Representations of ${}^{\ast }$-Algebras by Unbounded Operators: C${}^{\ast }$-Hulls, Local-Global Principle, and Induction

Abstract

We define a C${}^{\ast }$-hull for a ${}^{\ast }$-algebra, given a notion of integrability for its representations on Hilbert modules. We establish a local-global principle which, in many cases, characterises integrable representations on Hilbert modules through the integrable representations on Hilbert spaces. The induction theorem constructs a C${}^{\ast }$-hull for a certain class of integrable representations of a graded ${}^{\ast }$-algebra, given a C${}^{\ast }$-hull for its unit fibre.