Geometry and topology of operators on Hilbert \(C^*\)-modules (Q1576012)

The paper surveys the theory of operators on Hilbert \(C^*\)-modules. The author starts with multiplier \(C^*\)-algebras and discusses various topologies on the set of operators on a Hilbert \(C^*\)-module \(M\) over a \(C^*\)-algebra \(A\) and relations between various classes of operators on \(M\) and multipliers of the \(C^*\)-algebra of compact operators on \(M\). A number of examples is given and the commutative case is discussed in detail. The main interest of the author lies in topological properties of the full linear group of the algebra of bounded/adjointable operators on the standard Hilbert \(C^*\)-module \(\ell_2(A)\). A variant of the Dixmier-Douady theorem is proved, showing that every bounded set of invertible operators is contractible in the full linear group with respect to the left strict topology. The author gives a new short proof of the Cuntz--Higson theorem on the contractibility of the unitary group of operators on \(\ell_2(A)\) with respect to the norm topology and gives some new partial results on the contractibility of bounded sets of invertible bounded operators (non necessarily adjointable) with respect to the norm topology.