Khorshidvandpour, S., Aminpour, A. (2017). On the reducible $M$-ideals in Banach spaces. Sahand Communications in Mathematical Analysis, 7(1), 27-37.

Sajad Khorshidvandpour; Abdolmohammad Aminpour. "On the reducible $M$-ideals in Banach spaces". Sahand Communications in Mathematical Analysis, 7, 1, 2017, 27-37.

Khorshidvandpour, S., Aminpour, A. (2017). 'On the reducible $M$-ideals in Banach spaces', Sahand Communications in Mathematical Analysis, 7(1), pp. 27-37.

Khorshidvandpour, S., Aminpour, A. On the reducible $M$-ideals in Banach spaces. Sahand Communications in Mathematical Analysis, 2017; 7(1): 27-37.

^{}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.

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