A generalization of the \(n\)-weak module amenability of Banach algebras
type: Item
instance of: scholarly article
title: A generalization of the \(n\)-weak module amenability of Banach algebras
zbMATH Open document ID: 1341.46032
author: Berna Arslan
author: Hülya Inceboz
published in: Semigroup Forum
publication date: 2016-02-01T00:00:00Z
review text: Let \(A\) and \(U\) be Banach algebras and \(A\) be a Banach \(U\)-module with compatible module action: \[ \alpha\cdot (ab)=(\alpha\cdot a)\cdot b,\quad(ab)\cdot \alpha=a(b\cdot \alpha), \] where \(\alpha\in U\), \(a,b\in A\). Let \(X\) be a Banach \(A\)-bimodule and a Banach \(U\)-bimodule with compatible module actions as follows: \[ \alpha\cdot(a\cdot x)=(\alpha\cdot a)\cdot x, \quad \alpha\cdot (x\cdot a)=(\alpha\cdot x)\cdot a,\quad(a\cdot \alpha)\cdot x=a\cdot (\alpha\cdot x), \] where \(\alpha\in U\), \(a\in A\), \(x\in X\). Similarly, the right and two sided action can be defined. Then \(X\) is called an \(A\)-\(U\)-module. A Banach \(A\)-\(U\)-module is called commutative if \(\alpha\cdot x=x\cdot a\) for each \(x\in X\) and \(\alpha\in U\). It is easy to see that \(X^{*}, X^{**},\dots,X^{(n)}\) are also (commutative) \(A\)-\(U\)-modules, whenever \(X\) is a (commutative) \(A\)-\(U\)-module, respectively. Now suppose that \(A\) and \(B\) are Banach algebras and Banach \(U\)-bimodules. Then consider a (not necessarily linear) bounded map \(h:A\rightarrow B\) which satisfies \[ h(a+b)=h(a)+h(b),\quad h(a-b)=h(a)-h(b),\quad h(\alpha\cdot a)=\alpha\cdot h(a),\quad h(a\cdot \alpha)=h(a)\cdot \alpha, \] where \(a,b\in A\) and \(\alpha\in U\). Also, \(h\) is called a \(U\)-module morphism if \(h(ab)=h(a)h(b)\) for each \(a,b\in A\). The set of all bounded \(U\)-module morphisms from \(A\) into \(B\) is denoted by \(\operatorname{Hom}_{U}(A,B)\). Suppose that \(A,U\) and \(X\) are as above. A \(U\)-module map \(D:A\rightarrow X\) is called \((\sigma,\psi)\)-module derivation if \[ D(ab)=\sigma(a)\cdot D(b)+D(a)\cdot \psi(b)\quad(a,b\in A), \] where \(\sigma,\psi\in \operatorname{Hom}_{U}(A,A)\). For each \(x\in X\), define \[ D_{x}(a)=\sigma(a)\cdot x-x\cdot \psi(a), \] where \(a\in A\). If \(X\) is a commutative \(A\)-\(U\)-module, then \(D_{x}\) becomes an \(A\)-\(U\)-module derivation, which is called \((\sigma,\psi)\)-inner derivation. {Definition.} Let \(A\) …
reviewed by: Amir Sahami
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 43A07
zbMATH DE Number: 6537647
zbMATH Keywords: module derivation
zbMATH Keywords: module amenability
zbMATH Keywords: semigroup algebra
MaRDI profile type: Publication
full work available at URL: s00233-014-9668-y
OpenAlex ID: W1987575818
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DOI: 10.1007/S00233-014-9668-Y
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description: scientific article; zbMATH DE number 6537647