Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
function space; vector-valued Amir-Cambern theorem; scattered space; Banach-Mazur distance; closed boundary
function space; vector-valued Amir-Cambern theorem; scattered space; Banach-Mazur distance; closed boundary
Summary:
We study the dependence of the Banach-Mazur distance between two subspaces of vector-valued continuous functions on the scattered structure of their boundaries. In the spirit of a result of Y. Gordon (1970), we show that the constant $2$ appearing in the Amir-Cambern theorem may be replaced by $3$ for some class of subspaces. We achieve this by showing that the Banach-Mazur distance of two function spaces is at least 3, if the height of the set of weak peak points of one of the spaces differs from the height of a closed boundary of the second space. Next we show that this estimate can be improved if the considered heights are finite and significantly different. As a corollary, we obtain new results even for the case of $\mathcal C(K, E)$ spaces.
References:
[1] Amir, D.: On isomorphisms of continuous function spaces. Isr. J. Math. 3 (1965), 205-210. DOI 10.1007/BF03008398 | MR 0200708 | Zbl 0141.31301
[2] Cambern, M.: A generalized Banach-Stone theorem. Proc. Am. Math. Soc. 17 (1966), 396-400 \99999DOI99999 10.1090/S0002-9939-1966-0196471-9 . DOI 10.1090/S0002-9939-1966-0196471-9 | MR 0196471 | Zbl 0156.36902
[3] Candido, L.: On embeddings of $C_0(K)$ spaces into $C_0(L,X)$ spaces. Stud. Math. 232 (2016), 1-6. DOI 10.4064/sm7857-3-2016 | MR 3493285 | Zbl 1382.46024
[4] Candido, L.: On the distortion of a linear embedding of $C(K)$ into a $C_0(\Gamma,X)$ space. J. Math. Anal. Appl. 459 (2018), 1201-1207 \99999DOI99999 10.1016/j.jmaa.2017.11.039 . MR 3732581 | Zbl 1383.46020
[5] Candido, L., Galego, E. M.: How far is $C_0(\Gamma, X)$ with $\Gamma$ discrete from $C_0(K,X)$ spaces?. Fundam. Math. 218 (2012), 151-163 \99999DOI99999 10.4064/fm218-2-3 . MR 2957688 | Zbl 1258.46002
[6] Candido, L., Galego, E. M.: Embeddings of $C(K)$ spaces into $C(S,X)$ spaces with distortion strictly less than 3. Fundam. Math. 220 (2013), 83-92 \99999DOI99999 10.4064/fm220-1-5 . MR 3011771 | Zbl 1271.46005
[7] Candido, L., Galego, E. M.: How does the distortion of linear embedding of $C_0(K)$ into $C_0(\Gamma,X)$ spaces depend on the height of $K$?. J. Math. Anal. Appl. 402 (2013), 185-190 \99999DOI99999 10.1016/j.jmaa.2013.01.017 . MR 3023248 | Zbl 1271.46006
[8] Candido, L., Galego, E. M.: How far is $C(\omega)$ from the other $C(K)$ spaces?. Stud. Math. 217 (2013), 123-138 \99999DOI99999 10.4064/sm217-2-2 . MR 3117334 | Zbl 1288.46013
[9] Cengiz, B.: On topological isomorphisms of $C_{0}(X)$ and the cardinal number of $X$. Proc. Am. Math. Soc. 72 (1978), 105-108 \99999DOI99999 10.1090/S0002-9939-1978-0493291-0 . MR 0493291 | Zbl 0397.46022
[10] Cerpa-Torres, M. F., Rincón-Villamizar, M. A.: Isomorphisms from extremely regular subspaces of $C_0(K)$ into $C_0(S,X)$ spaces. Int. J. Math. Math. Sci. 2019 (2019), Article ID 7146073, 7 pages \99999DOI99999 10.1155/2019/7146073 . MR 4047607 | Zbl 1487.46025
[11] Chu, C.-H., Cohen, H. B.: Small-bound isomorphisms of function spaces. Function spaces Lecture Notes in Pure and Applied Mathematics 172. Marcel Dekker, New York (1995), 51-57 \99999MR99999 1352220 . MR 1352220 | Zbl 0888.46029
[12] Cohen, H. B.: A bound-two isomorphism between $C(X)$ Banach spaces. Proc. Am. Math. Soc. 50 (1975), 215-217 \99999DOI99999 10.1090/S0002-9939-1975-0380379-5 . MR 0380379 | Zbl 0317.46025
[13] Dostál, P., Spurný, J.: The minimum principle for affine functions and isomorphisms of continuous affine function spaces. Arch. Math. 114 (2020), 61-70 \99999DOI99999 10.1007/s00013-019-01371-0 . MR 4049228 | Zbl 1440.46003
[14] Fleming, R. J., Jamison, J. E.: Isometries on Banach Spaces: Function Spaces. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 129. Chapman & Hall/CRC, Boca Raton (2003),\99999DOI99999 10.1201/9781420026153 . MR 1957004 | Zbl 1011.46001
[15] Galego, E. M., Rincón-Villamizar, M. A.: Weak forms of Banach-Stone theorem for $C_0(K,X)$ spaces via the $\alpha$th derivatives of $K$. Bull. Sci. Math. 139 (2015), 880-891 \99999DOI99999 10.1016/j.bulsci.2015.04.002 . MR 3429497 | Zbl 1335.46005
[16] Gordon, Y.: On the distance coefficient between isomorphic function spaces. Isr. J. Math. 8 (1970), 391-397 \99999DOI99999 10.1007/BF02798685 . MR 0270128 | Zbl 0205.12401
[17] Hess, H. U.: On a theorem of Cambern. Proc. Am. Math. Soc. 71 (1978), 204-206 \99999DOI99999 10.1090/S0002-9939-1978-0500490-8 . MR 0500490 | Zbl 0394.46010
[18] Ludvík, P., Spurný, J.: Isomorphisms of spaces of continuous affine functions on compact convex sets with Lindelöf boundaries. Proc. Am. Math. Soc. 139 (2011), 1099-1104 \99999DOI99999 10.1090/S0002-9939-2010-10534-8 . MR 2745661 | Zbl 1225.46005
[19] Lukeš, J., Malý, J., Netuka, I., Spurný, J.: Integral Representation Theory: Applications to Convexity, Banach Spaces and Potential Theory. de Gruyter Studies in Mathematics 35. Walter de Gruyter, Berlin (2010),\99999DOI99999 10.1515/9783110203219 . MR 2589994 | Zbl 1216.46003
[20] Morrison, T. J.: Functional Analysis: An Introduction to Banach Space Theory. Pure and Applied Mathematics. A Wiley Series of Texts, Monographs and Tracts. Wiley, Chichester (2001),\99999MR99999 1885114 . MR 1885114 | Zbl 1005.46004
[21] Rondoš, J., Spurný, J.: Isomorphisms of subspaces of vector-valued continuous functions. Acta Math. Hung. 164 (2021), 200-231 \99999DOI99999 10.1007/s10474-020-01107-5 . MR 4264226 | Zbl 1488.46072
[22] Rondoš, J., Spurný, J.: Small-bound isomorphisms of function spaces. J. Aust. Math. Soc. 111 (2021), 412-429 \99999DOI99999 10.1017/S1446788720000129 . MR 4337946 | Zbl 1493.46016
[23] Semadeni, Z.: Banach Spaces of Continuous Functions. Vol. 1. Monografie matematyczne 55. PWN - Polish Scientific Publishers, Warszawa (1971). MR 0296671 | Zbl 0225.46030
Search
Partner of
