A note on Hilbert \(C^*\)-modules associated with a foliation (Q1076942)

Let (M,\({\mathfrak F})\) be a \(C^{\infty,0}\) \((C^{\infty}\) along leaves and \(C^ 0\) along transversal direction) foliation \(C^*\)-algebra with its holonomy group G Hausdorff. The author studies some properties of the Hilbert \(C^*\)-module \(E^{W_ 2}_{W_ 1}\) corresponding to two transversal submanifolds \(W_ 1\) and \(W_ 2\) of M and proves that \({\mathcal K}(E_ T^{W_ 1},E_ T^{W_ 2})\cong E^{W_ 2}_{W_ 1}\), where \({\mathcal K}(E_ T^{W_ 1},E_ T^{W_ 2})\) denotes the set of compact operators from \(E_ 1\) and \(E_ 2\). This reduces to a result of Hilsum and Skandali when \(W_ 1=W_ 2\).