Dilating Covariant Representations of a Semigroup Dynamical System Arising from Number Theory

In Cuntz et al. (Math Ann 355(4):1383–1423, 2013. doi:10.1007/s00208-012-0826-9), studied the \({C^*}\)-algebra \({\mathfrak {T}[R]}\) generated by the left-regular representation of the \({ax + b}\)-semigroup of a number ring R on \({\ell^2(R \rtimes R^\times)}\). They were able to describe it as a universal \({C^*}\)-algebra defined by generators and relations, and show that it has an interesting KMS-structure and that it is functorial for injective ring homomorphisms. In this paper we show that \({\mathfrak {T}[R]}\) can be realized as the \({C^*}\)-envelope of the isometric semicrossed product of a certain semigroup dynamical system \({(\mathcal {A}_R, \alpha, R^\times)}\). We do this by proving that a representation of \({\mathcal {A}_R \times_\alpha^{\rm is}R^\times}\) is maximal if it is also a representation of \({\mathfrak {T}[R]}\). We also show how to explicitly dilate any representation of \({\mathcal {A}_R \times_\alpha^{\rm is}R^\times}\) to a representation of \({\mathfrak {T}[R]}\).     

Crossed Product of a C *-Algebra by a Semigroup of Endomorphisms Generated by Partial Isometries

The paper presents a construction of the crossed product of a C^*- algebra by a semigroup of endomorphisms generated by partial isometries. This work was in part supported by Polish Ministry of Science and High Education grant number N N201 382634.     

Cantor极小动力系统生成交叉积C~*-代数的K_0群

设(X,α)为一个Cantor极小系统,C(X)×_αZ为相应的交叉积C~*-代数,U,V为X内的两个clopen集.证明了如果[j_α(1U)_0=[jα(1_v)]_0∈K_0(C(X)×_αZ),则存在α的一个拓扑全群元素σ,使得σ(U)=V.     

奇异变换半群中J类的幂乘数

首先给出奇异变换半群中J类的幂乘数的定义,然后讨论奇异变换半群中J类的幂乘数的性质.     

Crossed Product Orders Over Valuation Rings II: Tamely Ramified Crossed Product Algebras

Let V be a commutative valuation domain of arbitrary Krull-dimension,with quotient field F, let K be a finite Galois extension ofF with group G, and let S be the integral closure of V in K.Suppose that one has a 2-cocycle on G that takes values in thegroup of units of S. Then one can form the crossed product ofG over S, S*G, which is a V-order in the central simple F-algebraK*G. If S*G is assumed to be a Dubrovin valuation ring of K*G,then the main result of this paper is that, given a suitabledefinition of tameness for central simple algebras, K*G is tamelyramified and defectless over F if and only if K is tamely ramifiedand defectless over F. The residue structure of S*G is alsoconsidered in the paper, as well as its behaviour upon passageto Henselization. 2000 Mathematics Subject Classification 16H05,16S35.     

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.     

Twisted Semigroup Crossed Products and Twisted Toeplitz Algebras of Ordered Groups

Let г^＋ be the positive cone of a totally ordered abelian group г, and σa cocycle in г. We study the twisted crossed products by actions of г＋ as endomorphisms of C^＊-algebras, and use this to generalize the theorem of Ji.     

Crossed Product Orders over Valuation Rings

Let V be a commutative valuation domain of arbitrary Krull-dimension(rank), with quotient field F, and let K be a finite Galoisextension of F with group G, and S the integral closure of Vin K. If, in the crossed product algebra K * G, the 2-cocycletakes values in the group of units of S, then one can form,in a natural way, a ‘crossed product order’ S *G K * G. In the light of recent results by H. Marubayashi andZ. Yi on the homological dimension of crossed products, thispaper discusses necessary and/or sufficient valuation-theoreticconditions, on the extension K/F, for the V-order S * G to besemihereditary, maximal or Azumaya over V. 2000 MathematicsSubject Classification 16H05, 16S35.     

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.     

On the Number of Nilpotents in the Partial Symmetric Semigroup

Abstract

In this note, we obtain and discuss formulae for the total number of nilpotent partial and nilpotent partial one–one transformations of a finite set.     

The Crossed Product by a Partial Endomorphism

Given a closed ideal I in a C*-algebra A, an ideal J (not necessarily closed) in I , a *-homomorphism α : A → M(I ) and a map L : J → A with some properties, based on earlier works of Pimsner and Katsura, we define a C*-algebra which we call the Crossed Product by a Partial Endomorphism. We introduce the Crossed Product by a Partial Endomorphism induced by a local homeomorphism σ : U → X where X is a compact Hausdorff space and U is an open subset of X. A bijection between the gauge invariant ideals of and the σ, σ^-1- invariant open subsets of X is showed. If (X, σ) has the property that is topologically free for each closed σ, σ^-1-invariant subset X′ of X then we obtain a bijection between the ideals of and the open σ, σ^-1-invariant subsets of X. *Partially supported by CNPq. **Supported by CNPq.     

Theory of Grobner Basesin Semigroup Algebra k[A]

TheoryofGr6bnerbasesinpolynomialalgebrak[X]isthemostfundamentalcomputationaltheoryincompilationalalgebraicgeometyll--5].Nevertheless,efficienciesofthealgorithmsofGrobnerbasesareverylowlybecauseofspecialpropertyofsymboliccomputation.Algorithmsinsemigr...     

Semigroup Theory of Degenerate Linear Cauchy Problems

No Abstract. June 26, 1999     

A Probabilistic-Numerical Approximation for an Obstacle Problem Arising in Game Theory

We investigate a two-player zero-sum stochastic differential game in which one of the players has more information on the game than his opponent. We show how to construct numerical schemes for the value function of this game, which is given by the solution of a quasilinear partial differential equation with obstacle.     

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)).     

On the Exel Crossed Product of Topological Covering Maps

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C ^*-algebras C(X)⋊ _α,ℒℕ introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X)⋊ _α,ℒℕ is a maximal abelian C ^*-subalgebra of C(X)⋊ _α,ℒℕ; any nontrivial two sided ideal of C(X)⋊ _α,ℒℕ has non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)⋊ _α,ℒℕ is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C ^*-algebras of homeomorphism dynamical systems.     

Pentagon Identities Arising in Supersymmetric Gauge Theory Computations

Theoretical and Mathematical Physics - The partition functions of three-dimensional N=2 supersymmetric gauge theories on different manifolds can be expressed as q-hypergeometric integrals....     

On Some Iterated Means Arising in Homogenization Theory

We consider iteration of arithmetic and power means and discuss methods for determining their limit. These means appear naturally in connection with some problems in homogenization theory.