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We explore new implications of the $M(r,s)$ and $M^*(r,s)$ properties forBanach spaces. We show that a Banach space $X$ satisfying property $M(1,s)$ forsome $0<s\leq 1$, admitting a point $x_{0}$ in its unit sphere at which therelative weak and norm topologies agree, satisfies the generalized Gossez-LamiDozo property. We establish sufficient conditions, in terms of the $(r,s)$-Lipschitz weak$^*$ Kadec-Klee property on a Banach space $X$, to guaranteethat its dual space satisfies the UKK$^*$ property. We determine appropriateconditions to assure that a Banach space $X$ satisfies the $(r, s)$-Lipschitzweak$^*$ Kadec-Klee property. These results are applied to prove that everyspin factor satisfies the UKK property, and consequently, the KKP and the UKKproperties are equivalent for real and complex JB$^*$-triples.

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