On the cohomology of Banach A∞-module over admissible Banach A∞-algebra

In this paper we are concerned with Banach A_∞-module M over admissible Banach A_∞-algebra A. We give some properties of admissible modules and algebras. We study the cohomology of the complex C_∞(A, M). We show that the vanishing of cohomology of this complex in certain dimensions implies to the existence of the A_∞-module structure.     

Banach–Lie groupoids and generalized inversion

We study a few basic properties of Banach–Lie groupoids and algebroids, adapting some classical results on finite dimensional Lie groupoids. As an illustration of the general theory, we show that the notion of locally transitive Banach–Lie groupoid sheds fresh light on earlier research on some infinite-dimensional manifolds associated with Banach algebras.     

The deformed twisted Heisenberg–Virasoro type Lie bialgebra

Abstract

In this article, we investigate Lie bialgebra structures on the deformed twisted Heisenberg–Virasoro Lie algebra. Sufficient and necessary conditions for this type Lie bialgebra structures to be triangular coboundary are given.

Communicated by K. C. Misra     

K—very Smooth Banach Spaces

We introduce the notion of K-very smoothness which is a generalization of very smoothness in Banach spaces. A necessary and sufficient condition for a Banach space to be K-very smooth is obtained. We also consider some relations between K-very smoothness and other geometrical notions.     

Structures of Not-finitely Graded Lie Algebras Related to Generalized Heisenberg–Virasoro Algebras

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg–Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.     

Banach * 代数

Ｖ．Ｐｔáｋ［１］在有单位元的情形下详细研究了Ｂａｎａｃｈ代数．本文将推广他的结果到一般情形．特别，我们引入了可正开拓的正线性泛函，并对它进行了刻划．     

Representations and T*-Extensions of Hom–Jordan–Lie Algebras

The purpose of this article is to study representations and T*-extensions of δ-hom–Jordan–Lie algebras. In particular, adjoint representations, trivial representations, deformations, and many properties of T*-extensions of δ-hom–Jordan–Lie algebras are studied in detail. Derivations and central extensions of δ-hom–Jordan–Lie algebras are also discussed as an application.     

A Lie–Rinehart Algebra with No Antipode

The aim of this note is to communicate a simple example of a Lie–Rinehart algebra whose enveloping algebra is not a Hopf algebroid, neither in the sense of Böhm and Szlachányi, nor in the sense of Lu.     

Approximately Linear Mappings in Banach Modules over a C^＊-algebra

Let X and Y be vector spaces. The authors show that a mapping f ： X →Y satisfies the functional equation 2d f（∑^2d j=1（-1）^j＋1xj/2d）=∑^2dj=1（-1）^j＋1f（xj） with f（0） = 0 if and only if the mapping f ： X→ Y is Cauchy additive, and prove the stability of the functional equation （≠） in Banach modules over a unital C^＊-algebra, and in Poisson Banach modules over a unital Poisson C＊-algebra. Let A and B be unital C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊- algebras. As an application, the authors show that every almost homomorphism h ： A →B of A into is a homomorphism when h（（2d-1）^nuy） =- h（（2d-1）^nu）h（y） or h（（2d-1）^nuoy） = h（（2d-1）^nu）oh（y） for all unitaries u ∈A, all y ∈ A, n = 0, 1, 2,.... Moreover, the authors prove the stability of homomorphisms in C^＊-algebras, Poisson C^＊-algebras or Poisson JC^＊-algebras.     

Generalized Heisenberg equations on ℤ-graded Lie algebrasLie algebras

We study the integrable systems of the Heisenberg equation type that correspond to different decompositions of ℤ-graded Lie algebras into a direct sum of two subalgebras. We discover new non-Abelian generalizations of some known integrable models. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 2, pp. 248–255, August, 1999.     

A Banach–Zarecki Theorem for functions with values in Banach spaces

A Banach–Zarecki Theorem for a Banach space-valued function  \(F : [0,1] \rightarrow X\) with compact range is presented. We define the strong absolute continuity ( \(sAC_{||.||_{F}}\) ) and the bounded variation ( \(BV_{||.||_{F}}\) ) of \(F\) with respect to the Minkowski functional \(||.||_{F}\) associated to the closed absolutely convex hull \(C_{F}\) of \(F([0,1])\) . It is proved that \(F\) is \(sAC_{||.||_{F}}\) if and only if \(F\) is \(BV_{||.||_{F}}\) , weak continuous on \([0,1]\) and satisfies the weak property \((N)\) .     

Cycle modules and the intersection A ∞-algebra

In his paper Chow Group with Coefficients, M. Rost has developed a generalization of the classical Chow groups based on Milnor K-theory and an axiomatic generalization of it called cycles modules. For a cycle module M with a ring structure and a smooth scheme X of finite type over a field, we show that Rost’s cycle complex with coefficients ${C^*(X, M)_\mathbb{Q}}$ has a structure of an A _∞-algebra. In the case of Milnor K-theory it provides an homotopy model for the classical intersection theory of cycles.     

A continuous version of the Hausdorff–Banach–Tarski paradox

We come up with a simple proof for a continuous version of the Hausdorff–Banach–Tarski paradox, which does not make use of Robinson’s method of compatible congruences and fits in the case of finite and countable paradoxical decompositions. It is proved that there exists a free subgroup whose rank is of the power of the continuum in a rotation group of a three-dimensional Euclidean space. We also argue that unbounded subsets of Euclidean space containing inner points are denumerably equipollent.     

On the Banach ⁎-algebra crossed product associated with a topological dynamical system

Given a topological dynamical system $\Sigma =(X,\sigma )$, where X is a compact Hausdorff space and σ a homeomorphism of X, we introduce the Banach ^?-algebra crossed product ${?}^1(\Sigma )$ most naturally associated with Σ and initiate its study. It has a richer structure than its well investigated $C^{?}$-envelope, as becomes evident from the possible existence of non-self-adjoint closed ideals. We link its ideal structure to the dynamics, determining when the algebra is simple, or prime, and when there exists a non-self-adjoint closed ideal. A structure theorem is obtained when X consists of one finite orbit, and the algebra is shown to be Hermitian if X is finite. The key lies in analysing the commutant of $C(X)$ in the algebra, which is shown to be a maximal abelian subalgebra with non-zero intersection with each non-zero closed ideal.     

A Banach–Stone Theorem for Uniformly Continuous Functions

 In this note we prove that the uniformity of a complete metric space X is characterized by the vector lattice structure of the set U(X) of all uniformly continuous real functions on X.     

A Banach–Stone Theorem for Uniformly Continuous Functions

 In this note we prove that the uniformity of a complete metric space X is characterized by the vector lattice structure of the set U(X) of all uniformly continuous real functions on X. (Received 3 March 2000; in revised form 29 June 2000)     

三角Banach代数的Lie弱顺从性

设$\mathcal {A,\ B}$ 是含单位元的Banach代数, $\mathcal M$ 是一个Banach $\mathcal {A,\ B}$-双模. $\mathcal {T}=\left ( \begin{array}{cc} \mathcal {A} & \mathcal M \\ & \mathcal {B} \\ \end{array} \right )$按照通常矩阵加法和乘法,范数定义为$\|\left( \begin{array}{cc} a & m \\ & b\\ \end{array} \right)\|=\|a\|_{\mathcal A}+\|m\|_{\mathcal M}+\|b\|_{\mathcal B}$,构成三角Banach 代数.如果从$\mathcal T$到其$n$次对偶空间$\mathcal T^{n}$上的Lie导子都是标准的,则称$\mathcal T$是Lie $n$弱顺从的.本文研究了三角Banach代数$\mathcal T$上的Lie $n$弱顺从性,证明了有限维套代数是Lie $n$弱顺从的.     

Global Lie–Tresse theorem

We prove a global algebraic version of the Lie–Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.