A Group Algebra for Inductive Limit Groups. Continuity Problems of the Canonical Commutation Relations

Given an inductive limit group where each is locally compact, and a continuous two-cocycle , we construct a C^*-algebra group algebra is imbedded in its multiplier algebra , and the representations of are identified with the strong operator continuous of G. If any of these representations are faithful, the above imbedding is faithful. When G is locally compact, is precisely , the twisted group algebra of G, and for these reasons we regard in the general case as a twisted group algebra for G. Applying this construction to the CCR-algebra over an infinite dimensional symplectic space (S,\,B),we realise the regular representations as the representation space of the C^*-algebra , and show that pointwise continuous symplectic group actions on (S,\, B) produce pointwise continuous actions on , though not on the CCR-algebra. We also develop the theory to accommodate and classify 'partially regular' representations, i.e. representations which are strong operator continuous on some subgroup H of G (of suitable type) but not necessarily on G, given that such representations occur in constrained quantum systems.