Positive isometries of Orlicz–Kantorovich spaces

B. S. Zakirov   ; V. I. Chilin

Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 2, pp. 103-116

Résumé

Let $B$ be a complete Boolean algebra, $Q(B)$ the Stone compact of $B$, and let $C_\infty (Q(B))$ be the commutative unital algebra of all continuous functions $x: Q(B) \to [-\infty, +\infty]$, assuming possibly the values $\pm\infty$ on nowhere-dense subsets of $Q(B)$. We consider the Orlicz–Kantorovich spaces ${(L_{\Phi}(B,m), \|\cdot\|_{\Phi})\subset C_\infty (Q(B))}$ with the Luxembourg norm associated with an Orlicz function $\Phi$ and a vector-valued measure $m$, with values in the algebra of real-valued measurable functions. It is shown, that in the case when $\Phi$ satisfies the $(\Delta_2)$-condition, the norm $\|\cdot\|_{\Phi}$ is order continuous, that is, $\|x_n\|_{\Phi}\downarrow \mathbf{0}$ for every sequence $\{x_n\}\subset L_{\Phi}(B,m)$ with $x_n \downarrow \mathbf{0}$. Moreover, in this case, the norm $\|\cdot\|_{\Phi}$ is strictly monotone, that is, the conditions $|x|\lneqq |y|$, $x, y \in L_{\Phi}(B,m)$, imply $\|x\|_{\Phi} \lneqq \|y\|_{\Phi}$. In addition, for positive elements $x, y \in L_{\Phi}(B,m)$, the equality $\|x+y\|_{\Phi}=\|x-y\|_{\Phi}$ is valid if and only if $x\cdot y = 0$. Using these properties of the Luxembourg norm, we prove that for any positive linear isometry $V: L_{\Phi}(B,m) \to L_{\Phi}(B,m)$ there exists an injective normal homomorphisms $T : C_\infty (Q(B)) \to C_\infty (Q(B))$ and a positive element $y \in L_{\Phi}(B,m)$ such that $V(x ) =y\cdot T(x)$ for all $x\in L_{\Phi}(B,m)$.

Bibliographie

[1] Kusraev A. G., Dominated Operators, Mathematics and its Applications, 519, Kluwer Academic Publishers, Dordrecht, 2000 | MR | Zbl

[2] Kusraev A. G., Vekctor Duality and its Applications, Nauka, Novosibirsk, 1985 (in Russian) | MR

[3] Zakirov B. S., “The Luxemburg Norm on the Orlicz–Kantorovich Lattices”, Uzbek Mathematical Journal, 2007, no. 2, 32–44 (in Russian) | MR | Zbl

[4] Zakirov B. S., “The Orlicz–Kantorovich Lattices, Associated with $L^0$-Valued Measure”, Uzbek Mathematical Journal, 2007, no. 4, 18–34 (in Russian) | MR | Zbl

[5] Zakirov B. S., “An Analytical Representation of the $L_0$-Valued Homomorphisms in the Orlicz–Kantorovich Modules”, Siberian Advances in Mathematics, 19:2 (2009), 128–149 | DOI | MR

[6] Banach S., Theory of Linear Operations, North-Holland, Amsterdam–New-York–Oxford–Tokyo, 1987 | Zbl

[7] Lamperti J., “On the Isometries of Some Function Spaces”, Pacific Journal of Mathematics, 8 (1958), 459–466 | DOI | MR | Zbl

[8] Fleming R., Jamison J., Isometries on Banach Spaces: Function Spaces, Monographs and Surveys in Pure and Applied Mathematics, 129, Chapman and Hall, London, 2003 | MR | Zbl

[9] Lumer G., “On the Isometries of Reflexive Orlicz Spaces”, Annales de l'Institut Fourier, 13 (1963), 99–109 | DOI | MR | Zbl

[10] Zaidenberg M. G., “On Isometric Classification Of Symmetric Spaces”, Soviet Mathematics — Doklady, 18 (1977), 636–639 | MR

[11] Zaidenberg M. G., “A Representation of Isometries of Functional Spaces”, Mathematical Physics, Analysis, Geometry, 4:3 (1997), 339–347 | MR | Zbl

[12] Kalton N. J., Randrianantoanina B., “Surjective Isometries on Rearrangment-Invariant Spaces”, Quarterly Journal of Mathematics, 45:2 (1994), 301–327 | DOI | MR | Zbl

[13] Braverman M. Sh., Semenov E. M., “Isometries on Symmetric Spaces”, Soviet Mathematics — Doklady, 15 (1974), 1027–1029 | MR

[14] Braverman M. Sh., Semenov E. M., “Isometries on Symmetric Spaces”, Trudy Nauchno-Issledovatel'skogo Instituta Matematiki Voronezhskogo Universiteta, 17 (1975), 7–18 (in Russian) | MR

[15] Arazy J., “Isometries on Complex Symmetric Sequence Spaces”, Mathematische Zeitschrift, 188 (1985), 427–431 | DOI | MR

[16] Aminov B. R., Chilin V. I., “Isometries and Hermitian Operators on Complex Symmetric Sequence Spaces”, Siberian Advances in Mathematics, 27:4 (2017), 239–251 | DOI | MR

[17] Abramovich Y., “Isometries of Norm Latties”, Optimizatsiya, 43(60) (1988), 74–80 (in Russian) | MR | Zbl

[18] Veksler A., “Positive Isometries of Normed Solid Function Spaces”, Proceedings of the Tashkent State University “Mathematical Analysis and Probability Theory”, 1984, 7 pp. (in Russian)

[19] Abramovich Y., “Operators Preserving Disjointess on Rearrangement Invariant Spaces”, Pacific Journal of Mathematics, 148:2 (1991), 201–206 | DOI | MR | Zbl

[20] Abdullaev R., Chilin, V., “Positive Isometries of Orlicz Spaces”, Collection of Materials of the International Conference KROMSH-2020, Polyprint, Simferopol, 2020, 28–31

[21] Sukochev F., Veksler A., “Positive Linear Isometries in Symmetric Operator Spaces”, Integral Equations and Operator Theory, 90:5 (2018) | DOI | MR

[22] Vladimirov D. A., Boolean Algebras, Nauka, M., 1969 (in Russian) | MR

[23] Vulikh B. Z., Introduction to the Theory of Partially Ordered Spaces, Fizmatgiz, M., 1961 (in Russian) | MR | Zbl

[24] Zakirov B. S., Chilin V. I., “Decomposable Measures with Values in Order Complete Vector Lattices”, Vladikavkaz Mathematical Journal, 10:4 (2008), 31–38 (in Russian) | MR | Zbl

[25] Rubshtein B. A., Grabarnik G. Ya., Muratov M. A., Pashkova Yu. S., Foundations of Symmetric Spaces of Measurable Functions. Lorentz, Marcinkiewicz and Orlicz Spaces, Springer International Publishing, 2016 | MR | Zbl

[26] Edgar G. A., Sucheston L., Stoping Times and Directed Processes, Encyclopedia of Mathematics and its Applications, 47, Cambridge University Press, 1992 | MR

[27] Chilin V. I., Katz A. A., “A Note on Extensions of Homomorphisms of Boolean Algebras of Projections of Commutative AW*-Algebras”, Proceedings of the International Conference on Topological Algebras and Their Applications, ICTAA 2021, 62–73 | MR | Zbl