Exposed faces of the unit ball in a JBW*-triple

Measures of weak non-compactness in preduals of von Neumann algebras and JBW⁎-triples

“…Let p = sup n p n . Then p is σ-finite (see, e.g., [24,Theorem 3.4] or [10, Lemma 3.5]) and B ⊆ P 2 (p) * M * . Since P 2 (p) is a norm-one projection, an application of Lemma 2.1 shows that ω(B) = ω M2(p) (B).…”

“…Given a norm-one element a in a JBW * -triple M , there exists a smallest tripotent e ∈ M satisfying that a is a positive element in the JBW * -algebra M 2 (e), this tripotent is called the range tripotent of a, and it will be denoted by r(a) (see, for example, [23, comments before Proof. Assertion (a) is proved in [24,Theorem 3.2]. Statement (b) follows from the fact that any projection is also a tripotent, and from the property that a tripotent u is smaller than or equal to a projection p if and only if u is a projection and u ≤ p (cf.…”

Preduals of JBW*-triples are 1-Plichko spaces

“…Since M 2 (e) is a weak * -closed subtriple of M , assertion (i) follows from [19, Lemma 3.6(ii)]. Assertion (ii) follows from (i), [19,Theorem 4.4 (viii)-(ix)] and the fact that e is a complete tripotent in M 2 (e). Finally, assertion (iii) follows immediately from (i) and (ii).…”

“…Following standard notation, we shall say that M is σ-finite if every tripotent in M is σ-finite, equivalently, every orthogonal subset of tripotents in M is countable (cf. [19,Proposition 3.1]). It is also known that the sum of an orthogonal countable family of mutually orthogonal σ-finite tripotents in M is again a σ-finite tripotent (see [19, We will need the following properties of σ-finite tripotents which have been borrowed from [19].…”

Involutive gradings of JBW*-triple factors

“…Then, by Lemma 6.1, using [24,36], the increasing net ( j ∈ 0 v jj ) 0 ∈ , where is the directed set of finite subsets of , converges in the weak * -topology, and, hence, in the strong topology, to its supremum v in U(A). Notice that, by orthogonality, for 0 in ,…”