On real forms of JB*-triples

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

“…It is worth mentioning that a real JB * -triple is a norm-closed real subtriple of a JB * -triple [16,Definition 2.1]. Let E be a real JB * -triple.…”

“…Let E be a real JB * -triple. By [16,Proposition 2.8], there exists a unique complex JB * -triple structure on the algebraic complexification E ⊕ iE (denoted by E) and a conjugation τ on E + iE such that E = E τ := {z ∈ E : τ (z) = z}, i.e., every real JB * -triple is a real form of its complexification, which is a complex JB * -triple. Every real C * -algebra, every real Hilbert space, every complex JB * -triple (when is regarded as a real Banach space) and the Banach space of all bounded linear operators between real Hilbert spaces are examples of real JB * -triples (cf.…”

“…Every real C * -algebra, every real Hilbert space, every complex JB * -triple (when is regarded as a real Banach space) and the Banach space of all bounded linear operators between real Hilbert spaces are examples of real JB * -triples (cf. [16]). …”

“…The bidual E * * of every real or complex JB * -triple is a JBW * -triple with triple product extending the product of E (cf. [8] and [16,Lemma 4.2], respectively).…”

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

“…In the present paper we prove as the main result that every real J B * -triple X whose Banach space is not isomorphic to a Hilbert space satisfies Property P (Theorem 2.3). We note that the class of real J B * -triples, introduced in [15], contains that of complex J B * -triples (regarded as real Banach spaces), as well as that of J B-algebras. It is worth mentioning that the proof of Theorem 2.3 provided here is independent of the one given in [3] for complex J B * -triples, and has the advantage that it works autonomously in all relevant subclasses of the class of real J B * -triples.…”