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

Abstract

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 ≤ 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.It should be noted here that properties M(1, 1) and M * (1, 1) are precisely properties (M) and (M * ) in the terminology of [23].Properties M(r, s) have been successfully applied in fixed point theory (compare [20, 15, 10]).The fixed point theory motivated the developing of many interesting properties in Banach space theory. That is also the case of the Kadec-Klee (or Radon-Riesz) property (KKP in the sequel). We recall that a Banach space has the KKP if any sequence in the unit sphere whose weak limit is also in the unit sphere, is indeed norm convergent. The uniform Kadec-Klee property for the weak topology on a Banach space was introduced by R. Huff [24] as a useful substitute for uniform convexity, especially in many non-reflexive spaces. D. van Dulst and B. Sims [17]showed that the uniform Kadec-Klee property for weak and weak * topologies implies weak (resp. weak * ) normal structure.It is well-known (see [2,34]) that the Schatten p-classes C p , 1 ≤ p < ∞, have the KKP. It has been shown by C. Lennard [30] that the direct argument given by J. Arazy [2] for trace-class operators can be refined to show that L 1 (H), the space of trace class operators on an arbitrary infinite-dimensional Hilbert space H satisfies a stronger property called the uniformly Kadec-Klee in the weak * topology UKK * (see section 2 for the detailed definitions). From a somewhat different viewpoint, it is a classical theorem of F. Riesz that norm convergence for sequences in the unit sphere of L 1 [0, 1] coincides with convergence in measure. Appropriate uniform version of this theorem may be found in [31].Independently from the fixed point theory, The Kadec-Klee property has been deeply studied in certain particular classes of Banach spaces including C * -algebras and JB * -triples in connection with the Alternative Dunford-Pettis property (compare [1, 6, 3] and [8]). Proposition 2.13 in [6] provides a complete description of those JB * -triples satisfying the KKP, namely, a JB * -triple satisfies this property if and only if it is finite-dimensional or a Hilbert space or a spin factor. A similar conclusion holds for real JB * -triples (compare [3, Proposition 3.13]). It is a natural open problem to ask wether a JB * -triple satisfying t...

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