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Conditional \(L_{p}\)-spaces and the duality of modules over \(f\)-algebras. (English) Zbl 1359.46046
Summary: Motivated by dynamic asset pricing, we extend the dual pairs theory of J. Dieudonné [Ann. Sci. Éc. Norm. Supér. (3) 59, 107–139 (1942; JFM 68.0238.02)] and G. W. Mackey [Trans. Am. Math. Soc. 57, 155–207 (1945; Zbl 0061.24301)] to pairs of modules over a Dedekind complete \(f\)-algebra with multiplicative unit. The main tools are:
(1)
a Hahn-Banach Theorem for modules of this kind;
(2)
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 D. Filipović et al. [J. Funct. Anal. 256, No. 12, 3996–4029 (2009; Zbl 1180.46055)], when the algebra of all random variables on a probability space \((\operatorname{\Omega}, \mathcal{G}, P)\) is considered.
As a leading example, we study in some detail the duality of conditional \(L_p\)-spaces.

MSC:
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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