  Given a group-invariant quasi-free state on the algebra of canonical
commutation relations (CCR), we show how group averaging techniques can be used
to obtain a symmetry reduced CCR algebra and reduced quasi-free state. When the
group is compact this method of symmetry reduction leads to standard results
which can be obtained using other methods. When the group is non-compact the
group averaging prescription relies upon technically favorable conditions which
we delineate. As an example, we consider symmetry reduction of the usual vacuum
state for a Klein-Gordon field on Minkowski spacetime by a non-compact subgroup
of the Poincar\'e group consisting of a 1-parameter family of boosts, a
1-parameter family of spatial translations and a set of discrete translations.
We show that the symmetry reduced CCR algebra and vacuum state correspond to
that used by each of Berger, Husain, and Pierri for the polarized Gowdy ${\bf
T}^3$ quantum gravity model.
