Order type relations on the set of tripotents in a JB$^*$-triple

Abstract

We introduce, investigate and compare several order type relations on the set of tripotents in a JB * -triple. The main two relations we address are ≤ h and ≤n. We say that u ≤ h e (or u ≤n e) if u is a self-adjoint (or normal) element of the Peirce-2 subspace associated to e considered as a unital JB *algebra with unit e. It turns out that these relations need not be transitive, so we consider their transitive hulls as well. Properties of these transitive hulls appear to be closely connected with types of von Neumann algebras, with the results on products of symmetries, with determinants in finite-dimensional Cartan factors, with finiteness and other structural properties of JBW * -triples.2010 Mathematics Subject Classification. 17C65, 17C27, 17C10, 06A99, 46L10. Key words and phrases. JB * -triple, JBW * -triple, relations on tripotents, von Neumann algebra, partial isometries, products of symmetries.

Preservers of Triple Transition Pseudo-Probabilities in Connection with Orthogonality Preservers and Surjective Isometries

“…The natural partial order among partial isometries in B(H) and, more generally, among tripotents in a JB * -triple E is defined by e ≤ u in U(E) if u−e is a tripotent and u − e ⊥ e. This partial order is precisely the order considered by L. Molnár in [37], and it is a central notion in the theory of JB * -triples (cf., for example, the recent papers [24][25][26][27][28][29]). Thanks to the partial ordering we can consider tripotents which are minimal with respect to this ordering.…”

Maps preserving triple transition pseudo-probabilities

“…Building upon the relation "being orthogonal" we can define a canonical order "≤" on tripotents in E given by e ≤ u if and only if u − e is a tripotent and u − e ⊥ e. This partial ordering is precisely the order consider by L. Molnár in Theorem 1.2, and it provides an important tool in JB * -triples (see, for example, the recent papers [25,26,22,23,21,24] where it plays an important role). The partial order in U(E) enjoys several interesting properties; for example, e ≤ u if and only if e is a projection in the JB * -algebras E 2 (e) (cf.…”

Preservers of triple transition pseudo-probabilities in connection with orthogonality preservers and surjective isometries

“…The natural partial order among partial isometries in B(H) and, more generally, among tripotents in a JB * -triple E is defined by e ≤ u in U(E) if u − e is a tripotent and u − e ⊥ e. This partial order is precisely the order consider by L. Molnár in [37], and it is a central notion in the theory of JB * -triples (cf., for example, the recent papers [28,29,25,26,24,27]). Thanks to the partial ordering we can consider tripotents which are minimal with respect to this ordering.…”