A~triple of~infinite iterates of~the~functor of~positively homogeneous functionals

G. F. Djabbarov

Matematičeskie trudy, Tome 22 (2019) no. 1, pp. 101-118.

Résumé

The present article is devoted to the study of the space $OH(X)$ of all weakly additive order-preserving normalized positively homogeneous functionals on a metric compactum $X$. We prove the uniform metrizability of the functor $OH$ by means of the Kantorovich–Rubinshteĭn metric. We also show that the functor $O{H}_{+}$ is perfectly metrizable, where $$ OH_+(X)=\Big\{\mu\in OH(X): \big\vert\mu(\varphi) \big\vert\le\mu\big(|\varphi| \big), \varphi\in C(X) \Big\}. $$ Under natural assumptions on $X$, we show that the triple $$ \big(\mathcal{F}^\omega(X),\mathcal{F}^{++}(X),\mathcal{F}^+(X) \big) $$ is homeomorphic to $(Q,s,\mathrm{rint}Q)$, where $\mathcal{F}$ is a convex seminormal semimonadic subfunctor of $O{H}_{+}$.

Bibliographie

[1] Bessaga Cz., Pelczynski A., Selected Topics in Infinite-Dimensional Topology, Monografie Matematyczne, 58, Polish Scientific Publ., Warszawa, 1975 | MR | Zbl

[2] Brown M., “Some applications of an approximation theorem for inverse limits”, Proc. Amer. Math. Soc., 11:3 (1960), 478–483 | DOI | MR | Zbl

[3] Chapman T. A., Lectures on Hilbert Cube Manifolds, Conference Series in Mathematics, 28, Amer. Math. Soc., Providence, RI, 1976 | DOI | MR | Zbl

[4] Davletov D. E., Djabbarov G. F., “Functor of semiadditive functionals”, Methods Funct. Anal. Topology, 14:4 (2008), 314–322 | MR | Zbl

[5] Djabbarov G. F., “Description of extremal points of the space of weakly additive positively-homogeneous functionals of the two-point set”, Uzbek. Mat. Zh., 2005, no. 3, 17–25 | MR

[6] Djabbarov G. F., “Categorical properties of the functor of weakly additive positively-homogeneous functionals”, Uzbek. Mat. Zh., 2006, no. 2, 20–28 | MR

[7] Djabbarov G. F., “Metrization of the space of weakly additive positively-homogeneous functionals”, Topology Appl., 221 (2017), 133–143 | DOI | MR | Zbl

[8] Fedorchuk V. V., “Triples of infinite iterates of metrizable functors”, Math. USSR, Izv., 36:2 (1991), 411–433 ; Fedorchuk V. V., “Troiki beskonechnykh iteratsii metrizuemykh funktorov”, Izv. AN SSSR. Ser. matem., 54:2 (1990), 396–417 | DOI | MR | Zbl

[9] Georganopoulos G., “Sur l'approximation des fonctions continues par des fonctions lipschitziennes”, C. R. Acad. Sci. Paris, Ser A, 264:7 (1967), 319–321 | MR | Zbl

[10] Karchevska L., Radul T., “On extension of functors”, Comment. Math. Univ. Carolin., 53:2 (2012), 269–279 | MR | Zbl

[11] Radul T., “On the functor of order-preserving functionals”, Comment. Math. Univ. Carolin., 39:3 (1998), 609–615 | MR | Zbl

[12] Sadovnichij Yu. V., “Lifting the functors $U_\tau$ and $U_R$ to the categories of bounded metric spaces and uniform spaces”, Sb. Math., 191:11 (2000), 1667–1691 | DOI | MR | Zbl

[13] Shapiro L. B., “On function extension operators and normal functors”, Mosc. Univ. Math. Bull., 47:1 (1992), 34–38 | MR | Zbl

[14] Toruńchyk H., West J., “A Hilbert space limit for the iterated hyperspace functor”, Proc. Amer. Math. Soc., 89 (1983), 329–335 | DOI | MR

[15] Valov V., “Extenders and $\kappa$-metrizable compacta”, Math. Notes, 89:3 (2011), 319–327 | DOI | MR | Zbl

[16] Zaitov A. A., “Open mapping theorem for spaces of weakly additive homogeneous functionals”, Math. Notes, 88:5–6 (2010), 655–660 | DOI | MR | Zbl

[17] Zarichnyi M. M., “Iterated superextensions”, General Topology, Moscow State Univ. Press, M., 1986, 45–59 (in Russian) | MR