A generalization of the \(n\)-weak module amenability of Banach algebras
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 \(A\) be a Banach algebra and \(\sigma\in \operatorname{Hom}_{U}(A,A)\). \(A\) is called \((\sigma)\)-\(n\)-weakly amenable if \((A/J)^{n}\) is a commutative \(A\)-\(U\)-module and each \((\sigma,\sigma)\)-module derivation \(D:A\rightarrow (A/J)^{n}\) is \((\sigma,\sigma)\)-inner. Here, \(J\) is a certain ideal. The paper is devoted to study some hereditary property of this notion and also the extension of classical results in the realm of module amenability. We mention that the semigroup algebra \(\ell^{1}(S)\) has a module structure over \(\ell^{1}(E)\), where \(E\) is the set of all idempotents of \(S\). As an application of their results, the authors show that for the inverse semigroup \(S\), \(\ell^{1}(S)\) is \((\sigma)\)-\((2n+1)\)-weakly module amenable. For the cyclic semigroup \(S\), they show that \(\ell^{1}(S)\) is \((\sigma)\)-\(n\)-weakly module amenable provided that \(\sigma^{2}=\operatorname{id}_{\ell^{1}(S)}\). At the end, some results about \((\sigma)\)-\(n\)-weakly module amenability of \(C^{*}(S)\) (\(C^{*}\)-algebra enveloping \(\ell^{1}(S)\)) are given.
scientific article; zbMATH DE number 6537647
scientific article