On the Banach-Mazur distance between continuous function spaces with scattered boundaries

Jakub Rondoš

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 2, page 367-393
  • ISSN: 0011-4642

Abstract

top
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 𝒞 ( K , E ) spaces.

How to cite

top

Rondoš, Jakub. "On the Banach-Mazur distance between continuous function spaces with scattered boundaries." Czechoslovak Mathematical Journal 73.2 (2023): 367-393. <http://eudml.org/doc/299409>.

@article{Rondoš2023,
abstract = {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.},
author = {Rondoš, Jakub},
journal = {Czechoslovak Mathematical Journal},
keywords = {function space; vector-valued Amir-Cambern theorem; scattered space; Banach-Mazur distance; closed boundary},
language = {eng},
number = {2},
pages = {367-393},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Banach-Mazur distance between continuous function spaces with scattered boundaries},
url = {http://eudml.org/doc/299409},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Rondoš, Jakub
TI - On the Banach-Mazur distance between continuous function spaces with scattered boundaries
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 2
SP - 367
EP - 393
AB - 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.
LA - eng
KW - function space; vector-valued Amir-Cambern theorem; scattered space; Banach-Mazur distance; closed boundary
UR - http://eudml.org/doc/299409
ER -

References

top
  1. Amir, D., 10.1007/BF03008398, Isr. J. Math. 3 (1965), 205-210. (1965) Zbl0141.31301MR0200708DOI10.1007/BF03008398
  2. Cambern, M., 10.1090/S0002-9939-1966-0196471-9, Proc. Am. Math. Soc. 17 (1966), 396-400 9999DOI99999 10.1090/S0002-9939-1966-0196471-9 . (1966) Zbl0156.36902MR0196471DOI10.1090/S0002-9939-1966-0196471-9
  3. Candido, L., 10.4064/sm7857-3-2016, Stud. Math. 232 (2016), 1-6. (2016) Zbl1382.46024MR3493285DOI10.4064/sm7857-3-2016
  4. Candido, L., On the distortion of a linear embedding of C ( K ) into a C 0 ( Γ , X ) space, J. Math. Anal. Appl. 459 (2018), 1201-1207 9999DOI99999 10.1016/j.jmaa.2017.11.039 . (2018) Zbl1383.46020MR3732581
  5. Candido, L., Galego, E. M., How far is C 0 ( Γ , X ) with Γ discrete from C 0 ( K , X ) spaces?, Fundam. Math. 218 (2012), 151-163 9999DOI99999 10.4064/fm218-2-3 . (2012) Zbl1258.46002MR2957688
  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 9999DOI99999 10.4064/fm220-1-5 . (2013) Zbl1271.46005MR3011771
  7. Candido, L., Galego, E. M., How does the distortion of linear embedding of C 0 ( K ) into C 0 ( Γ , X ) spaces depend on the height of K ?, J. Math. Anal. Appl. 402 (2013), 185-190 9999DOI99999 10.1016/j.jmaa.2013.01.017 . (2013) Zbl1271.46006MR3023248
  8. Candido, L., Galego, E. M., How far is C ( ω ) from the other C ( K ) spaces?, Stud. Math. 217 (2013), 123-138 9999DOI99999 10.4064/sm217-2-2 . (2013) Zbl1288.46013MR3117334
  9. Cengiz, B., On topological isomorphisms of C 0 ( X ) and the cardinal number of X , Proc. Am. Math. Soc. 72 (1978), 105-108 9999DOI99999 10.1090/S0002-9939-1978-0493291-0 . (1978) Zbl0397.46022MR0493291
  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 9999DOI99999 10.1155/2019/7146073 . (2019) Zbl1487.46025MR4047607
  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 9999MR99999 1352220 . (1995) Zbl0888.46029MR1352220
  12. Cohen, H. B., A bound-two isomorphism between C ( X ) Banach spaces, Proc. Am. Math. Soc. 50 (1975), 215-217 9999DOI99999 10.1090/S0002-9939-1975-0380379-5 . (1975) Zbl0317.46025MR0380379
  13. Dostál, P., Spurný, J., The minimum principle for affine functions and isomorphisms of continuous affine function spaces, Arch. Math. 114 (2020), 61-70 9999DOI99999 10.1007/s00013-019-01371-0 . (2020) Zbl1440.46003MR4049228
  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),9999DOI99999 10.1201/9781420026153 . (2003) Zbl1011.46001MR1957004
  15. Galego, E. M., Rincón-Villamizar, M. A., Weak forms of Banach-Stone theorem for C 0 ( K , X ) spaces via the α th derivatives of K , Bull. Sci. Math. 139 (2015), 880-891 9999DOI99999 10.1016/j.bulsci.2015.04.002 . (2015) Zbl1335.46005MR3429497
  16. Gordon, Y., On the distance coefficient between isomorphic function spaces, Isr. J. Math. 8 (1970), 391-397 9999DOI99999 10.1007/BF02798685 . (1970) Zbl0205.12401MR0270128
  17. Hess, H. U., On a theorem of Cambern, Proc. Am. Math. Soc. 71 (1978), 204-206 9999DOI99999 10.1090/S0002-9939-1978-0500490-8 . (1978) Zbl0394.46010MR0500490
  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 9999DOI99999 10.1090/S0002-9939-2010-10534-8 . (2011) Zbl1225.46005MR2745661
  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),9999DOI99999 10.1515/9783110203219 . (2010) Zbl1216.46003MR2589994
  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),9999MR99999 1885114 . (2001) Zbl1005.46004MR1885114
  21. Rondoš, J., Spurný, J., Isomorphisms of subspaces of vector-valued continuous functions, Acta Math. Hung. 164 (2021), 200-231 9999DOI99999 10.1007/s10474-020-01107-5 . (2021) Zbl1488.46072MR4264226
  22. Rondoš, J., Spurný, J., Small-bound isomorphisms of function spaces, J. Aust. Math. Soc. 111 (2021), 412-429 9999DOI99999 10.1017/S1446788720000129 . (2021) Zbl1493.46016MR4337946
  23. Semadeni, Z., Banach Spaces of Continuous Functions. Vol. 1, Monografie matematyczne 55. PWN - Polish Scientific Publishers, Warszawa (1971). (1971) Zbl0225.46030MR0296671

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.