The GL-l.u.st. constant and asymmetry of the Kalton-Peck twisted sum in finite dimensions

Authors:
Y. Gordon, M. Junge, M. Meyer and S. Reisner

Journal:
Proc. Amer. Math. Soc. **139** (2011), 2793-2805

MSC (2010):
Primary 46B20; Secondary 46B07

DOI:
https://doi.org/10.1090/S0002-9939-2011-10715-9

Published electronically:
February 22, 2011

MathSciNet review:
2801620

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the Kalton-Peck twisted sum of -dimensional Hilbert spaces has a GL-l.u.st. constant of order and bounded GL constant. This is the first concrete example which shows different explicit orders of growth in the GL and GL-l.u.st. constants. We also discuss the asymmetry constants of .

**[CK]**Casazza P. G. and Kalton N. J.: Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces. Israel J. Math. 95 (1996), 349-373. MR**1418300 (97k:46010)****[FJT]**Figiel T., Johnson W.B. and Tzafriri L.: On Banach lattices and spaces having local unconditional structure, with applications to Lorentz function spaces. Collection of articles dedicated to G. G. Lorentz on the occasion of his sixty-fifth birthday, IV. J. Approximation Theory 13 (1975), 395-412. MR**0367624 (51:3866)****[GG]**Garling D. J. H. and Gordon Y.: Relations between some constants associated with finite dimensional Banach spaces. Israel J. Math. 9 (1971), 346-361. MR**0412775 (54:896)****[GL1]**Gordon Y. and Lewis D. R.: Absolutely summing operators and local unconditional structures. Acta Math. 133 (1974), 27-48. MR**0410341 (53:14091)****[GL2]**Gordon Y. and Lewis D. R.: Banach ideals on Hilbert spaces. Studia Math. 54 (1975), no. 2, 161-172. MR**0388149 (52:8986)****[JLS]**Johnson W. B., Lindenstrauss J. and Schechtman G.: On the relation between several notions of unconditional structure. Israel J. Math. 37 (1980), 120-129. MR**599307 (83f:46015)****[KP]**Kalton N. J. and Peck N. T.: Twisted sums of sequence spaces and the three space problem. Trans. Amer. Math. Soc. 255 (1979), 1-30. MR**542869 (82g:46021)****[KT]**Komorowski R.A. and Tomczak-Jaegermann N.: Subspaces of and without local unconditional structure. Studia Math. 149 (2002), no. 1, 1-21. MR**1881713 (2003c:46016)****[LT1]**Lindenstrauss J. and Tzafriri L.: Classical Banach spaces. I. Springer-Verlag, Berlin-Heidelberg-New York, 1977. MR**0500056 (58:17766)****[LT2]**Lindenstrauss J. and Tzafriri L.: Classical Banach spaces. II. Springer-Verlag, Berlin-Heidelberg-New York, 1979. MR**540367 (81c:46001)****[PW]**Pelczynski A. and Wojciechowski M.: Spaces of functions with bounded variation and Sobolev spaces without local unconditional structure. J. Reine Angew. Math. 558 (2003), 109-157. MR**1979184 (2004c:46058)****[P]**Pietsch A.: Operator Ideals. VEB Deutsch. Ver. Wiss., Berlin, 1978; and North-Holland Publ. Co., Amsterdam-New York, 1980. MR**582655 (81j:47001)****[R]**Reisner S.: Operators which factor through convex Banach lattices. Canad. J. Math. 32 (1980), no. 6, 1482-1500. MR**604702 (82k:47031)****[T]**Tomczak-Jaegermann N.: Banach-Mazur distances and finite-dimensional operator ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics, 38. Longman Scientific and Technical, Harlow; copublished in the United States with John Wiley and Sons, Inc., New York, 1989. MR**993774 (90k:46039)**

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Additional Information

**Y. Gordon**

Affiliation:
Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel

Email:
gordon@techunix.technion.ac.il

**M. Junge**

Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Email:
junge@math.uiuc.edu

**M. Meyer**

Affiliation:
Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Université Paris-Est-Marne-la-Vallée, Cité Descartes-5, Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France

Email:
Mathieu.Meyer@univ-mlv.fr

**S. Reisner**

Affiliation:
Department of Mathematics, University of Haifa, Haifa 31905, Israel

Email:
reisner@math.haifa.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-2011-10715-9

Keywords:
Banach spaces,
local unconditional structure,
asymmetry

Received by editor(s):
March 12, 2010

Received by editor(s) in revised form:
July 21, 2010

Published electronically:
February 22, 2011

Additional Notes:
The first, third and fourth authors were supported in part by the France-Israel Research Network Program in Mathematics contract #3-4301.

The second author was supported in part by NSF grant DMS-0901457.

Dedicated:
Dedicated to the memory of Nigel J. Kalton

Communicated by:
Nigel J. Kalton

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.