Proceedings of the American Mathematical Society

On the classes of $\mathcal{L}^\lambda$ , quasi- $\mathcal{L}^E$ and $\mathcal{L}^{\lambda,g}$ spaces

Author(s): María J. Rivera
Journal: Proc. Amer. Math. Soc. 133 (2005), 2035-2044.
MSC (2000): Primary 46M05, 46A32
Posted: January 21, 2005
Retrieve article in: PDF DVI PostScript

Abstract: The two better-known ways of understanding the notion of local unconditional structure allow us to define successive extensions of the well-known class of the $\mathcal{L}^p$ spaces of Lindenstrauss and Pelczynski. This paper also studies stability properties of these classes under ultrapowers, biduals and complemented subspaces.

1.: S.J. Bernau, E. Lacey, A local characterization of Banach lattices with order continuous norm, Studia Math. LVIII (1976), 101-128. MR 0625322 (58:30020)
2.: E. Dubinsky, A. Pelczynski, H.P. Rosenthal, On Banach spaces for which $\Pi_2(\mathcal{L}_\infty, X)=B(\mathcal{L}_\infty, X).$ Studia Math. t. XLIV (1972), 617-648. MR 0365097 (51:1350)
3.: T. Figiel, W.B. Johnson, L. Tzafriri, On Banach lattices and spaces having local unconditional structure with applications to Lorentz function spaces. J. Approximation Theory 13 (1995), 395-412. MR 0367624 (51:3866)
4.: Y. Gordon, D.R. Lewis, Absolutely summing operators and local unconditional structures. Acta Math. 133 (1974), 27-48. MR 0410341 (53:14091)
5.: R. Haydon, M. Levy, Y. Raynaud, Randomly normed spaces. Hermann, 1991. MR 1760188 (2001e:46017)
6.: S. Heinrich, Ultraproducts in Banach space theory. J. reine angew. Math. 313 (1980), 72-104. MR 0552464 (82b:46013)
7.: S. Heinrich, Finite representability and super-ideals of operators. Dissertationes Mathematicae, 172, 1980. MR 0575755 (81g:47047)
8.: W.D. Johnson, On finite dimensional subspaces of Banach spaces with local unconditional structure. Studia Math. 51 (1974), 225-240. MR 0358306 (50:10772)
9.: P.W. Jones, BMO and the Banach space approximation problem. Amer. J. Math. 107 (1985), 853-893. MR 0796906 (86m:46017)
10.: H.E. Lacey, Local unconditional structure in Banach spaces, Lecture Notes in Math. v. 604 (1977), 44-56. MR 0454596 (56:12845)
11.: H.E. Lacey, The isometric theory of Classical Banach spaces, Springer Verlag. Berlin, Heidelberg, New York, (1974). MR 0493279 (58:12308)
12.: J. Lindenstrauss, A. Pelczynski, Absolutely summing operators in $\mathcal{L}_p$ spaces and their applications. Studia Math. 29 (1968), 275-326. MR 0231188 (37:6743)
13.: J. Lindenstrauss, H.P. Rosenthal, The $\mathcal{L}_p$ spaces. Israel J. Math. 7 (1969), 325-349. MR 0270119 (42:5012)
14.: J. Lindenstrauss, L. Tzafriri, The uniform approximation property in Orlicz spaces. Israel J. Math. 23 (1976), 142-155. MR 0399806 (53:3648)
15.: J.A. López Molina, M.J. Rivera, Operator ideals and tensor norms defined by a sequence space. To appear in Bulletin of the Australian Math. Soc.
16.: A. Pelczynski, H.P. Rosenthal, Localization techniques in spaces. Studia Math. 52 (1975), 263-289. MR 0361729 (50:14174)
17.: A. Pietsch Operator Ideals. North Holland Math. Library. Amsterdam, New York. 1980. MR 0582655 (81j:47001)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46M05, 46A32

María J. Rivera
Affiliation: Departamento de Matemática Aplicada, E.T.S.I. Agrónomos, Universidad Politécnica de Valencia, Camino Vera s/n, E-46022 Valencia, Spain
Email: mjrivera@mat.upv.es

DOI: 10.1090/S0002-9939-05-07761-0
PII: S 0002-9939(05)07761-0
Keywords: Local theory, ultraproducts
Received by editor(s): September 30, 2003
Received by editor(s) in revised form: March 2, 2004
Posted: January 21, 2005
Additional Notes: This research was supported in part by MCYT DGI Project BFM 2001-2670.
Communicated by: N. Tomczak-Jaegermann
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS