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We explore new implications of theM  (  r  ,  s  )andM    *    (  r  ,  s  )properties for Banach spaces. We show that a Banach spaceXsatisfying propertyM  (  1  ,  s  )for some0  &lt;  s  ≤  1, admitting a pointx  0in 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 spaceX, to guarantee that its dual space satisfies the uniform Kadec-Klee property in the weak*topology. We determine appropriate conditions to assure that a Banach spaceXsatisfies the(  r  ,  s  )-Lipschitz weak*KKP. These results are applied to prove that every spin factor satisfies the UKK property, and consequently, the KKP and the UKK property are equivalent for real and complex JB*-triples.

the quarterly journal of mathematics

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