here N$sub:1$esub:, N$sub:2$esub:, Z are, respectively, the number of spin up electrons, spin down electrons, and protons in the system.) We establish existence for this problem. Uniqueness remains open.     

压缩变换半群CT_n中一类元素的基数

压缩变换半群CT_n是全变换半群T_n中的压缩元组成的子半群.主要讨论了当n≥5时CT_n中一类元素{α∈CT_n|1α=1,((n-2)α,(n-1)α,nα)=(3,2,1))的基数并得到了递推公式:LN_n~((1,321))=d-d_1LN_(n-1)~(((1,321))-…-d_(k-3)LN_(n-(k-3))~((1,321)).     

迁移理论中一类非线性积分方程的解的个数

在文献[1—3]里,讨论了迁移理论中一类非线性积分方程存在唯一解和多解的充分条件。本文将对方程     

Theory and Computation of Covariant Lyapunov Vectors

Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.