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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
##### Keywords:
Banach algebra; Lau product; amenability; weak amenability
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##### References:
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