Module amenability for Banach modules (Q2904096)

Let \(A\) be a Banach algebra and let us fix a Banach \(A\)-bimodule \(E\) with a bounded module map \(\Delta:E\to A\). For each Banach \(A\)-bimodule \(X\), the authors define a \(\Delta\)-derivation \(D:A\to X\) as a bounded linear map such that NEWLINE\[NEWLINE D(\Delta(ae))=aD(\Delta(e))+D(a)\Delta(e)\quad\text{and}\quad D(\Delta(ea))=D(\Delta(e))a+\Delta(e)D(a)\,, NEWLINE\]NEWLINE for each \(a\in A\) and \(e\in E\). A \(\Delta\)-derivation \(D:A\to X\) is defined to be \(\Delta\)-inner if there exists \(x_0\in X\) such that NEWLINE\[NEWLINE D(\Delta(e))=x_0\Delta(e)-\Delta(e)x_0\quad(e\in E)\,. NEWLINE\]NEWLINE They then define a bimodule \(E\) to be \(\Delta\)-amenable if for each Banach \(A\)-bimodule \(X\) every \(\Delta\)-derivation \(D:A\to X^*\) is \(\Delta\)-inner. This generalizes the well-known notion of amenable Banach algebras introduced long ago by B. E. Johnson.NEWLINENEWLINEThe authors then prove various properties of \(\Delta\)-amenability, including an equivalence characterization of it with the existence of module (approximate/virtual) diagonals (these again generalize the well-known concepts related to amenable Banach algebras).