A Hahn-Jordan decomposition and Riesz-Frechet representation theorem in Riesz spaces
Abstract
We give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of [{{\sc B. A. Watson}, {An And\^o-Douglas type theorem in Riesz spaces with a conditional expectation,} {\em Positivity,} {\bf 13} (2009), 543 - 558}] and a Riesz-Frechet representation theorem for the -strong dual, where is a Riesz space conditional expectation operator. The result of Watson was formulated specifically to assist in the proof of the existence of Riesz space conditional expectation operators with given range space, i.e., a result of And\^{o}-Douglas type. This was needed in the study of Markov processes and martingale theory in Riesz spaces. In the current work, our interest is a Riesz-Frechet representation theorem, for which another variant of the Hahn-Jordan decomposition is required.
Keywords
Cite
@article{arxiv.2209.00715,
title = {A Hahn-Jordan decomposition and Riesz-Frechet representation theorem in Riesz spaces},
author = {Anke Kalauch and Wen-Chi Kuo and Bruce Alastair Watson},
journal= {arXiv preprint arXiv:2209.00715},
year = {2022}
}