Surjective Fischer--Muszély maps on positive cones of JB-algebras
Abstract
We prove that every surjective Fischer--Muszély map between positive cones of arbitrary JB-algebras is additive and positively homogeneous, and extends uniquely to a bounded positive linear surjection. In particular, this gives a complete affirmative answer to Molnár's problem for arbitrary C*-algebras. The theorem includes nonunital and exceptional JB-algebras and assumes neither injectivity nor continuity. The proof connects the norm-valued functional equation to positive-cone rigidity through translation invariance of an induced pseudometric. For a surjective map satisfying the Fischer--Muszély identity with uniform error ε, we also construct a unique bounded positive linear map L at uniform distance at most 3ε. The set L(A+) is dense in B+, although L need not be surjective, even when the original map is continuous and bijective. For bijective FM maps, we obtain weighted Jordan representations on the biduals. We also extend the characterization of surjective norm-sum preservers to JB-algebras and characterize norm arithmetic-midpoint and harmonic-mean identities on positive invertible cones of unital JB-algebras.
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BibTeX
@misc{hatori2026surjective,
title = {{Surjective Fischer--Muszély maps on positive cones of JB-algebras}},
author = {Osamu Hatori and Shiho Oi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19668},
eprint = {2609.19668},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19668},
}
RIS
TY - UNPB
TI - Surjective Fischer--Muszély maps on positive cones of JB-algebras
AU - Osamu Hatori
AU - Shiho Oi
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19668
UR - https://arxiv.org/abs/2609.19668
AB - We prove that every surjective Fischer--Muszély map between positive cones of arbitrary JB-algebras is additive and positively homogeneous, and extends uniquely to a bounded positive linear surjection. In particular, this gives a complete affirmative answer to Molnár's problem for arbitrary C^*-algebras. The theorem includes nonunital and exceptional JB-algebras and assumes neither injectivity nor continuity. The proof connects the norm-valued functional equation to positive-cone rigidity through translation invariance of an induced pseudometric. For a surjective map satisfying the Fischer--Muszély identity with uniform error ε, we also construct a unique bounded positive linear map L at uniform distance at most 3ε. The set L(A_+) is dense in B_+, although L need not be surjective, even when the original map is continuous and bijective. For bijective FM maps, we obtain weighted Jordan representations on the biduals. We also extend the characterization of surjective norm-sum preservers to JB-algebras and characterize norm arithmetic-midpoint and harmonic-mean identities on positive invertible cones of unital JB-algebras.
ER -