  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.
