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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.
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