Finite tripotents and finite JBW$^*$-triples

TL;DR

This paper explores the structure of tripotents in JBW*-triples, introduces a notion of finiteness for these tripotents, and analyzes their properties and decompositions, extending concepts from von Neumann algebra theory.

Contribution

It introduces a new notion of finiteness for tripotents in JBW*-triples and provides a detailed analysis of preorders and decompositions related to these elements.

Findings

Defined and studied two preorders on tripotents in JB*-triples.

Introduced and analyzed the concept of finiteness for tripotents in JBW*-triples.

Provided a decomposition of JBW*-triples into finite and infinite parts.

Abstract

We study two natural preorders on the set of tripotents in a JB$^*$-triple defined in terms of their Peirce decomposition and weaker than the standard partial order. We further introduce and investigate the notion of finiteness for tripotents in JBW$^*$-triples which is a natural generalization of finiteness for projections in von Neumann algebras. We analyze the preorders in detail using the standard representation of JBW$^*$-triples. We also provide a refined version of this representation - in particular a decomposition of any JBW$^*$-triple into its finite and properly infinite parts. Since a JBW$^*$-algebra is finite if and only if the extreme points of its unit ball are just unitaries, our notion of finiteness differs from the concept of modularity widely used in Jordan structures so far. The exact relationship of these two notions is clarified in the last section.