## The third dual of a Banach algebra.(English)Zbl 1174.46022

Let $$A$$ be a Banach algebra and $$A'$$, $$A''$$, $$A'''$$ the first, second and third duals of $$A$$, respectively. Let $$D:A\to A''$$ be a continuous derivation, and $$D'':A''\to A'''$$ its second transpose. Let $$A''$$ be the second dual of $$A$$ with the first Arens product $$\square$$. There are two $$(A'',\square$$)-bimodule structures on $$A'''$$ that are not always equal.
One of the problems in Banach algebra theory is the relation between amenability and weak amenability of $$A$$ and $$(A'',\square)$$. In this paper, the author determines conditions for two $$(A'',\square)$$-bimodule structures on $$A''$$ to coincide. Then, the following results are proved.
If the two $$(A'',\square)$$-bimodule structures on $$A'''$$ coincide, then $$D'':A''\to (A'')'$$ is a continuous derivation.
If $$(A'',\square)$$ is weakly amenable, and if the two $$(A'',\square)$$-bimodule structures on $$A'''$$ coincide, then $$A$$ is weakly amenable.

### MSC:

 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46H20 Structure, classification of topological algebras
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