On Extreme Extension of Positive Operators

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/s3201-6067-0570-n On Extreme Extension of Positive Operators Kusraev, A. G. Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 2. Abstract: Given vector lattices \(E\), \(F\) and a positive operator \(S\) from a majorzing subspace \(D\) of \(E\) to \(F\), denote by \(\mathcal{E}(S)\) the collection of all positive extensions of \(S\) to all of \(E\). This note aims to describe the collection of extreme points of the convex set \(\mathcal{E}(T\circ S)\). It is proved, in particular, that \(\mathcal{E}(T\circ S)\) and \(T\circ\mathcal{E}(S)\) coincide and every extreme point of \(\mathcal{E}(T\circ S)\) is an extreme point of \(T\circ\mathcal{E}(S)\), whenever \(T:F\to G\) is a Maharam operator between Dedekind complete vector lattices. The proofs of the main results are based on the three ingredients: a characterization of extreme points of subdifferentials, abstract disintegration in Kantorovich spaces, and an intrinsic characterization of subdifferentials. Keywords: vector lattice, positive operator, extreme extension, subdifferential, Maharam operator Language: English Download the full text For citation: Kusraev, A. G. On Extreme Extension of Positive Operators, Vladikavkaz Math. J., 2024, vol. 26, no. 2, pp. 47-53. DOI 10.46698/s3201-6067-0570-n + References 1. Aliprantis, C. D. and Burkinshaw, O. Positive Operators, Dordrecht, Springer, 2006. 2. Lipecki, Z., Plachky, D. and Thomsen, W. Extension of Positive Operators and Extreme Points. I, Colloquium Mathematicum, 1979, vol. 42, pp. 279-284. DOI: 10.4064/cm-42-1-279-284. 3. Kutateladze, S. S. Extreme Points of Subdifferentials, Doklady Akademii Nauk SSSR, 1978, vol. 242, no. 5, pp. 1001-1003 (in Russian). 4. Kutateladze, S. S. The Krein-Mil'man Theorem and its Inverse, Siberian Mathematical Journal, 1980, vol. 21, no. 1, pp. 97-103. DOI: 10.1007/BF00970127. 5. Kusraev, A. G. and Kutateladze, S. S. Analysis of Subdifferentials via Boolean-Valued Models, Doklady Akademii Nauk SSSR, 1982, vol. 265, no. 5, pp. 1061-1064 (in Russian). 6. Kusraev, A. G. General Desintegration Formulas, Doklady Akademii Nauk SSSR, 1982, vol. 265, no. 6, pp. 1312-1316. 7. Lipecki, Z. Compactness and Extreme Points of the Set of Quasi-Measure Extensions of a Quasi-Measure, Dissertationes Mathematicae, 2013, vol. 493, pp. 1-59. DOI: 10.4064/dm493-0-1. 8. Holmes, R. B. Geometric Functional Analysis and Its Applications, Springer-Verlag, Berlin etc., 1975. 9. Kusraev, A. G. and Kutateladze, S. S. Subdifferentials: Theory and Applications, Dordrecht, Kluwer Academic Publishers, 1995. 10. Lipecki, Z. Extensions of Positive Operators and Extreme Points. III, Colloquium Mathematicum, 1982, vol. 46, pp. 263-268. DOI: 10.4064/cm-46-2-263-268 The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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