Representations of *-algebras by unbounded operators: C*-hulls, local-global principle, and induction

TL;DR

This paper introduces a framework for representing *-algebras with unbounded operators using C*-hulls, establishing a local-global principle and an induction theorem to connect representations on Hilbert modules and spaces.

Contribution

It defines a notion of integrability for *-algebra representations, proves a local-global principle, and develops an induction theorem for constructing C*-hulls in graded *-algebras.

Findings

Established a local-global principle for integrable representations.

Constructed C*-hulls for classes of integrable representations.

Connected representations on Hilbert modules with those on Hilbert spaces.

Abstract

We define a C*-hull for a *-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*-hull for a certain class of integrable representations of a graded *-algebra, given a C*-hull for its unit fibre.