scholarly journals Theory of B(X)-Module: Algebraic Module Structure of Generally Unbounded Infinitesimal Generators

2020 ◽  
Vol 2020 ◽  
pp. 1-27
Author(s):  
Yoritaka Iwata
Keyword(s):  
Banach Algebra ◽  
Algebraic Structure ◽  
Differential Operators ◽  
Mathematical Physics ◽  
Rotation Group ◽  
Module Structure ◽  
Unbounded Operators ◽  
Infinitesimal Generators ◽  
Rigorous Formulation

The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally unbounded infinitesimal generators. In conclusion, the concept of module over a Banach algebra is proposed as the generalization of the Banach algebra. As an application to mathematical physics, the rigorous formulation of a rotation group, which consists of unbounded operators being written by differential operators, is provided using the module over a Banach algebra.

First cohomology of 𝔞𝔣𝔣(1) and 𝔞𝔣𝔣(1|1) acting on linear differential operators

2016 ◽  
Vol 13 (01) ◽  
pp. 1550130 ◽  
Author(s):  
Imed Basdouri ◽  
Maha Boujelben ◽  
Ammar Derbali
Keyword(s):  
Differential Operators ◽  
Lie Superalgebra ◽  
Module Structure ◽  
Differential Cohomology ◽  
Linear Differential Operators ◽  
Linear Differential

We consider the [Formula: see text]-module structure on the spaces of differential operators acting on the spaces of weighted densities. We compute the first differential cohomology of the Lie superalgebra [Formula: see text] with coefficients in differential operators acting on the spaces of weighted densities. We study also the super analogue of this problem getting the same results.

scholarly journals On a Chaotic Weighted Shiftzpdp+1/dzp+1of Orderpin Bargmann Space

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Abdelkader Intissar
Keyword(s):  
Differential Operators ◽  
Unbounded Operators ◽  
Bargmann Space ◽  
Backward Shift ◽  
Creation Operators

This paper is devoted to the study of the chaotic properties of some specific backward shift unbounded operatorsHp=A*pAP+1;p=0,1,…realized as differential operators in Bargmann space, whereAandA*are the standard Bose annihilation and creation operators such that[A,A*]=I.

Cohomology and deformation of 𝔞𝔣𝔣(1|1) acting on differential operators

2018 ◽  
Vol 15 (05) ◽  
pp. 1850072
Author(s):  
Khaled Basdouri ◽  
Salem Omri
Keyword(s):  
Differential Operators ◽  
Lie Superalgebra ◽  
Module Structure ◽  
Infinitesimal Deformation ◽  
Differential Cohomology ◽  
Linear Differential Operators ◽  
Formal Deformation ◽  
Pseudo Differential Operators ◽  
Linear Differential

We consider the [Formula: see text]-module structure on the spaces of differential operators acting on the spaces of weighted densities. We compute the second differential cohomology of the Lie superalgebra [Formula: see text] with coefficients in differential operators acting on the spaces of weighted densities. We classify formal deformations of the [Formula: see text]-module structure on the superspaces of symbols of differential operators. We prove that any formal deformation of a given infinitesimal deformation of this structure is equivalent to its infinitesimal part. This work is the simplest superization of a result by Basdouri [Deformation of [Formula: see text]-modules of pseudo-differential operators and symbols, J. Pseudo-differ. Oper. Appl. 7(2) (2016) 157–179] and application of work by Basdouri et al. [First cohomology of [Formula: see text] and [Formula: see text] acting on linear differential operators, Int. J. Geom. Methods Mod. Phys. 13(1) (2016)].

The first module (σ,τ)-cohomology group of triangular Banach algebras of order three

2018 ◽  
Vol 17 (12) ◽  
pp. 1850225
Author(s):  
Hülya İnceboz ◽  
Berna Arslan
Keyword(s):  
Semigroup Forum ◽  
Banach Algebra ◽  
Cohomology Group ◽  
Banach Algebras ◽  
Module Structure ◽  
Semigroup Algebras ◽  
Module Amenability ◽  
Weak Module ◽  
Triangular Banach Algebras

The notion of module amenability for a class of Banach algebras, which could be considered as a generalization of Johnson’s amenability, was introduced by Amini in [Module amenability for semigroup algebras, Semigroup Forum 69 (2004) 243–254]. The weak module amenability of the triangular Banach algebra [Formula: see text], where [Formula: see text] and [Formula: see text] are Banach algebras (with [Formula: see text]-module structure) and [Formula: see text] is a Banach [Formula: see text]-module, is studied by Pourabbas and Nasrabadi in [Weak module amenability of triangular Banach algebras, Math. Slovaca 61(6) (2011) 949–958], and they showed that the weak module amenability of [Formula: see text] triangular Banach algebra [Formula: see text] (as an [Formula: see text]-bimodule) is equivalent with the weak module amenability of the corner algebras [Formula: see text] and [Formula: see text] (as Banach [Formula: see text]-bimodules). The main aim of this paper is to investigate the module [Formula: see text]-amenability and weak module [Formula: see text]-amenability of the triangular Banach algebra [Formula: see text] of order three, where [Formula: see text] and [Formula: see text] are [Formula: see text]-module morphisms on [Formula: see text]. Also, we give some results for semigroup algebras.

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