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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.
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.
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