Hendrik Grundling
Group Algebras for Groups which are not Locally Compact.
(107K, Plain TEX)

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.