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

TL;DR

This paper explores order relations among tripotents in JB*-triples, analyzing their properties, transitive hulls, and connections to von Neumann algebra types and structural properties of JBW*-triples.

Contribution

It introduces and compares new order relations on tripotents, examines their transitive hulls, and links these properties to von Neumann algebra types and structural aspects of JBW*-triples.

Findings

Relations $_h$ and $_n$ are not necessarily transitive.

Transitive hulls of these relations are studied and characterized.

Connections established between order relations and von Neumann algebra types.

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 $\le_h$ and $\le_n$. We say that $u\le_h e$ (or $u\le_n e$) 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.