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Order isomorphisms between cones of JB-algebras

Volume 254 / 2020

Hendrik van Imhoff, Mark Roelands 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.

Authors

  • Hendrik van ImhoffMathematical Institute
    Leiden University
    P.O. Box 9512
    2300 RA Leiden, The Netherlands
    e-mail
  • Mark RoelandsSchool of Mathematics
    Statistics & Actuarial Science
    University of Kent
    Canterbury, CT2 7NX, United Kingdom
    e-mail

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