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DOI: 10.46698/z3076-9333-9133-l
Vector Lattice Powers: Continuous and Measurable Vector Functions
Kusraeva, Z. A.
Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 2.
Abstract: In the study of order properties of homogeneous polynomials in vector lattices two constructions are of fundamental importance: the symmetric positive tensor product and the vector lattice power. Both associate a canonical \(n\)-homogeneous polynomial with each Archimedean vector lattice, such that any other homogeneous polynomial of an appropriate class defined on the same vector lattice is the composition of the canonical polynomial with a linear operator. With this so called "linearization" in hand, various tools of the theory of positive linear operators can be used to study homogeneous polynomials. Thus, the problem of description of the Fremlin symmetric tensor products and the vector lattice powers for special vector lattices arises. The former enables one to study a large class of order bounded homogeneous polynomials, but has a very complicated structure; the latter has a much more transparent structure, but handles a narrower class of homogeneous polynomials, namely orthogonally additive ones. The purpose of this note is to describe the power of the vector lattice of continuous or Bochner measurable vector functions with values in a Banach lattice and to apply this result to the representation of homogeneous orthogonally additive polynomials.
For citation: Kusraeva, Z. A. Vector Lattice Powers: Continuous and Measurable Vector Functions, Vladikavkaz Math. J., 2025, vol. 27, no. 2, pp. 84-92. DOI 10.46698/z3076-9333-9133-l
1. Fremlin, D. H. Tensor Products of Banach Lattices, Mathematische Annalen , 1974, vol. 211, pp. 87-106. DOI: 10.1007/BF01344164.
2. Boulabiar, K. and Buskes, G. Vector Lattice Powers: \(f\)-Algebras and Functional Calculus, Communications in Algebra, 2006, vol. 34, no. 4, pp. 1435-1442. DOI: 10.1080/00927870500454885.
3. Bu, Q. and Buskes, G. Polynomials on Banach Lattices and Positive Tensor Products, Journal of Mathematical Analysis and Applications, 2012, vol. 388, no. 2, pp. 845-862. DOI: 10.1016/j.jmaa.2011.10.001.
4. Troitsky, V. G. and Zabeti, O. Fremlin Tensor Products of Concavifications of Banach Lattices, Positivity, 2014, vol. 18, no. 1, pp. 191-200. DOI: 10.1007/s11117-013-0239-3.
5. Kusraeva, Z. A. Powers of Quasi-Banach Lattices and Orthogonally Additive Polynomials, Journal of Mathematical Analysis and Applications, 2018, vol. 458, no. 1, pp. 767-780. DOI: 10.1016/j.jmaa.2017.09.019.
6. Kusraeva, Z. A. Order Properties of Homogeneous Orthogonally Additive Polynomials, Vladikavkaz Mathematical Journal, 2021, vol. 23, no. 3, pp. 91-112 (in Russian). DOI: 10.46698/l0779-9998-4272-b.
7. Aliprantis, C. D. and Burkinshaw, O. Positive Operators New York, Academic Press, 1985.
8. Fremlin, D. H. Measure Algebras, Broad Foundations. Vol. 2, Colchester, Torres Fremlin, 2001.
9. Lin, P.-K. Kothe-Bochner Function Spaces, Springer, 2004.
10. Buskes, G., de Pagter, B. and van Rooij, A. Functional Calculus on Riesz Spaces, Indagationes Mathematicae, N.S., 1991, vol. 2, no. 4, pp. 423-436. DOI: 10.1016/0019-3577(91)90028-6.
11. Maligranda, L. Type, Cotype and Convexity Properties of Quasi-Banach Spaces, Proceedings of the International Symposium on Banach and Function Spaces, Yokohama, Yokohama Publ., 2004, pp. 83-120.
12. Lindenstrauss, J. and Tzafriri, L. Classical Banach Spaces. Vol. 2. Function Spaces, Springer-Verlag, Berlin etc, 1979, 243 p.
13. Kusraeva, Z. A. Representation of Orthogonally Additive Polynomials, Siberian Mathematical Journal, 2011, vol. 52, no. 2, pp. 248-255. DOI: 10.1134/S003744661102008X.
14. Ben Amor, F. Orthogonally Additive Homogenous Polynomials on Vector Lattices, Communications in Algebra, 2015, vol. 43, no. 3, pp. 1118-1134. DOI: 10.1080/00927872.2013.865038.
15. Benyamini, Y., Lassalle, S. and Llavona, J. G. Homogeneous Orthogonally Additive Polynomials on Banach Lattices,
Bulletin of the London Mathematical Society, 2006, vol. 38, no. 3, pp. 459-469. DOI: 10.1112/S0024609306018364.
16. Ibort, A., Linares, P. and Llavona, J. G. A Representation Theorem for Orthogonally Additive Polynomials on Riesz Spaces, Revista Matematica Complutense, 2012, vol. 25, pp. 21-30. DOI: 10.1007/s13163-010-0053-4.
17. Gutman, A. E. Banach Bundles in the Theory of Lattice-Normed Spaces. II. Measurable Banach Bundles, Siberian Advances in Mathematics, 1993, vol. 3, no. 4, pp. 8-40.
18. Gutman, A. E. Banach Bundles in the Theory of Lattice-Normed Spaces. I. Continuous Banach Bundles, Siberian Advances in Mathematics, 1993, vol. 3, no. 3, p. 1-55.
19. Kusraev, A. G. Functional Calculus and Minkowski Duality on Vector Lattices, Vladikavkaz Mathematical Journal, 2009, vol. 11, no. 2, pp. 31-42.
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