On Stein extensions of real symmetric spaces.
Consider the four pairs of groups , , and , where , are locally compact second countable abelian groups, is a dense subgroup of with inclusion map from to continuous; is a closed subgroup of ; , are the duals of and respectively, and is the annihilator of in . Let the first co-ordinate of each pair act on the second by translation. We connect, by a commutative diagram, the systems of imprimitivity which arise in a natural fashion on each pair, starting with a system...
We consider the unitary group U of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever U acts by isometries on a metric space, every orbit is bounded. Equivalently, U is not the union of a countable chain of proper subgroups, and whenever E ⊆ U generates U, it does so by words of a fixed finite length.