Quasi-units as orthogonal projections

TL;DR

This paper characterizes quasi-units as orthogonal projections in Hilbert spaces, explores their lattice properties, and applies these findings to nonnegative sesquilinear forms, providing new insights into their extremal structure and bounds.

Contribution

It identifies quasi-units as orthogonal projections and applies this to characterize extremal forms and their bounds, connecting operator theory with form theory.

Findings

Quasi-units are orthogonal projections on a Hilbert space.

Quasi-units are extremal points of effect algebra.

Conditions for the greatest lower bound of forms are established.

Abstract

The notion of quasi-unit has been introduced by Yosida in unital Riesz spaces. Later on, a fruitful potential theoretic generalization was obtained by Arsove and Leutwiler. Due to the work of Eriksson and Leutwiler, this notion also turned out to be an effective tool by investigating the extreme structure of operator segments. This paper has multiple purposes which are interwoven, and are intended to be equally important. On the one hand, we identify quasi-units as orthogonal projections acting on an appropriate auxiliary Hilbert space. As projections form a lattice and are extremal points of the effect algebra, we conclude the same properties for quasi-units. Our second aim is to apply these results for nonnegative sesquilinear forms. Constructing an order preserving bijection between operator- and form segments, we provide a characterization of being extremal in the convexity sense, and we give a necessary and sufficient condition for the existence of the greatest lower bound of two forms. Closing the paper we revisit some statements by using the machinery developed by Hassi, Sebesty\'en, and de Snoo. It will turn out that quasi-units are exactly the closed elements with respect to the antitone Galois connection induced by parallel addition and subtraction.