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On systems of imprimitivity on locally compact abelian groups with dense actions

J. Mathew, M. G. Nadkarni (1978)

Annales de l'institut Fourier

Consider the four pairs of groups ( Γ , R ) , ( Γ / Γ 0 , R / Γ 0 ) , ( K S , P ) and ( S , B ) , where Γ , R are locally compact second countable abelian groups, Γ is a dense subgroup of R with inclusion map from Γ to R continuous; Γ 0 Γ R is a closed subgroup of R ; S , B are the duals of R and Γ respectively, and K is the annihilator of Γ 0 in B . 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...

On the algebraic structure of the unitary group.

Éric Ricard, Christian Rosendal (2007)

Collectanea Mathematica

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.

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