- 93-329 Grundling H.
- A Group Algebra for Inductive Limit Groups.
Continuity Problems of the Canonical Commutation Relations.
Dec 17, 93
(auto. generated ps),
of related papers
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.