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.


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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