Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T07:05:29.391Z Has data issue: false hasContentIssue false
  • English
  • Français

ALMOST ISOMETRIC IDEALS IN BANACH SPACES

Published online by Cambridge University Press:  13 August 2013

TROND A. ABRAHAMSEN ,
VEGARD LIMA  and
OLAV NYGAARD
TROND A. ABRAHAMSEN
Affiliation:
Department of Mathematics, Agder University, Servicebox 422, 4604 Kristiansand, Norway e-mail: Trond.A.Abrahamsen@uia.no
VEGARD LIMA
Affiliation:
Ålesund University College, Postboks 1517, 6025 Ålesund, Norway e-mail: Vegard.Lima@gmail.com
OLAV NYGAARD
Affiliation:
Department of Mathematics, Agder University, Servicebox 422, 4604 Kristiansand, Norway. e-mail: Olav.Nygaard@uia.no
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A natural class of ideals, almost isometric ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observation that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces that are ideals in every superspace; we show that being an almost isometric ideal in every superspace characterizes the class of Gurariy spaces.

Keywords

46B2046B04
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 2 , May 2014 , pp. 395 - 407
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Abrahamsen, T. A., Lima, V. and Nygaard, O., Remarks on diameter $2$ properties, J. Conv. Anal. 20 (2) (2013), 439–452.Google Scholar
2.Acosta, M. D. and Guerrero, J. Becerra, Weakly open sets in the unit ball of some Banach spaces and the centralizer, J. Funct. Anal. 259 (4) (2010), 842–856. MR 2652174 (2011e:46078).Google Scholar
3.Acosta, M. D., Becerra Guerrero, J. and López-Pérez, G., Stability results of diameter two properties, J. Convex Anal. (to appear).Google Scholar
4.Albiac, F. and Kalton, N. J., Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233 (Springer, New York, 2006). MR 2192298 (2006h:46005).Google Scholar
5.Fakhoury, H., Sélections linéaires associées au théorème de Hahn-Banach, J. Funct. Anal. 11 (1972), 436–452. MR 0348457 (50#955).Google Scholar
6.Garbulinśka, J. and Kubiś, W., Remarks on Gurarii spaces, Extracta Math. 26 (2) (2011), 235–269.Google Scholar
7.Godefroy, G., Kalton, N. J. and Saphar, P. D., Unconditional ideals in Banach spaces, Studia Math. 104 (1) (1993), 13–59. MR 1208038 (94k:46024).CrossRefGoogle Scholar
8.Gurarii, V. I., Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces, Sibirsk. Mat. Ž. 7 (1966), 1002–1013. MR 0200697 (34#585).Google Scholar
9.Harmand, P. and Lima, Å., Banach spaces which are M-ideals in their biduals, Trans. Amer. Math. Soc. 283 (1) (1984), 253–264. MR 735420 (86b:46016).Google Scholar
10.Harmand, P., Werner, D. and Werner, W., M-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547 (Springer-Verlag, Berlin, Germany, 1993). MR 1238713 (94k:46022).Google Scholar
11.Heinrich, S. and Mankiewicz, P., Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (3) (1982), 225–251. MR 675426 (84h:46026).CrossRefGoogle Scholar
12.Johnson, W. B., Rosenthal, H. P. and Zippin, M., On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506. MR 0280983 (43#6702).Google Scholar
13.Kadets, V. M., Shvidkoy, R. V., Sirotkin, G. G. and Werner, D., Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2) (2000), 855–873. MR 1621757 (2000c:46023).CrossRefGoogle Scholar
14.Kalton, N. J., Locally complemented subspaces and ${\mathcal{L}}_{p}$-spaces for 0<p<1, Math. Nachr. 115 (1984), 71–97. MR 755269 (86h:46006).Google Scholar
15.Lacey, H. E., The isometric theory of classical Banach spaces, Die Grundlehren der mathematischen Wissenschaften, Band 208 (Springer-Verlag, New York, 1974). MR 0493279 (58#12308).CrossRefGoogle Scholar
16.Lima, V. and Lima, Å., Strict u-ideals in Banach spaces, Studia Math. 195 (3) (2009), 275–285. MR 2559177 (2010g:46014).CrossRefGoogle Scholar
17.Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), vol. 97 (Springer-Verlag, Berlin, Germany, 1979), Function spaces MR 540367 (81c:46001).CrossRefGoogle Scholar
18.Lusky, W., The Gurarij spaces are unique, Arch. Math. (Basel) 27 (6) (1976), 627–635. MR 0433198 (55#6177).CrossRefGoogle Scholar
19.Nygaard, O. and Werner, D., Slices in the unit ball of a uniform algebra, Arch. Math. (Basel) 76 (6) (2001), 441–444. MR 1831500 (2002e:46057).Google Scholar
20.Oja, E. and Põldvere, M., Principle of local reflexivity revisited, Proc. Amer. Math. Soc. 135 (4) (2007), 1081–1088 (electronics). MR 2262909.Google Scholar
21.Rao, T. S. S. R. K., On ideals in Banach spaces, Rocky Mountain J. Math. 31 (2) (2001), 595–609. MR 1840956 (2002d:46018).Google Scholar
22.Shvydkoy, R. V., Geometric aspects of the Daugavet property, J. Funct. Anal. 176 (2) (2000), 198–212. MR 1784413 (2001h:46019).CrossRefGoogle Scholar
23.Werner, D., The Daugavet equation for operators on function spaces, J. Funct. Anal. 143 (1) (1997), 117–128. MR 1428119 (98c:47025).CrossRefGoogle Scholar
24.Werner, D., Recent progress on the Daugavet property, Irish Math. Soc. Bull. (46) (2001), 77–97. MR 1856978 (2002i:46014).Google Scholar
25.Wojtaszczyk, P., Some remarks on the Daugavet equation, Proc. Amer. Math. Soc. 115 (4) (1992), 1047–1052. MR 1126202 (92k:47041).Google Scholar