A tensor norm preserving unconditionality for ℒ_{𝓅}-spaces

Defant, Andreas; Pérez-García, David

doi:10.1090/S0002-9947-08-04428-0

A tensor norm preserving unconditionality for $\mathcal {L}_p$-spaces
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by Andreas Defant and David Pérez-García PDF

Trans. Amer. Math. Soc. 360 (2008), 3287-3306 Request permission

Abstract:

We show that, for each $n\in \mathbb {N}$, there is an $n$-tensor norm $\alpha$ (in the sense of Grothendieck) with the surprising property that the $\alpha$-tensor product $\tilde {\bigotimes }_\alpha (Y_1, \ldots , Y_n)$ has local unconditional structure for each choice of $n$ arbitrary $\mathcal {L}_{p_j}$-spaces $Y_j$. In fact, $\alpha$ is the tensor norm associated to the ideal of multiple $1$-summing $n$-linear forms on Banach spaces.

References

Lev Aizenberg, Multidimensional analogues of Bohr’s theorem on power series, Proc. Amer. Math. Soc. 128 (2000), no. 4, 1147–1155. MR 1636918, DOI 10.1090/S0002-9939-99-05084-4
G. Bennett, Schur multipliers, Duke Math. J. 44 (1977), no. 3, 603–639. MR 493490
Harold P. Boas, Majorant series, J. Korean Math. Soc. 37 (2000), no. 2, 321–337. Several complex variables (Seoul, 1998). MR 1775963
Harold P. Boas and Dmitry Khavinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2975–2979. MR 1443371, DOI 10.1090/S0002-9939-97-04270-6
H. Bohr, A theorem concerning power series, Proc. London Math. Soc. 13 (1914), 1–5.
Fernando Bombal, David Pérez-García, and Ignacio Villanueva, Multilinear extensions of Grothendieck’s theorem, Q. J. Math. 55 (2004), no. 4, 441–450. MR 2104683, DOI 10.1093/qjmath/55.4.441
T. K. Carne, Tensor products and Banach algebras, J. London Math. Soc. (2) 17 (1978), no. 3, 480–488. MR 493403, DOI 10.1112/jlms/s2-17.3.480
Andreas Defant, Juan Carlos Díaz, Domingo García, and Manuel Maestre, Unconditional basis and Gordon-Lewis constants for spaces of polynomials, J. Funct. Anal. 181 (2001), no. 1, 119–145. MR 1818112, DOI 10.1006/jfan.2000.3702
Andreas Defant, Domingo García, and Manuel Maestre, Bohr’s power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197. MR 1978407, DOI 10.1515/crll.2003.030
Andreas Defant, Domingo García, Manuel Maestre, and David Pérez-García, Extension of multilinear forms and polynomials from subspaces of $\scr L_1$-spaces, Houston J. Math. 33 (2007), no. 3, 839–860. MR 2335739
Andreas Defant and Nigel Kalton, Unconditionality in spaces of $m$-homogeneous polynomials, Q. J. Math. 56 (2005), no. 1, 53–64. MR 2124579, DOI 10.1093/qmath/hah022
A. Defant, M. Maestre, and C. Prengel, Domains of convergence for monomial series expansions of holomorphic functions in infinitely many variables, preprint.
Andreas Defant, Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces, Positivity 5 (2001), no. 2, 153–175. MR 1825653, DOI 10.1023/A:1011466509838
Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR 1209438
Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR 1342297, DOI 10.1017/CBO9780511526138
Seán Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MR 1705327, DOI 10.1007/978-1-4471-0869-6
Seán Dineen and Richard M. Timoney, Absolute bases, tensor products and a theorem of Bohr, Studia Math. 94 (1989), no. 3, 227–234. MR 1019790, DOI 10.4064/sm-94-3-227-234
Klaus Floret and Stephan Hunfeld, Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces, Proc. Amer. Math. Soc. 130 (2002), no. 5, 1425–1435. MR 1879966, DOI 10.1090/S0002-9939-01-06228-1
B. R. Gelbaum and J. Gil de Lamadrid, Bases of tensor products of Banach spaces, Pacific J. Math. 11 (1961), 1281–1286. MR 147881
Y. Gordon and D. R. Lewis, Absolutely summing operators and local unconditional structures, Acta Math. 133 (1974), 27–48. MR 410341, DOI 10.1007/BF02392140
Y. Gordon, D. R. Lewis, and J. R. Retherford, Banach ideals of operators with applications, J. Functional Analysis 14 (1973), 85–129. MR 0380488, DOI 10.1016/0022-1236(73)90031-1
A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79 (French). MR 94682
S. Kwapień and A. Pełczyński, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43–68. MR 270118, DOI 10.4064/sm-34-1-43-67
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
Mário C. Matos, Fully absolutely summing and Hilbert-Schmidt multilinear mappings, Collect. Math. 54 (2003), no. 2, 111–136. MR 1995136
David Pérez-García, The inclusion theorem for multiple summing operators, Studia Math. 165 (2004), no. 3, 275–290. MR 2110152, DOI 10.4064/sm165-3-5
David Pérez-García, Comparing different classes of absolutely summing multilinear operators, Arch. Math. (Basel) 85 (2005), no. 3, 258–267. MR 2172384, DOI 10.1007/s00013-005-1125-4
David Pérez-García and Ignacio Villanueva, Unconditional bases in tensor products of Hilbert spaces, Math. Scand. 96 (2005), no. 2, 280–288. MR 2153415, DOI 10.7146/math.scand.a-14957
David Pérez-García and Ignacio Villanueva, Multiple summing operators on Banach spaces, J. Math. Anal. Appl. 285 (2003), no. 1, 86–96. MR 2000141, DOI 10.1016/S0022-247X(03)00352-4
David Pérez-García and Ignacio Villanueva, Multiple summing operators on $C(K)$ spaces, Ark. Mat. 42 (2004), no. 1, 153–171. MR 2056549, DOI 10.1007/BF02432914
David Pérez-García and Ignacio Villanueva, A composition theorem for multiple summing operators, Monatsh. Math. 146 (2005), no. 3, 257–261. MR 2184227, DOI 10.1007/s00605-005-0316-1
David Pérez-García and Ignacio Villanueva, There is no lattice preserving natural tensor norm, Quaest. Math. 27 (2004), no. 3, 267–273. MR 2109666, DOI 10.2989/16073600409486099
Albrecht Pietsch, Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. MR 582655
Gilles Pisier, Some results on Banach spaces without local unconditional structure, Compositio Math. 37 (1978), no. 1, 3–19. MR 501916
Carsten Schütt, Unconditionality in tensor products, Israel J. Math. 31 (1978), no. 3-4, 209–216. MR 516148, DOI 10.1007/BF02761492
I. Schütt, Banachräume absolut $p$-summierender Operatoren, Ph.D. Thesis, Kiel, 1986.
N. Tomczak-Jaegermann, Banach–Mazur distances and finite-dimensional operator ideals, Longman Scientific & Technical, 1989.