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Consider a (fixed) unitary operator $U$ acting on the Hilbert space $\mathcal{H}$. Because the unit ball is compact in the weak topology, it is not hard to see that there exists a (smallest) compact (semi-)topological group $G$ (contained in the set of unitary operators) that contains $U$ (note: the multiplication is continuous separately, but not jointly). I am uncertain if $G$ is just the closure of the powers of $U$, but this is perhaps a reasonable first approximation. What interests me are projections (i.e. $P^2 = P$) contained in $G$

In the strong topology, a limit of isometries is an isometry, so if we were talking about strong topology, $G$ could not contain any projections at all (except identity). However, this is not the case in the weak topology, and I know for a fact that $G$ will generally contain projections. I am trying to get some intuitive feeling of the situation: get the feeling of "how" it happens that projections land in $G$. I think I can show that there are no (non-trivial) projections when $\mathcal{H}$ is spanned by eigenvectors of $U$. Could someone show me an example when projections can be found? Most appreciated would be examples when one can "see what is happening", projection can be "explicitly" given, and one can get a hold on "how many" projections there are.

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