2008/06/15 by Yu. Savchuk, Savchuk, Yu., Konrad Schmuedgen +1 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.0806.2428
Induced representations of ∗-algebras by unbounded operators in Hilbert space are investigated. Conditional expectations of a ∗-algebra \cA onto a unital ∗-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 ∗-algebras \cA=⊕g∈ G\cAg for which the *-subalgebra \cB:=\cAe is commutative. Then the canonical projection p:\cA→\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 ∑\cA2∩\cB. The complete Mackey theory is developed for ∗-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 ∗-representations of such ∗-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 ∗-algebras generated by dynamical systems our theory is carried out in great detail.