## The second transpose of a derivation.(English)Zbl 1023.46051

For normed linear spaces $$X,Y,Z$$ and a continuous bilinear map $$f: X\times Y\to Z$$ the transpose $$f^*:Z^*\times X\to Y^*$$ is defined by $$(z^*, x)\mapsto y^*$$ such that $$y^*(y):= \langle z^*,f(x,y) \rangle$$ for each $$y\in Y$$. Now let $$A$$ be a Banach algebra with product $$\pi$$. Then $$A^{**}$$ becomes a Banach algebra with product $$\pi^{***}$$. The inner derivations $$D:A\to A^*$$ on $$A$$ are the maps $$a\mapsto x^*\cdot a-a\cdot x^*$$ for fixed $$x^*\in A^*$$ (where $$\langle x^*\cdot a,b\rangle: =\langle x^*,ab\rangle$$ and $$\langle a\cdot x^*,b\rangle: =\langle x^*,ba\rangle$$ for $$b\in A)$$. The question whether $$D^{**}$$ is a derivation when $$D:A\to A^*$$ is a continuous derivation or an inner derivation is discussed. For continuous derivations a necessary and sufficient condition is $$D^{**}(A^{**}) \cdot A^{**}\subset A^*$$. $$A$$ is called weakly amenable if every continuous derivation from $$A$$ to $$A^*$$ is inner. $$A$$ and $$A^*$$ are compared with respect to this property.
Reviewer: G.Garske (Hagen)

### MSC:

 46H40 Automatic continuity

### Keywords:

Banach algebra; derivation; weakly amenable
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