Document Type : Research Paper
Authors
Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
Abstract
The object of the investigation is to study reducible $M$-ideals in Banach spaces. It is shown that if the number of $M$-ideals in a Banach space $X$ is $n(<\infty)$, then the number of reducible $M$-ideals does not exceed of $\frac{(n-2)(n-3)}{2}$. Moreover, given a compact metric space $X$, we obtain a general form of a reducible $M$-ideal in the space $C(X)$ of continuous functions on $X$. The intersection of two $M$-ideals is not necessarily reducible. We construct a subset of the set of all $M$-ideals in a Banach space $X$ such that the intersection of any pair of it's elements is reducible. Also, some Banach spaces $X$ and $Y$ for which $K(X,Y)$ is not a reducible $M$-ideal in $L(X,Y)$, are presented. Finally, a weak version of reducible $M$-ideal called semi reducible $M$-ideal is introduced.
Keywords
Main Subjects
[2] P. Bandyopadhyay and S. Dutta, Almost Constrained Subspaces of Banach Spaces-II, Houston Journal of Mathematics., 35(3) (2009) 945--957.
[3] S. Basu and T.S.S.R.K. Rao, Some Stability Results for Asymptotic Norming Properties of Banach Spaces, Colloquium Mathematicum., 75(2) (1998) 271-284.
[4] E. Behrands, M-structure and the Banach-Stone Theorem, Lecture Notes in Math, 736, Springer, Berlin Heidelberg-New York, 1979.
[5] P. Harmand and A. Lima, Banach spaces which are $M$-ideals in their biduals, Trans. Amer. Math. Soc., 283 (1984) 253-264.
[6] P. Harmand, D. Werner, and W. Werner, $M$-ideals in Banach spaces and Banach algebras, Lecture notes in Mathematics, vol. 1547, Springer, Berlin, 1993.
[7] J. Johnson, Remarks on Banach Spaces of Compact Operators, Journal of Functional Analysis., 32 (1979) 304-311.
[8] H.E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer-Verlag, Berlin, 1974.
[9] A. Lima and D. Yost, Absolutely Chebyshev subspaces, In: S. Fitzpatrick and J. Giles, editors, Workshop/Miniconference Funct. Analysis/Optimization. Canberra, Proc. Cent. Math. Anal. Austral. Nat. Univ., 20 (1988) 116-127.
[10] R.R. Phelps, Uniqueness of Hahn-Banach extensions and unique best approximation, Trans. Amer. Math. Soc., 95 (1960) 238-255.
[11] N.M. Roy, An $M$-ideal characterization of $G$-spaces, Pacific journal of mathematics., 92 (1981) 151-160.
[12] R.R. Smith and J.D. Ward, $M$-ideal structure in Banach algebras, J. Functional Analysis., 27 (1978) 337-349.
[13] R.R. Smith and J.D. Ward, $M$-ideals in $B(l_p)$, Pacific Journal of Mathematics., 81(1) (1979) 227-237.
[14] U. Uttersrud, On $M$-ideals and the Alfsen-Effros structure topology, Math. Scand., 43 (1978) 369-381.
[15] S. Willard, General Topology, Addison-Wesley, Reading, MA, 1970.
)