Representations of *-Algebras by Unbounded Operators: C*-Hulls, Local--Global Principle, and Induction

We define a \Cstar\nb-hull for a \Star{}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 \Cstar\nb-hull for a certain class of integrable representations of a graded \Star{}algebra, given a \Cstar\nb-hull for its unit fibre.

2010 Mathematics Subject Classification: Primary 47L60; Secondary 46L55

Keywords and Phrases: unbounded operator; regular Hilbert module operator; integrable representation; induction of representations; graded \Star{}algebra; Fell bundle; \Cstar\nb-algebra generated by unbounded operators; \Cstar\nb-envelope; \Cstar\nb-hull; host algebra; Weyl algebra; canonical commutation relations; Local--Global Principle; Rieffel deformation

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