Additive mappings on operator algebras preserving absolute values Original Research Article

It is shown that an additive map phi:B(H) → B(K) is the sum of two *-homomorphisms, one of which is image -linear and the other is image -antilinear provided that 1. [(a)] phi(A)=phi(A) for all Aset membership, variantB(H), 2. [(b)] phi(I) is an orthogonal projection, and 3. [(c)] phi(iI)Ksubset ofphi(I)K. The structure of phi is more refined when it is injective. The paper also studies the properties of phi in the absence of condition (b). Here, B(H) and B(K) denote the algebras of all (bounded linear) operators on Hilbert spaces H and K, respectively. These extend a result of L. Molnár [Bull Austral. Math. Soc. 53 (1996) 391] saying an additive map phi:B(H) → B(H) is a constant multiple of an either image -linear or image -antilinear *-homomorphism provided that 1. [(a′)] phi(A)=phi(A) for all Aset membership, variantB(H), and 2. [(b′)] phi(B(H)) contains all finite-rank operators.