On selfadjoint extensions of semigroups of partial isometries

Abstract

Let 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 T generated by S consists of partial isometries as well. Amongst other things, we show that this is the case when the set Q(S) of final projections of elements of S generates an abelian von Neumann algebra of uniform finite multiplicity.