Banach spaces which are $M$-ideals in their biduals
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- by Peter Harmand and Åsvald Lima PDF
- Trans. Amer. Math. Soc. 283 (1984), 253-264 Request permission
Abstract:
We investigate Banach spaces $X$ such that $X$ is an $M$-ideal in ${X^{{\ast }{\ast }}}$. Subspaces, quotients and ${c_0}$-sums of spaces which are $M$-ideals in their biduals are again of this type. A nonreflexive space $X$ which is an $M$-ideal in ${X^{{\ast }{\ast }}}$ contains a copy of ${c_0}$. Recently Lima has shown that if $K(X)$ is an $M$-ideal in $L(X)$ then $X$ is an $M$-ideal in ${X^{{\ast }{\ast }}}$. Here we show that if $X$ is reflexive and $K(X)$ is an $M$-ideal in $L(X)$, then $K{(X)^{{\ast }{\ast }}}$ is isometric to $L(X)$, i.e. $K(X)$ is an $M$-ideal in its bidual. Moreover, for real such spaces, we show that $K(X)$ contains a proper $M$-ideal if and only if $X$ or ${X^{\ast }}$ contains a proper $M$-ideal.References
- T. Ando, On the predual of $H^{\infty }$, Comment. Math. Special Issue 1 (1978), 33–40. Special issue dedicated to Władysław Orlicz on the occasion of his seventy-fifth birthday. MR 504151
- Ehrhard Behrends, $M$-structure and the Banach-Stone theorem, Lecture Notes in Mathematics, vol. 736, Springer, Berlin, 1979. MR 547509, DOI 10.1007/BFb0063153
- Ehrhard Behrends, Rainer Danckwerts, Richard Evans, Silke Göbel, Peter Greim, Konrad Meyfarth, and Winfried Müller, $L^{p}$-structure in real Banach spaces, Lecture Notes in Mathematics, Vol. 613, Springer-Verlag, Berlin-New York, 1977. MR 0626051, DOI 10.1007/BFb0068175
- 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, DOI 10.1090/surv/015
- Moshe Feder and Pierre Saphar, Spaces of compact operators and their dual spaces, Israel J. Math. 21 (1975), no. 1, 38–49. MR 377591, DOI 10.1007/BF02757132
- Patrick Flinn, A characterization of $M$-ideals in $B(l_{p})$ for $1<p<\infty$, Pacific J. Math. 98 (1982), no. 1, 73–80. MR 644939
- Gilles Godefroy, Espaces de Banach: existence et unicité de certains préduaux, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 3, x, 87–105 (French, with English summary). MR 511815
- Julien Hennefeld, A decomposition for $B(X)^{\ast }$ and unique Hahn-Banach extensions, Pacific J. Math. 46 (1973), 197–199. MR 370265, DOI 10.2140/pjm.1973.46.197
- Julien Hennefeld, $M$-ideals, HB-subspaces, and compact operators, Indiana Univ. Math. J. 28 (1979), no. 6, 927–934. MR 551156, DOI 10.1512/iumj.1979.28.28065
- Richard B. Holmes, Geometric functional analysis and its applications, Graduate Texts in Mathematics, No. 24, Springer-Verlag, New York-Heidelberg, 1975. MR 0410335, DOI 10.1007/978-1-4684-9369-6
- Åsvald Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1–62. MR 430747, DOI 10.1090/S0002-9947-1977-0430747-4
- Asvald Lima, Intersection properties of balls in spaces of compact operators, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 3, v, 35–65. MR 511813
- Åsvald Lima, $M$-ideals of compact operators in classical Banach spaces, Math. Scand. 44 (1979), no. 1, 207–217. MR 544588, DOI 10.7146/math.scand.a-11804
- Åsvald Lima, On $M$-ideals and best approximation, Indiana Univ. Math. J. 31 (1982), no. 1, 27–36. MR 642613, DOI 10.1512/iumj.1982.31.31004
- 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, DOI 10.1007/978-3-642-66557-8 H. P. Rosenthal, On relative disjoint families of measures, Studia Math. 37 (1970), 13-36. 311-313.
- Wolfgang M. Ruess and Charles P. Stegall, Extreme points in duals of operator spaces, Math. Ann. 261 (1982), no. 4, 535–546. MR 682665, DOI 10.1007/BF01457455
- Klaus Saatkamp, $M$-ideals of compact operators, Math. Z. 158 (1978), no. 3, 253–263. MR 470752, DOI 10.1007/BF01214796 —, Schnitteigenschaften und Beste Approximation, Dissertation, Bonn, 1979.
- R. R. Smith and J. D. Ward, $M$-ideal structure in Banach algebras, J. Functional Analysis 27 (1978), no. 3, 337–349. MR 0467316, DOI 10.1016/0022-1236(78)90012-5
- R. R. Smith and J. D. Ward, $M$-ideals in $B(l_{p})$, Pacific J. Math. 81 (1979), no. 1, 227–237. MR 543746
- R. R. Smith and J. D. Ward, Applications of convexity and $M$-ideal theory to quotient Banach algebras, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 119, 365–384. MR 545071, DOI 10.1093/qmath/30.3.365
- Masamichi Takesaki, On the conjugate space of operator algebra, Tohoku Math. J. (2) 10 (1958), 194–203. MR 100799, DOI 10.2748/tmj/1178244713
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 283 (1984), 253-264
- MSC: Primary 46B10; Secondary 47D30
- DOI: https://doi.org/10.1090/S0002-9947-1984-0735420-4
- MathSciNet review: 735420