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.