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Some notes on amenability and weak amenability of Lau product of Banach algebras defined by a Banach algebra morphism. (English) Zbl 1335.46040
Let \(A\) be a Banach algebra. \(A\) is called amenable if every bounded linear derivation \[ D:A\rightarrow X^{*} \] is inner for every Banach \(A\)-bimodule \(X\), i.e., there exists \(x_{0}\) in \(X^{*}\) such that \[ D(a)=a\cdot x_{0}-x_{0}\cdot a \] for each \(a\in A\). \(A\) is called weakly amenable (\(n\)-weakly amenable) if every bounded linear derivation \(D:A\rightarrow A^{*}\) \((A^{(n)})\) is inner, respectively. Let \(A\) and \(B\) be Banach algebras. Suppose that \(T:B\rightarrow A\) is an algebra homomorphism with \(||T||\leq 1\) and \(A\) is commutative. Equip \(A\times B\) with the product \[ (a,b)(a^{\prime},b^{\prime})=(aa^{\prime}+T(b)a^{\prime}+T(b^{\prime})a,bb^{\prime})\quad (a,a^{\prime},b,b^{\prime}\in A) \] and with the norm \[ ||(a,b)||=||a||_{A}+||b||_{B}, \] then \(A\times B\) becomes a Banach algebra which is denoted by \(A\times_{T} B\). In this paper, the authors study weak amenability (\(n\)-weak amenability) of \(A\times_{T}B\). They also define a new version of \(A\times_{T}B\) via a new product \[ (a,b)(a^{\prime},b^{\prime})=(aa^{\prime}+T(b)a^{\prime}+aT(b^{\prime}),bb^{\prime})\quad (a,a^{\prime},b,b^{\prime}\in A). \] They investigate amenability, weak amenability and Arens regularity of these new Banach algebras.

MSC:
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46H05 General theory of topological algebras
46J05 General theory of commutative topological algebras
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