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)


46H40 Automatic continuity
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