93-329 Grundling H.

A Group Algebra for Inductive Limit Groups. Continuity Problems of the Canonical Commutation Relations. (104K, TeX) Dec 17, 93

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Abstract. Given a group G which is an inductive limit of locally compact subgroups, and a continuous two-cocycle $\rho$ on it with values in the circle group, we construct a C*-algebra L for which the twisted discrete group algebra $C^*_\rho(G_d)$ is imbedded in its multiplier algebra, and the representations of L are identified with the strong operator continuous projective representations of G, (with cocycle $\rho$). 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 as a twisted group algebra for G when G is not locally compact. 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, regarding S as an Abelian group, 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.

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