Isometries of combinatorial Banach spaces

Brech, C.; Ferenczi, V.; Tcaciuc, A.

doi:10.1090/proc/15122

Isometries of combinatorial Banach spaces
HTML articles powered by AMS MathViewer

by C. Brech, V. Ferenczi and A. Tcaciuc

Proc. Amer. Math. Soc. 148 (2020), 4845-4854

DOI: https://doi.org/10.1090/proc/15122

Published electronically: July 29, 2020

PDF | Request permission

Abstract:

We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.

References

L. Antunes, K. Beanland, and H. Viet Chu, On the geometry of higher order Schreier spaces, preprint.
Spiros A. Argyros and Stevo Todorcevic, Ramsey methods in analysis, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2005. MR 2145246, DOI 10.1007/3-7643-7360-1
Steven F. Bellenot, Isometries of James space, Banach space theory (Iowa City, IA, 1987) Contemp. Math., vol. 85, Amer. Math. Soc., Providence, RI, 1989, pp. 1–18. MR 983378, DOI 10.1090/conm/085/983378
Dong-Ni Tan, Some new properties and isometries on the unit spheres of generalized James spaces $J_p$, J. Math. Anal. Appl. 393 (2012), no. 2, 457–469. MR 2921688, DOI 10.1016/j.jmaa.2012.03.024
Richard J. Fleming and James E. Jamison, Isometries on Banach spaces: function spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 129, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1957004
Richard J. Fleming and James E. Jamison, Isometries on Banach spaces. Vol. 2, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 138, Chapman & Hall/CRC, Boca Raton, FL, 2008. Vector-valued function spaces. MR 2361284
V. P. Fonf, The massiveness of the set of extreme points of the conjugate ball of a Banach space and polyhedral spaces, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 91–92 (Russian). MR 509400
W. Gowers, Must an “explicitly defined” Banach space contain $c_0$ or $l_p$?, Feb. 17, 2009, Gowers’s Weblog: Mathematics related discussions.
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
Stefan Rolewicz, Metric linear spaces, Monografie Matematyczne, Tom 56. [Mathematical Monographs, Vol. 56], PWN—Polish Scientific Publishers, Warsaw, 1972. MR 0438074
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815