On selfadjoint extensions of semigroups of partial isometries
  Let $\mathcal S$ be a semigroup of partial isometries acting on a complex, infinite-dimensional, separable Hilbert space. In this paper we seek criteria which will guarantee that the selfadjoint semigroup $\mathcal T$ generated by $\mathcal S$ consists of partial isometries as well. Amongst other things, we show that this is the case when the set of final projections of elements of $\mathcal S$ generates an abelian von Neumann algebra of uniform finite multiplicity. 
Janez Bernik, Laurent W. Marcoux, Alexey I. Popov, and Heydar Radjavi
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