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Derivations into various duals of Lau product of Banach algebras. (English) Zbl 1399.46064
Summary: For two Banach algebras $$A$$ and $$B$$, an interesting product $$A\times_{\vartheta}B$$, called the $$\vartheta$$-Lau product, was recently introduced and studied for some non-zero multiplicative linear functional $$\vartheta$$ on $$B$$. In this paper, by discussing general necessary and sufficient conditions for $$n$$-weak amenability of $$A\times_{\vartheta}B$$, we extend some results on the $$n$$-weak amenability of the unitization $$A^{\natural}$$ of $$A$$, to the $$\vartheta$$-Lau product $$A\times_{\vartheta}B$$. In particular, we improve several known results on $$n$$-weak amenability of $$A\times_{\vartheta}B$$ and answer some questions on this topic.
##### MSC:
 46H05 General theory of topological algebras 46H20 Structure, classification of topological algebras 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 47B47 Commutators, derivations, elementary operators, etc. 43A20 $$L^1$$-algebras on groups, semigroups, etc.
##### Keywords:
Banach algebra; derivation; $$n$$-weak amenability
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