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The super order dual of an ordered vector space and the Riesz-Kantorovich formula


Authors: Charalambos D. Aliprantis and Rabee Tourky
Journal: Trans. Amer. Math. Soc. 354 (2002), 2055-2077
MSC (2000): Primary 46A40, 46E99, 47B60; Secondary 91B50
DOI: https://doi.org/10.1090/S0002-9947-01-02925-7
Published electronically: December 27, 2001
MathSciNet review: 1881030
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Abstract | References | Similar Articles | Additional Information

Abstract: A classical theorem of F. Riesz and L. V. Kantorovich asserts that if $L$ is a vector lattice and $f$ and $g$are order bounded linear functionals on $L$, then their supremum (least upper bound) $f\lor g$ exists in $L^\sim$ and for each $x\in L_+$ it satisfies the so-called Riesz-Kantorovich formula:

\begin{displaymath}\bigl[f\lor g\bigr](x)=\sup\bigl\{f(y)+g(z)\colon y,z\in L_+ \,\hbox{and} \, y+z=x\bigr\}\,. \end{displaymath}

Related to the Riesz-Kantorovich formula is the following long-standing problem: If the supremum of two order bounded linear functionals $f$ and $g$ on an ordered vector space exists, does it then satisfy the Riesz-Kantorovich formula?

In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz-Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the above-mentioned problem and to the properties of the Riesz-Kantorovich formula.


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Additional Information

Charalambos D. Aliprantis
Affiliation: Department of Economics and Department of Mathematics, Purdue University, West Lafayette, Indiana 47907–1310
Email: aliprantis@mgmt.purdue.edu

Rabee Tourky
Affiliation: Department of Economics, University of Melbourne, Parkville, Victoria 3052, Australia
Email: rtourky@unimelb.edu.au

DOI: https://doi.org/10.1090/S0002-9947-01-02925-7
Keywords: Ordered vector space, super order dual, Riesz--Kantorovich formula, decentralizing prices
Received by editor(s): April 20, 2000
Received by editor(s) in revised form: August 16, 2001
Published electronically: December 27, 2001
Additional Notes: The research of C. D. Aliprantis is supported by NSF Grant EIA-007506, and the research of R. Tourky is funded by Australian Research Council Grant A00103450.
Article copyright: © Copyright 2001 American Mathematical Society

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