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

Abstract

Given an inductive limit group $$G = \underrightarrow {\lim }G_\beta ,\beta \in \Gamma$$ where each $$G_\beta$$ is locally compact, and a continuous two-cocycle $$\rho \in Z^2 (G,T)$$ , we construct a C*-algebra $$ L $$ group algebra $$ C_\rho ^* (G_d )$$ is imbedded in its multiplier algebra $$ M(L) $$ , and the representations of $$ L $$ are identified with the strong operator continuous $$ \rho - {\text{representation}} $$ of G. If any of these representations are faithful, the above imbedding is faithful. When G is locally compact, $$ L $$ is precisely $$ C_\rho ^* (G) $$ , 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 systems.