A GROUP-ALGEBRA FOR INDUCTIVE LIMIT GROUPS - CONTINUITY PROBLEMS OF THE CANONICAL COMMUTATION RELATIONS

Given an inductive limit group G = (lim)under right arrow G(beta), bet a is an element of Gamma where each G(beta) is locally compact, and a continuous two-cocycle rho is an element of Z(2)(G, T), we construct a C-algebra L for which the twisted discrete group algebra C*(G(d)) is imbedded in its multiplier algebra M(L), and the representations of L are identified with the strong operator continuous rho-representation s of G. If any of these representations are faithful, the above imbedd ing 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 t he general case as a twisted group algebra for G. Applying this constr uction to the CCR-algebra over an infinite dimensional symplectic spac e (S, B), we realise the regular representations as the representation space of the C-algebra C, and show that pointwise continuous symplec tic group actions on (S, B) produce pointwise continuous actions on L, though not on the CCR-algebra. We also develop the theory to accommod ate and classify 'partially regular' representations, i.e. representat ions which are strong operator continuous on some subgroup H of G (of suitable type) but not necessarily on G, given that such representatio ns occur in constrained quantum systems.