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Strong Subdifferentiability of the Norm on J B-Triples

# Strong Subdifferentiability of the Norm on J B-Triples,10.1093/qjmath/54.4.381,Quarterly Journal of Mathematics,J. B. Guerrero

Strong Subdifferentiability of the Norm on J B-Triples
Journal: Quarterly Journal of Mathematics - QUART J MATH , vol. 54, no. 4, pp. 381-390, 2003
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## Citation Context (1)

• ...Recently, Becerra-Guerrero and Rodr´ õguez-Palacios have completely described the points of strong subdifferentiability for the norm of a (complex) JB ∗ triple [3]...
• ...in its complexification and the strong subdifferentiability at a norm-one point is inherited by real subspaces, the next corollary is a consequence of Theorem [3, Theorem 2.7]...
• ...Having the above facts in mind, the same arguments given in [3, Theorem 2.5] can be adapted to obtain the following corollary...
• ...the strong subdifferentiability of its norm at a point of the unit sphere, similar to those obtained for (complex) JB ∗ -triples in [3, Theorem 2.7]...
• ...The following corollary is an extension to real JB ∗ -triples of the main result of [10] (see also [3]) for complex JB ∗ -triples...
• ...Therefore, by [23, Lemma 3], we have D(C, e) = D(C, a). Finally, let f in D(E, a). It is clear that f |C lies in D(C, a) = D(C, e) and hence f( e) = 1. This assures that D(E, a) ⊆ D(E, e). (⇐) Suppose now that D(E, a) = D(E, e). By [3, Theorem 2.7] we conclude that the norm of E is strongly subdifferentiable at every tripotent element of E. Since the strong subdifferentiability of the norm of E at a norm-one element x depends only on the ...
• ...From [3, Theorem 2.7] we conclude that there is a tripotent u in E such that a lies in Eu. By the first part of the proof we have D(E, u) = D(E, a) = D(E, e). By Lemma 3.1 we get u = e, which gives a ∈ Ee. (b) Let C be the JB ∗ -subtriple generated by e and x = e + z0 (z0 ∈ E0(e), � z0� < 1). As we have seen in the first part of the proof, C is an abelian C ∗ algebra...
• ...of the one given in the complex case but replacing [3, Theorem 2.7] by Corollary 2.5...

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