Non-commutative generalisations of Urysohn's lemma and hereditary inner ideals

We establish several generalisations of Urysohn's lemma in the setting of JB∗-triples which provide full answers to Problems 1.12 and 1.13 in Fernández-Polo and Peralta (2007) [22]. These results extend the previous generalisations obtained by C.A. Akemann, G.K. Pedersen and L.G. Brown in the setting of C∗-algebras. A generalised Kadison's transitivity theorem is established for finite sums of pairwise orthogonal compact tripotents in JBW∗-triples. We introduce the notion of positively open tripotent in the bidual of a JB∗-triple as an extension of a concept which was already considered in the setting of ternary rings of operators. We investigate the connections appearing between positively open tripotents and hereditary inner ideals.