Atomic decomposition of real JBW-triples

Relatively weakly open sets in closed balls of Banach spaces, and real JB*-triples of finite rank

“…The name "minimal" is in agreement with the fact that, in the case that X is actually a real or complex J BW * -triple, minimal tripotents of X are precisely those tripotents of X which are minimal relative to the order defined in the set of all tripotents of X by u ≤ v if and only if v = u + w for some tripotent w orthogonal to u [27, Proposition 2.2]. Now, by arguing as in the proof of [27,Lemma 3.2], we obtain the following corollary. To conclude the proof it is enough to show that X is of finite rank whenever the Banach space of X is isomorphic to a Hilbert space.…”

Subdifferentiability of the norm and the Banach-Stone theorem for real and complex JB*-triples

“…Therefore, every element in E can be approximated in norm by linear combinations of minimal tripotents in E. Since E = E + iE we conclude, by [22,Lemma 3.2] (see also [2,Corollary 3.5]), that every element in E can be approximated in norm by linear combinations of minimal tripotents in E. By [4, Theorem 3.4, (i) ⇔ (vi)] it follows that E is weakly compact.…”

Local Triple Derivations on Real C $$^*$$ ∗ -Algebras and JB $$^*$$ ∗ -Triples

“…[27]) and the weak * -density of E in E * * . The atomic decomposition established in [29,Theorem 3.6] assures that every JBW * -triple W decomposes as an orthogonal sum…”

“…It is also proved in [29,Theorem 3.6] that A is an orthogonal sum of weak * -closed triple ideals which are generalized real Cartan factors.…”