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New classes of Banach spaces which are M-ideals in their biduals

Published online by Cambridge University Press:  24 October 2008

Dirk Werner
Dirk Werner
Affiliation:
I. Mathematisches Institut, Freie Universitt Berlin, Arnimallee 3, D-W-1000 Berlin 33, Germany
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Extract

A subspace X of a Banach space Y is called an M-ideal if there is an L-projection P on Y* whose kernel is X1, the annihilator of X; that is, we have

In the case that Y is the bidual of X, a natural linear projection on X*** = Y* with kernel X1 is available, namely .

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Type
Research Article
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Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 2 , March 1992 , pp. 337 - 354
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Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

1Anderson, J. M. and Duncan, J.. Duals of Banach spaces of entire functions. Glasgow Math. J. 32 (1990), 215220.Google Scholar
2Arazy, J.. More on convergence in unitary matrix spaces. Proc. Amer. Math. Soc. 83 (1981), 4448.Google Scholar
3Arazy, J. and Fisher, S. D.. Some aspects of the minimal Mbius invariant space of analytic functions on the unit disk. In Interpolation Spaces and Allied Topics in Analysis, Proc. Conf. Lund 1983 (eds. Cwikel, M. and Peetre, J.), Lecture Notes in Math. vol. 1070 (Springer-Verlag, 1984), pp. 2444.Google Scholar
4Arazy, J., Fisher, S. D. and Peetre, J.. Mbius invariant function spaces. J. Reine Angew. Math. 363 (1985), 110145.Google Scholar
5Axler, S.. Bergman spaces and their operators. In Surveys of Some Recent Results in Operator Theory, vol. 1 (eds. Conway, J. B. and Morrell, B. B.), Pitman Research Notes in Math. Series no. 171 (Longman, 1988), pp. 150.,Google Scholar
6Beauzamy, B. and Laprest, J.-T.. Modles tals des Espaces de Banach (Hermann, 1984).Google Scholar
7Behrends, E.. M-Structure and the Banach-Stone Theorem. Lecture Notes in Math. vol. 736 (Springer-Verlag, 1979).CrossRefGoogle Scholar
8Behrends, E.. Sur les M-idaux des espaces d'oprateurs compacts. In Sminaire d'Analyse Fonctionnelle 198485. Universit Paris VIVII, (eds. Beauzamy, B., Maurey, B. and Pisier, G.), (1985), pp. 1118.Google Scholar
9Behrends, E.. Points of symmetry of convex sets in the two-dimensional complex space a counterexample to D. Yost's problem. Math. Ann. 290 (1991), 463471.Google Scholar
10Behrends, E. and Harmand, P.. Banach spaces which are proper M-ideals. Studia Math. 81 (1985), 159169.Google Scholar
11Bennett, C. D. and Sharpley, R. S.. Interpolation of Operators (Academic Press, 1988).Google Scholar
12Bierstedt, K. D. and Summers, W. H.. Biduals of weighted Banach spaces of analytic functions. Preprint (1990).Google Scholar
13Bretagnolle, J. and Dacunha-Castelle, D.. Application de l'tude de certaines formes linaires alatoires au plongement d'espaces de Banach dans les espaces Lp. Ann. Sci. cole Norm. Sup. 2 (1969), 437480.Google Scholar
14Carothers, N. L. and Dilworth, S. J.. Subspaces of Lp, q. Proc. Amer. Math. Soc. 104 (1988), 537545.Google Scholar
15Casazza, P. G. and Shura, T. J.. Tsirelson's Space. Lecture Notes in Math. vol. 1363 (Springer-Verlag, 1989).CrossRefGoogle Scholar
16Cho, C.-M. and Johnson, W. B.. A characterization of subspaces X of lp for which K(X) is an M-ideal in L(X). Proc. Amer. Math. Soc. 93 (1985), 466470.Google Scholar
17Cima, J. A.. The basic properties of Bloch functions. Internat. J.Math. Sci. 2 (1979), 369413.Google Scholar
18Cima, J. A. and Wogen, W. R.. Extreme points of the unit ball of the Bloch space *0. Michigan Math. J. 25 (1978), 213222.Google Scholar
19Cima, J. A. and Wogen, W. R.. On isometries of the Bloch space. Illinois J. Math. 24 (1980), 313316.Google Scholar
20Dixmier, J.. Les fonctionnelles linaires sur l'ensemble des oprateurs borns d'un espace de Hilbert. Ann. of Math. 51 (1950), 387408.CrossRefGoogle Scholar
21Fabian, M. and Godefroy, G.. The dual of every Asplund space admits a projectional resolution of the identity. Studia Math. 91 (1988), 141151.CrossRefGoogle Scholar
22Garling, D. J. H.. On symmetric sequence spaces. Proc. London Math. Soc. 16 (1966), 85106.CrossRefGoogle Scholar
23Godefroy, G.. Espaces de Banach: existence et unicit de certains prduaux. Ann. Inst. Fourier 283 (1978), 87105.CrossRefGoogle Scholar
24Godefroy, G.. Sous-espaces bien disposs de L1-Applications. Trans. Amer. Math. Soc. 286 (1984), 227249.Google Scholar
25Godefroy, G.. On Riesz subsets of abelian discrete groups. Israel J. Math. 61 (1988), 301331.CrossRefGoogle Scholar
26Godefroy, G. and Li, D.. Banach spaces which are M-ideals in their bidual have property (u). Ann. Inst. Fourier 392 (1989), 361371.CrossRefGoogle Scholar
27Godefroy, G. and Li, D.. Some natural families of M-ideals. Math. Scand. 66 (1990), 249263.CrossRefGoogle Scholar
28Godefroy, G. and Saab, P.. Weakly unconditionally convergent series in. M-ideals. Math. Scand. 64 (1989), 307318.CrossRefGoogle Scholar
29Godefroy, G. and Saphar, P.. Duality in spaces of operators and smooth norms on Banach spaces. Illinois J. Math. 32 (1988), 672695.Google Scholar
30Gohberg, I. C. andKrein, M. G.. Introduction to the Theory of Linear Non-selfadjoint Operators. Translations Math. Monogr. vol. 18 (American Mathematical Society, 1969).Google Scholar
31Gowers, W. T.. Symmetric block bases of sequences with large average growth. Israel J. Math. 69 (1990), 129151.Google Scholar
32Harmand, P. and Lima, A.. Banach spaces which are M-ideals in their biduals. Trans. Amer. Math. Soc. 283 (1983), 253264.Google Scholar
33Harmand, P. and Rao, T. S. S. R. K.. An intersection property of balls and relations with M-ideals. Math. Z. 197 (1988), 277290.CrossRefGoogle Scholar
34Harmand, P., Werner, D. and Werner, W.. M-ideals in Banach Spaces and Banach Algebras. In preparation.Google Scholar
35Jarosz, K.. A generalization of the BanachStone theorem. Studia Math. 73 (1982), 3339.Google Scholar
36Lima, .. Intersection properties of balls and subspaces in Banach spaces. Trans. Amer. Math. Soc. 227 (1977), 162.CrossRefGoogle Scholar
37Lima, .. M-ideals of compact operators in classical Banach spaces. Math. Scand. 44 (1979), 207217.Google Scholar
38Lima, .. On M-ideals and best approximation. Indiana Univ. Math. J. 31 (1982), 2736.Google Scholar
39Lindenstrauss, J. and Tzafriri, L.. Classical Banach Spaces, vol. 1 (Springer-Verlag, 1977).CrossRefGoogle Scholar
40Lindenstrauss, J. and Tzafriri, L.. Classical Banach Spaces, vol. 2 (Springer-Verlag, 1979).Google Scholar
41Lorentz, G. G.. Some new functional spaces. Ann. of Math. 51 (1950), 3755.CrossRefGoogle Scholar
42Luecking, D. H.. The compact Hankel operators form an M-ideal in the space of Hankel operators. Proc. Amer. Math. Soc. 79 (1980), 222224.Google Scholar
43Nara, C.. Uniqueness of the predual of the Bloch space and its strongly exposed points. Illinois J. Math. 34 (1990), 98107.Google Scholar
44Odell, E.. On quotients of Banach spaces having shrinking unconditional bases. Preprint (1990).Google Scholar
45Oja, E.. Dual de l'espace des oprateurs linaires continus. C. R. Acad. Sci. Paris Sr. A 309 (1989), 983986.Google Scholar
46Oja, E.. Remarks on the dual of the space of continuous linear operators. Acta Comment. Univ. Tartuensis (1991), to appear.Google Scholar
47Oja, E. and Werner, D.. Remarks on M-ideals of compact operators on X p X. Math. Nachr. 152 (1991), 101111.Google Scholar
48Rubel, L. A. and Shields, A. L.. The second duals of certain spaces of analytic function. J. Austral. Math. Soc. 11 (1970), 276280.Google Scholar
49Saatkamp, K.. Schnitteigenschaften und beste Approximation. Dissertation, Universitat Bonn (1979).Google Scholar
50Saatkamp, K.. Best approximation in the space of bounded operators and its applications. Math. Ann. 250 (1980), 3554.CrossRefGoogle Scholar
51Schreier, J.. Ein Gegenbeispiel zur schwachen Konvergenz. Studia Math. 2 (1930), 5862.Google Scholar
52Shields, A. L. and Williams, D. L.. Bounded projections, duality, and multipliers in spaces of harmonic functions. J. Reine Angew. Math. 299300 (1978), 256279.Google Scholar
53Shura, T. J. and Trautman, D.. The -property in Schreier's space S and the Lorentz space d(a, 1). Glasgow Math. J. 32 (1990), 277284.Google Scholar
54Simon, B.. Trace Ideals and Their Applications. London Math. Soc. Lecture Note Series no. 35 (Cambridge University Press, 1979).Google Scholar
55Werner, D.. De nouveau: M-idaux des espaces d'oprateurs compacts. Sminaire Initiation l'Analyse, anne 1988/89, expos 17, Universit Paris VI (1989).Google Scholar
56Werner, D.. Remarks on. M-ideals of compact operators. Quart. J. Math. Oxford Ser. (2) 41 (1990), 501507.CrossRefGoogle Scholar