On property \((M)\) and its generalizations (Q5946946)

In 1993, \textit{N. I. Kalton} [Ill. J. Math. 37, No. 1, 147-169 (1993; Zbl 0824.46029)] introduced property \((M)\): A Banach space \(X\) has property \((M)\) if whenver \(u,v\in X\) are such \(\|u\|=\|v\|\) and \((x_n)\) is weakly null sequence in \(X\) we have NEWLINE\[NEWLINE\limsup_{n\to\infty} \|u+ x_n\|= \limsup_{n\to\infty} \|v+ x_n\|.NEWLINE\]NEWLINE In the present paper the authors introduced properties strict \((M)\) and uniform \((M)\) and discuss the relations among this properties and other geometrical properties of Banach spaces, for example; Opial's property, generalized Gossez-Lami Dozo property, weak uniform normal structure. They shown, among others, that if \(X\) has property \((M)\) and is uniformly convex in every direction, then \(X\) has both strict \((M)\) and uniform \((M)\). If \(X^*\) is separable, then strict \((M)\) implies uniform \((M)\) and property \((M)\) implies weak uniform normal structure.