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arXiv · 2305.19086

Order theoretical structures in atomic JBW-algebras: disjointness, bands, and centres

Onno van Gaans·Anke Kalauch·Mark Roelands
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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.

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Onno van Gaans, Anke Kalauch, Mark Roelands. 2023-05-30. Order theoretical structures in atomic JBW-algebras: disjointness, bands, and centres. https://arxiv.org/abs/2305.19086

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