Unbounded Induced Representations of *-Algebras

Abstract

Induced representations of $\ast$-algebras by unbounded operators in Hilbert space are investigated. Conditional expectations of a $\ast$-algebra $\cA$ onto a unital $\ast$-subalgebra $\cB$ are introduced and used to define inner products on the corresponding induced modules. The main part of the paper is concerned with group graded $\ast$-algebras $\cA=\oplus_{g\in G}\cA_g$ for which the *-subalgebra $\cB:=\cA_e$ is commutative. Then the canonical projection $p:\cA\to\cB$ is a conditional expectation and there is a partial action of the group $G$ on the set $\cBp$ of all characters of $\cB$ which are nonnegative on the cone $\sum\cA^2\cap\cB.$ The complete Mackey theory is developed for $\ast$-representations of $\cA$ which are induced from characters of $\cBp.$ Systems of imprimitivity are defined and two versions of the imprimitivity theorem are proved in this context. A concept is well-behaved $\ast$-representations of such $\ast$-algebras $\cA$ 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 $\ast$-algebras generated by dynamical systems our theory is carried out in great detail.