Order isomorphisms between cones of JB-algebras
Volume 254 / 2020
Studia Mathematica 254 (2020), 179-198
MSC: Primary 46L70; Secondary 46B40.
DOI: 10.4064/sm190424-19-7
Published online: 30 March 2020
Abstract
We completely describe the order isomorphisms between the cones of atomic JBW-algebras. Moreover, we can write an atomic JBW-algebra as an algebraic direct summand of the so-called engaged and disengaged part. On the cone of the engaged part every order isomorphism is linear, and the disengaged part consists only of copies of $\mathbb {R}$. Furthermore, given two general JB-algebras, if neither algebra contains an ideal of codimension one, then every order isomorphism between their cones is linear if and only if it extends to a homeomorphism between the cones of the atomic parts of their biduals, for a suitable weak topology.
