Revistes Catalanes amb Accés Obert (RACO)

On the algebraic structure of the unitary group

Éric Ricard, Christian Rosendal


We consider the unitary group $\mathbb{U}$ of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever $\mathbb{U}$ acts by isometries on a metric space, every orbit is bounded. Equivalently, $\mathbb{U}$ is not the union of a countable chain of proper subgroups, and whenever $\mathbb{E}\subseteq \mathbb{U}$ generates $\mathbb{U}$, it does so by words of a fixed finite length.

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