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

Given an inductive limit group G = lim → Gβ , β ∈ Γ where each Gβ is locally compact, and a continuous two–cocycle ρ ∈ Z(G, T) , we construct a C*–algebra L for which the twisted discrete group algebra C∗ ρ (Gd) is imbedded in its multiplier algebra M(L) , and the representations of L are identified with the strong operator continuous ρ–representations of G . If any of these representations are faithful, the above imbedding is faithful. When G is locally compact, L is precisely C∗ ρ(G) , the twisted group algebra of G , and for these reasons we regard L 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 L , and show that pointwise continuous symplectic group actions on (S, B) produce pointwise continuous actions on L , 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 sys-