Unbounded Induced Representations of ∗-Algebras

Abstract

Induced representations of *-algebras by unbounded operators in Hilbert space are investigated. Conditional expectations of a *-algebra \({{\mathcal{A}}}\) onto a unital *-subalgebra \({{\mathcal{B}}}\) are introduced and used to define inner products on the corresponding induced modules. The main part of the paper is concerned with group graded *-algebras \({{\mathcal{A}}}=\oplus_{g\in G}{{\mathcal{A}}}_g\) for which the *-subalgebra \({{\mathcal{B}}}:={{\mathcal{A}}}_e\) is commutative. Then the canonical projection \(p:{{\mathcal{A}}}\to{{\mathcal{B}}}\) is a conditional expectation and there is a partial action of the group G on the set \({{\mathcal{B}}}p\) of all characters of \({{\mathcal{B}}}\) which are nonnegative on the cone \(\sum{{\mathcal{A}}}^2{{\mathcal{A}}}p{{\mathcal{B}}}.\) The complete Mackey theory is developed for *-representations of \({{\mathcal{A}}}\) which are induced from characters of \({{\widehat{{{\mathcal{B}}}}^+}}.\) Systems of imprimitivity are defined and two versions of the Imprimitivity Theorem are proved in this context. A concept of well-behaved *-representations of such *-algebras \({{\mathcal{A}}}\) is introduced and studied. It is shown that well-behaved representations are direct sums of cyclic well-behaved representations and that induced representations of well-behaved representations are again well-behaved. The theory applies to a large variety of examples. For important examples such as the Weyl algebra, enveloping algebras of the Lie algebras su(2), su(1,1), and of the Virasoro algebra, and *-algebras generated by dynamical systems our theory is carried out in great detail.