A note on the Banach–Mazur distances between \(c_0\) and other \(\ell_1\)-preduals

Agnieszka Gergont

Abstract


We prove that if \(X\) is an \(\ell_{1}\)-predual isomorphic to the space \(c_{0}\) of sequences converging to zero, then for any isomorphism \(T:X\rightarrow c_{0}\) we have \(\Vert T\Vert\, \Vert T^{-1}\Vert\ge1+2r^{*}(X)\), where \(r^{*}(X)\) is the smallest radius of the closed ball of the dual space \(X^{*}\) containing  all the weak\(^{*}\) cluster points of the set of all extreme points of the closed unit ball of  \(X^*\).

Keywords


\(\ell_1\)-preduals; Banach--Mazur distance; \(c_0\) space

Full Text:

PDF

References


Alspach, D. E., Quotients of c0 are almost isometric to subspaces of c0, Proc. Amer. Math. Soc. 79 (1979), 285–288.

Alspach, D. E., A l1-predual which is not isometric to a quotient of C(alpha), arXiv:math/9204215v1 (1992).

Banach, S., Theorie des operations lineaires, Warszawa, 1932.

Cambern, M., On mappings of sequence spaces, Studia Math. 30 (1968), 73–77.

Casini, E., Miglierina, E., Piasecki, Ł, Hyperplanes in the space of convergent sequences and preduals of l1, Canad. Math. Bull. 58 (2015), 459–470.

Casini, E., Miglierina, E., Piasecki, Ł, Popescu, R., Stability constants of the weak* fixed point property in the space l1, J. Math. Anal. Appl. 452(1) (2017), 673–684.

Durier, R., Papini, P. L., Polyhedral norms in an infinite dimensional space, Rocky Mountain J. Math. 23 (1993), 863–875.

Gergont, A., Piasecki, Ł, On isomorphic embeddings of c into L1-preduals and some applications, J. Math. Anal. Appl. 492(1) (2020), 124431, 11 pp.

Gergont, A., Piasecki, Ł, Some topological and metric properties of the space of l1-predual hyperplanes in c, Colloq. Math. 168(2) (2022), 229–247.

Megginson, R. E., An Introduction to Banach Space Theory, Springer-Verlag, New York, 1998.




DOI: http://dx.doi.org/10.17951/a.2022.76.1.25-30
Date of publication: 2022-10-05 20:39:32
Date of submission: 2022-10-04 19:08:19


Statistics


Total abstract view - 696
Downloads (from 2020-06-17) - PDF - 577

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2022 Agnieszka Gergont