Let A and B be C^-algebras, and let T:A_+→ B_+ be surjective and satisfy the Fischer--Muszély identity |T(a+b)|=|T(a)+T(b)| for all a,b∈ A_+. Must T be additive and positively homogeneous, hence extend uniquely to a bounded positive linear surjection? In the bijective setting, must T moreover have the corresponding Jordan-isomorphism form?

Molnár's problem on Fischer–Muszély maps

Molnár’s problem asks whether surjective maps on positive cones satisfying the Fischer–Muszély norm equation $\|T(a+b)\|=\|T(a)+T(b)\|$ must be additive and have a Jordan-isomorphism form.

Known results

Unital $C^{*}$ -algebras: a March 2024 preprint claimed a positive solution, with maps extending to Jordan $*$ -isomorphisms.
Commutative $C^{*}$ -algebras: additivity and positive homogeneity were proved in November 2025.
General noncommutative $C^{*}$ -algebras were explicitly described as open in November 2025.
A June 2026 preprint claimed the required form under bijectivity, plus a conditional surjective result.

September 2026 JB-algebra preprint

Hatori and Oi claim a theorem for arbitrary $C^{*}$ -algebras via JB-algebras, removing injectivity and continuity assumptions and covering nonunital and exceptional cases. The result is a claimed resolution, not yet independently verified.

Current status (as of September 2026): A September preprint claims the general problem is solved, but independent verification is pending.

Molnár's problem on Fischer–Muszély maps

Primary source

Additional references

Progress summary

Solutions 0