FULL REGULARITY FOR A C*-ALGEBRA OF THE CANONICAL COMMUTATION RELATIONS

The Weyl algebra-the usual C*-algebra employed to model the canonical commutation relations (CCRs), has a well-known defect, in that it has a large number of representations which are not regular and these cannot model physical fields. Here, we construct explicitly a C*-algebra which can reproduce the CCRs of a countably dimensional symplectic space ( S, B) and such that its representation set is exactly the full set of regular representations of the CCRs. This construction uses Blackadar's version of infinite tensor products of nonunital C*-algebras, and it produces a "host algebra" (i.e. a generalized group algebra, explained below) for the s-representation theory of the Abelian group S where sigma(.,.) := e(iB)(.,.)/2. As an easy application, it then follows that for every regular representation of Delta(S, B) on a separable Hilbert space, there is a direct integral decomposition of it into irreducible regular representations (a known result).