For every monothetic unitary group without co-compact subgroups there exists an affine compactification by a semitopological semigroup such that the set of idempotents of the semigroup is not closed. This is a negative answer to a question asked by Berglund (see Problem 29 in Ruppert's list). Since invertible elements of the compactification are not locally compact, this result also solves Problem 39 in Ruppert's list. The answer of the Berglund's question was independently obtained by \textit{B.~Bordbar} and \textit{J.~Pym} [Trans. Am. Math. Soc. 352, 823-842 (2000; Zbl 0929.22003)].