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

Abstract

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.

Local triple derivations on C∗-algebras and JB∗-triples

“…The notions of range, support or compact-G δ , open, closed and compact tripotents also make sense in the bidual of every real JB * -triple. When real JB * -triples are regarded as real forms of complex JB * -triples, the generalized Urysohn's lemmas proved by the second and the third authors of this note in [22] remain valid for real JB * -triples. Furthermore, an appropriate local Gelfand theory for single-generated real JB * -subtriples is also available in the real setting (cf.…”

“…Let F be a JB * -subtriple of a JB * -triple E and let e be a tripotent in F * * ≡F σ(E * * ,E * ) ⊆ E * * . Corollary 2.9 in [20] and [22,Proposition 3.3] prove that u is compact in F * * if, and only if, u is compact in E * * .…”

“…Proof. Since e and u are two orthogonal compact tripotents in E * * , it follows from [22,Proposition 3.7] that e ± u is a compact tripotent in E * * . It is easy to see that the sum and the difference of two orthogonal range tripotents in E * * is again a range tripotent in E * * .…”

A solution to Tingley’s problem for isometries between the unit spheres of compact C*-algebras and JB*-triples

“…The relation ≤ T is consistent with the natural partial order on the set of tripotents, that is, for any two tripotents e and u we have e ≤ u if and only if e ≤ T u. Following the same reference, a tripotent e in E * * satisfying that E * * 0 (e) ∩ E is weak * dense in E * * 0 (e) is called closed relative to E. The following characterization of compact tripotents in the second dual of a JB * -triple has been borrowed from [25,Theorem 2.6] (see also [26,Theorem 3.2]), and will be applied later. After having introduced the necessary concepts, the norm closed faces of the closed unit ball of a JB * -triple E can be characterized in terms of the compact tripotents in E * * via the next theorem, which due to Edwards, Fernández-Polo, Hoskin and the first author of this note (see [18]).…”

Inner ideals, compact tripotents and Čebyšëv subtriples of JB $$^{*}$$ ∗ -triples and C $$^*$$ ∗ -algebras

“…[16]). Theorem 2.6 in [16] (see also [19,Theorem 3.2]) asserts that a tripotent e in E * * is compact if, and only if, e is closed and bounded.…”