Bounded approximation properties via integral and nuclear operators

Lima, Åsvald; Lima, Vegard; Oja, Eve

doi:10.1090/S0002-9939-09-10061-8

Bounded approximation properties via integral and nuclear operators
HTML articles powered by AMS MathViewer

by Åsvald Lima, Vegard Lima and Eve Oja PDF

Proc. Amer. Math. Soc. 138 (2010), 287-297 Request permission

Abstract:

Let $X$ be a Banach space and let $\mathcal A$ be a Banach operator ideal. We say that $X$ has the $\lambda$-bounded approximation property for $\mathcal A$ ($\lambda$-BAP for $\mathcal A$) if for every Banach space $Y$ and every operator $T\in \mathcal A(X,Y)$, there exists a net $(S_\alpha )$ of finite rank operators on $X$ such that $S_\alpha \to I_X$ uniformly on compact subsets of $X$ and \[ \limsup _\alpha \|TS_\alpha \|_{\mathcal A}\leq \lambda \|T\|_{\mathcal A}.\] We prove that the (classical) $\lambda$-BAP is precisely the $\lambda$-BAP for the ideal $\mathcal I$ of integral operators, or equivalently, for the ideal ${\mathcal {S{\kern -0.15em}I}}$ of strictly integral operators. We also prove that the weak $\lambda$-BAP is precisely the $\lambda$-BAP for the ideal $\mathcal N$ of nuclear operators.

References

T.A. Abrahamsen, V. Lima, and Å. Lima. Unconditional ideals of finite rank operators. II. Houston J. Math. 35 (2009) 627–646.
Peter G. Casazza, Approximation properties, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 271–316. MR 1863695, DOI 10.1016/S1874-5849(01)80009-7
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
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
G. Godefroy, N. J. Kalton, and P. D. Saphar, Unconditional ideals in Banach spaces, Studia Math. 104 (1993), no. 1, 13–59. MR 1208038
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (1982), no. 3, 225–251. MR 675426, DOI 10.4064/sm-73-3-225-251
William B. Johnson, A complementary universal conjugate Banach space and its relation to the approximation problem, Israel J. Math. 13 (1972), 301–310 (1973). MR 326356, DOI 10.1007/BF02762804
Åsvald Lima and Eve Oja, Ideals of finite rank operators, intersection properties of balls, and the approximation property, Studia Math. 133 (1999), no. 2, 175–186. MR 1686696
Åsvald Lima and Eve Oja, The weak metric approximation property, Math. Ann. 333 (2005), no. 3, 471–484. MR 2198796, DOI 10.1007/s00208-005-0656-0
Vegard Lima, The weak metric approximation property and ideals of operators, J. Math. Anal. Appl. 334 (2007), no. 1, 593–603. MR 2332578, DOI 10.1016/j.jmaa.2007.01.007
Vegard Lima and Åsvald Lima, Ideals of operators and the metric approximation property, J. Funct. Anal. 210 (2004), no. 1, 148–170. MR 2052117, DOI 10.1016/j.jfa.2003.10.001
A. Lissitsin, K. Mikkor, and E. Oja. Approximation properties defined by spaces of operators and approximability in operator topologies. Illinois J. Math. 52 (2008).
Eve Oja, Operators that are nuclear whenever they are nuclear for a larger range space, Proc. Edinb. Math. Soc. (2) 47 (2004), no. 3, 679–694. MR 2097268, DOI 10.1017/S0013091502001165
Eve Oja, Lifting bounded approximation properties from Banach spaces to their dual spaces, J. Math. Anal. Appl. 323 (2006), no. 1, 666–679. MR 2262236, DOI 10.1016/j.jmaa.2005.10.076
Eve Oja, The impact of the Radon-Nikodým property on the weak bounded approximation property, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100 (2006), no. 1-2, 325–331 (English, with English and Spanish summaries). MR 2267414
Eve Oja, The strong approximation property, J. Math. Anal. Appl. 338 (2008), no. 1, 407–415. MR 2386425, DOI 10.1016/j.jmaa.2007.05.038
E. Oja. Inner and outer inequalities with applications to approximation properties. Preprint (2008).
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
Narcisse Randrianantoanina, Complemented copies of $l^1$ and Pełczyński’s property $(\textrm {V}^*)$ in Bochner function spaces, Canad. J. Math. 48 (1996), no. 3, 625–640. MR 1402332, DOI 10.4153/CJM-1996-033-2
Raymond A. Ryan, Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2002. MR 1888309, DOI 10.1007/978-1-4471-3903-4
Brailey Sims and David Yost, Linear Hahn-Banach extension operators, Proc. Edinburgh Math. Soc. (2) 32 (1989), no. 1, 53–57. MR 981992, DOI 10.1017/S0013091500006908