Conditional Lp-spaces and the duality of modules over f-algebras

Abstract

Motivated by dynamic asset pricing, we extend the dual pairs' theory of Dieudonné (1942) and Mackey (1945) to pairs of modules over a Dedekind complete f-algebra with multiplicative unit. The main tools are:•a Hahn–Banach Theorem for modules of this kind;•a topology on the f-algebra that has the special feature of coinciding with the norm topology when the algebra is a Banach algebra and with the strong order topology of Filipovic, Kupper, and Vogelpoth (2009), when the algebra of all random variables on a probability space (Ω,G,P) is considered. As a leading example, we study in some detail the duality of conditional Lp-spaces.