  We explore new implications of the $M(r,s)$ and $M^*(r,s)$ properties for
Banach spaces. We show that a Banach space $X$ satisfying property $M(1,s)$ for
some $0<s\leq 1$, 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 $(r,
s)$-Lipschitz weak$^*$ Kadec-Klee property on a Banach space $X$, to guarantee
that its dual space satisfies the UKK$^*$ property. We determine appropriate
conditions to assure that a Banach space $X$ satisfies the $(r, s)$-Lipschitz
weak$^*$ 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$^*$-triples.
