Geometric implications of the M(r,s)-properties and the uniform Kadec-Klee property in JB*-triples

We explore new implications of the $M(r,s)$ and $M^{\ast }(r,s)$ properties for Banach spaces. We show that a Banach space $X$ satisfying property $M(1,s)$ for some $0<s\le 1$, admitting a point $x_0$ in its unit sphere at which the relative weak and norm topologies agree, satisfies the generalized Gossez-Lami Dozo property. We establish sufficient conditions, in terms of the $(r, s)$-Lipschitzweak$^*$Kadec-KleepropertyonaBanachspace$X$, to guarantee that its dual space satisfies the UKK${}^{\ast }$ property. We determine appropriate conditions to assure that a Banach space $X$ satisfies the $(r,s)$-Lipschitz weak${}^{\ast }$ Kadec-Klee property. These results are applied to prove that every spin factor satisfies the UKK property, and consequently, the KKP and the UKK properties are equivalent for real and complex JB${}^{\ast }$-triples.