On selfadjoint extensions of semigroups of partial isometries

TL;DR

This paper investigates conditions under which the selfadjoint semigroup generated by a semigroup of partial isometries on an infinite-dimensional Hilbert space also consists of partial isometries, focusing on the structure of their projections.

Contribution

It provides criteria ensuring the selfadjoint extension of a semigroup of partial isometries remains within partial isometries, especially when projections generate an abelian von Neumann algebra.

Findings

Selfadjoint semigroup of partial isometries can be characterized by the structure of their projections.

The set of final projections generating an abelian von Neumann algebra of finite multiplicity guarantees the property.

Conditions identified extend understanding of the structure of semigroups of partial isometries.

Abstract

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.