- 04-118 Hendrik Grundling
- Group Algebras for Groups which are not Locally Compact.
(107K, Plain TEX)
Apr 17, 04
(auto. generated ps),
of related papers
Abstract. We generalise the definition of a group algebra so that
it makes sense for non--locally compact topological groups,
in particular, we require that the representation theory
of the group algebra is isomorphic (in the sense of Gelfand--Raikov)
to the continuous representation theory of the group,
or to some other important subset of representations.
We prove that a group algebra if it exists, is always unique
up to isomorphism. From examples, group algebras do not always
exist for non--locally compact groups, but they do exist for some.
We define a convolution on the dual of the Fourier--Stieltjes
algebra making it into a Banach *-algebra, we prove that a group algebra
if it exists, can always be embedded in this convolution algebra,
and we find sufficient conditions for a subalgebra to be a group
algebra. When the group is locally compact, we obtain a new characterisation of its group algebra which does not involve the Haar measure, nor behaviour of measures on compact sets.