Additive mappings on C*-algebras preserving absolute values

Abstract

Let H and K be two complex Hilbert spaces and 𝒜 ⊂ B(H) and ℬ ⊂ B(K) be two unital C*-algebras. It is shown that if φ : 𝒜 → ℬ is an additive surjective mapping satisfying φ(|A|) = |φ(A)| for every A ∈ 𝒜 and φ(I) is a projection, then the restriction of mapping φ to both 𝒜_s and 𝒜_sk is a Jordan *-homomorphism onto corresponding set in ℬ, where 𝒜_s and 𝒜_sk denote the set of all self-adjoint and skew-self-adjoint elements, respectively. Furthermore, if ℬ is a C*-algebra of real-rank zero then φ is a ℂ-linear or ℂ-antilinear *-homomorphism.

Keywords:

• ℂ-linear
• ℂ-antilinear
• homomorphism
• linear preserver problem
• real-rank zero

AMS Subject Classifications::

• primary: 47B49
• secondary: 46L05, 47L30

Acknowledgements

This research is partially supported by the Research Center in Algebraic Hyperstructures and Fuzzy Mathematics, University of Mazandaran, Babolsar, Iran.