On Bilinear Forms from the Point of View of Generalized Effect Algebras

TL;DR

This paper investigates positive bilinear forms on Hilbert spaces, exploring their structure as generalized effect algebras and analyzing their completeness properties in the context of operator theory.

Contribution

It characterizes when families of bilinear forms form generalized effect algebras and examines their monotone completeness properties.

Findings

Certain families of bilinear forms can be described as generalized effect algebras.

Some families are monotone downwards $\sigma$-complete, others are not.

The paper provides conditions for these algebraic structures to be complete.

Abstract

We study positive bilinear forms on a Hilbert space which are neither not necessarily bounded nor induced by some positive operator. We show when different families of bilinear forms can be described as a generalized effect algebra. In addition, we present families which are or are not monotone downwards (Dedekind upwards) $\sigma$-complete generalized effect algebras.