Order theoretical structures in atomic JBW-algebras: disjointness, bands, and centres
Functional Analysis
2023-05-31 v1
Abstract
Every atomic JBW-algebra is known to be a direct sum of JBW-algebra factors of type I. Extending Kadison's anti-lattice theorem, we show that each of these factors is a disjointness free anti-lattice. We characterise disjointness, bands, and disjointness preserving bijections with disjointness preserving inverses in direct sums of disjointness free anti-lattices and, therefore, in atomic JBW-algebras. We show that in unital JB-algebras the algebraic centre and the order theoretical centre are isomorphic. Moreover, the order theoretical centre is a Riesz space of multiplication operators. A survey of JBW-algebra factors of type I is included.
Cite
@article{arxiv.2305.19086,
title = {Order theoretical structures in atomic JBW-algebras: disjointness, bands, and centres},
author = {Onno van Gaans and Anke Kalauch and Mark Roelands},
journal= {arXiv preprint arXiv:2305.19086},
year = {2023}
}