A local global principle for regular operators in Hilbert \(C^*\)-modules (Q425733)

Unbounded operators on Hilbert modules play an important role for Kasparov theory. In this context, it is often difficult to check the technical condition of regularity. This article provides efficient criteria for this purpose. The main idea is that a Hilbert \(B\)-module \(E\) and a representation of \(B\) on a Hilbert space \(H\) produce a Hilbert space \(E\otimes_B H\) in a natural way. An unbounded operator on \(E\) induces unbounded operators on \(E\otimes_B H\), called localizations. It is shown that an unbounded operator on \(E\) is self-adjoint and regular if and only if all its localizations are self-adjoint.NEWLINENEWLINEThe proof of this criterion is based on a separation theorem for Hilbert submodules: if \(F\) is a proper Hilbert submodule of \(E\), then there is a Hilbert space representation such that \(F\otimes_B H\) is a proper subspace of \(E\otimes_B H\).NEWLINENEWLINEAn application of the criterion is a generalization of Wüst's Theorem to Hilbert module operators: Let \(T\) be a self-adjoint regular operator and let \(V\) be a symmetric operator with larger domain. If \(V^*V\leq T^*T+b\) for some scalar \(b\), then \(T+V\) is essentially self-adjoint and regular.NEWLINENEWLINEInstead of Hilbert space representations, we may use states above, which correspond to cyclic representations by the GNS-construction. The authors show that in the commutative case, it is enough to work with pure states (or irreducible representations). They formulate some conjectures about which of their results remain true with pure states instead of states for general \(C^*\)-algebras. In addition, they examine some examples of irregular operators to illustrate the difficulties that can occur.