In this chapter we make a few remarks about non-abelian locally compact Hausdorff groups.

For compact Hausdorff (not necessarily abelian) groups there is a duality theory due to M.G. Krein and T. Tannaka. The dual object of a compact Hausdorff group G is not another topological group, as in the abelian case, but rather the class of continuous finite-dimensional unitary representations of G (or a Krein algebra). For full details, see E. Hewitt and K.A. Ross, Abstract harmonic analysis Vol.11.

Let H be a complex vector space and T(H) the group

of all one-one linear transformations of H onto itself.

A representation of a group G is a map of G

into T(H) such that, for each x and y

in G, with the identity operator. A representation U of a topological group is said to be a continuous irreducible unitary representation if (a) H is a Hilbert space, (b) every transformation is unitary on

H, (c) for every and η in H, the function

of G into the complex numbers is continuous, and

(d) there are no proper closed subspaces of H carried into

themselves by every.

The central theorem in representation theory of topological groups is due to I.M. Gelfand and D.A. Raikov.

Theorem (Gelfand-Raikov). Every locally compact Hausdorff group G has enough continuous irreducible unitary representations to separate points.