Abstract: 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 \(x\in L_{\Phi}(B,m)\).
Keywords: the Banach-Kantorovich space, the Orlicz function, vector-valued measure, positive isometry, normal homomorphism.
1. Kusraev, A. G. Dominated Operators, Mathematics and its Applications,
vol. 519, Dordrecht, Kluwer Academic Publishers, 2000.
2. Kusraev, A. G. Vektornaya dvoystvennost' i ee prilozheniya [Vekctor Duality
and its Applications], Novosibirsk, Nauka, 1985 (in Russian).
3. Zakirov, B. S. The Luxemburg Norm on the Orlicz-Kantorovich Lattices,
Uzbek Mathematical Journal, 2007, no. 2, pp. 32-44 (in Russian).
4. Zakirov, B. S.
The Orlicz-Kantorovich Lattices, Associated with \(L^0\)-Valued Measure,
Uzbek Mathematical Journal, 2007, no. 4, pp. 18-34 (in Russian).
5. Zakirov, B. S. An Analytical Representation of the \(L_0\)-Valued Homomorphisms
in the Orlicz-Kantorovich Modules, Siberian Advances in Mathematics, 2009, vol. 19, no. 2, pp. 128-149.
DOI: 10.3103/S1055134409020047.
6. Banach, S. Theory of Linear Operations,
North-Holland, Amsterdam-New-York-Oxford-Tokyo, 1987.
7. Lamperti, J. On the Isometries of Some Function Spaces,
Pacific Journal of Mathematics, 1958, vol. 8. pp. 459-466.
8. Fleming, R. and Jamison, J. Isometries on Banach Spaces: Function Spaces,
Monographs and Surveys in Pure and Applied Mathematics, vol. 129,
London, Chapman and Hall, 2003.
9. Lumer, G. On the Isometries of Reflexive Orlicz Spaces,
Annales de l'Institut Fourier, 1963, vol. 13, pp. 99-109.
10. Zaidenberg, M. G. On Isometric Classification Of Symmetric Spaces,
Soviet Mathematics - Doklady, 1977, vol. 18, pp. 636-639.
11. Zaidenberg, M. G.
A Representation of Isometries of Functional Spaces,
Matematicheskaya Fizika, Analiz, Geometriya [Mathematical Physics, Analysis, Geometry], 1997, vol. 4, no. 3, pp. 339-347.
12. Kalton, N. J. and Randrianantoanina, B. Surjective Isometries on Rearrangment-Invariant Spaces,
Quarterly Journal of Mathematics, 1994, vol. 45, no. 2, pp. 301-327.
DOI: 10.48550/arXiv.math/9211208.
13. Braverman, M. Sh. and Semenov, E. M. Isometries on Symmetric Spaces,
Soviet Mathematics - Doklady, 1974, vol. 15. pp. 1027-1029.
14. Braverman, M. Sh. and Semenov, E. M. Isometries on Symmetric Spaces,
Trudy Nauchno-Issledovatel'skogo Instituta Matematiki Voronezhskogo Universiteta,
1975, vol. 17, pp. 7-18 (in Russian).
15. Arazy, J. Isometries on Complex Symmetric Sequence Spaces,
Mathematische Zeitschrift, 1985, vol. 188, pp. 427-431.
DOI: 10.1007/BF01159187.
16. Aminov, B. R. and Chilin, V. I. Isometries and Hermitian Operators
on Complex Symmetric Sequence Spaces,
Siberian Advances in Mathematics, 2017, vol. 27, no. 4, pp. 239-251.
DOI: 10.3103/S1055134417040022.
17. Abramovich, Y. Isometries of Norm Latties,
Optimizatsiya, 1988, vol. 43(60), pp. 74-80 (in Russian).
18. Veksler, A. Positive Isometries of Normed Solid Function Spaces,
Proceedings of the Tashkent State University "Mathematical Analysis and Probability Theory", 7 p. 1984 (in Russian).
19. Abramovich, Y. Operators Preserving Disjointess on Rearrangement Invariant Spaces,
Pacific Journal of Mathematics, 1991, vol. 148, no. 2, pp. 201-206.
DOI: 10.2140/pjm.1991.148.201.
20. Abdullaev, R. and Chilin, V. Positive Isometries of Orlicz Spaces,
Collection of Materials of the International Conference KROMSH-2020, Simferopol, Polyprint, 2020, pp. 28-31.
21. Sukochev, F. and Veksler, A. Positive Linear Isometries in Symmetric Operator Spaces,
Integral Equations and Operator Theory, 2018, vol. 90, no. 5.
DOI: 10.1007/s00020-018-2483-1.
22. Vladimirov, D. A. Bulevy algebry [Boolean Algebras], Moscow, Nauka, 1969 (in Russian).
23. Vulikh, B. Z. Vvedenie v teoriyu poluuporyadochennykh prostranstv
[Introduction to the Theory of Partially Ordered Spaces], Moscow, Fizmatgiz, 1961 (in Russian).
24. Zakirov, B. S. and Chilin, V. I. Decomposable Measures with Values in Order Complete Vector Lattices,
Vladikavkaz Mathematical Journal, 2008, vol. 10, no. 4, pp. 31-38 (in Russian).
25. Rubshtein, B. A., Grabarnik, G. Ya., Muratov, M. A. and Pashkova, Yu. S. Foundations
of Symmetric Spaces of Measurable Functions. Lorentz, Marcinkiewicz and Orlicz Spaces,
Springer International Publishing, 2016.
26. Edgar, G. A. and Sucheston, L. Stoping Times and Directed Processes,
Encyclopedia of Mathematics and its Applications, vol. 47,
Cambridge University Press, 1992.
27. Chilin, V. I. and 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, pp. 62-73.
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