Regularity of unbounded operators in \(C^*\)-modules.
Regular operators between Hilbert \(C^*\)-modules are studied. Sufficient conditions for the regularity of the product and of the sum of such operators are provided. It is shown that, if \(d\) is a closed densely defined operator on a \(C^*\)-module \(E\), with densely defined adjoint \(d^*\), such that \(\text{Im}\, d \subset \text{Dom}\,d\), and there exist compact operators \(a,b,k\) on \(E\) whose images are included in \(\text{Dom}\,d\) and satisfy \(da+bd=1+k\) on \(\text{Dom}\,d\), then the operators \(d\) and \(d+d^*\) are regular, and the resolvent \((d+d^*\pm i)^{-1}\) is compact. As an application, the signature operator on a Lip\-schitz manifold endowed with a proper action of a discrete group is shown to define a \(K\)-theory element on the reduced group \(C^*\)-algebra.
scientific article; zbMATH DE number 5178519