Additive mappings on operator algebras preserving absolute values (Q5935371)

It is shown that an additive map \(f: B(H)\to B(K)\) is the sum of two *-homomorphisms, one of which is \(\mathbb{C}\)-linear and the other is \(\mathbb{C}\)-antilinear provided thatNEWLINENEWLINENEWLINE(a) \(|f(A)|= f(|A|)\) for all \(A\in B(H)\),NEWLINENEWLINENEWLINE(b) \(f(I)\) is an orthogonal projection, andNEWLINENEWLINENEWLINE(c) \(f(iI)K\subset f(I)K\).NEWLINENEWLINENEWLINEThe structure of \(f\) is more refined when it is injective. The paper also studies the properties of \(f\) 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.