Group Algebras for Groups which are not Locally Compact

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.