Order type relations on the set of tripotents in a JB$^*$-triple
Jan Hamhalter, Ondřej F. K. Kalenda, Antonio M. Peralta
TL;DR
The paper develops a systematic framework for order-type relations on tripotents in JB$^*$-triples, introducing and analyzing several intermediate relations between the standard partial order and the natural order $≤_2$, notably $≤_h$ and $≤_n$ and their transitive hulls. It leverages the structure theory of JB$^*$-triples and their summands (Cartan factors, spins, and von Neumann-like components) to derive characterizations, transitivity results, and chain-length bounds, with sharp distinctions between finite and infinite settings. Key contributions include translations of these relations into concrete C$^*$-algebraic criteria, determinant-type invariants for unitary tripotents in finite-dimensional Cartan factors, and a comprehensive treatment across JBW$^*$- and von Neumann algebra contexts, identifying where transitivity holds and where it fails. The work connects order relations on tripotents to deep structural properties such as finiteness, type decomposition, and tensor product representations, providing both general results and detailed case analyses for spin factors and exceptional Cartan factors, and it highlights open problems for continuous summands and certain infinite-dimensional settings. This framework advances understanding of the interplay between algebraic, geometric, and order-theoretic aspects of JB$^*$-triples and their operator-algebraic realizations.
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.