A Hahn-Jordan decomposition and Riesz-Frechet representation theorem in Riesz spaces

1AbstractWe give a Hahn-Jordan decomposition in Riesz spaces which generalizes that of B.A. Watson, An Andô-Douglas type theorem in Riesz spaces with a conditional expectation, Positivity, 13 (2009), 543–558 and a Riesz-Frechet representation theorem for the T-strong dual, where T 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 Andô-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.Mathematics Subject Classification (2020): 47B6547B6046C5046E4046A4046A2046B40Key words: Riesz spacesAndô-Douglas theoremRadon-Nikodým theoremHahn-Jordan decompositionRiesz-Frechet representationstrong dualgeneralized inner products