Additive mappings on operator algebras preserving square absolute values (Q2764625)

Let \({\mathcal B}(H)\) and \({\mathcal B}(K)\) denote the algebras of all bounded linear operators on Hilbert space \(H\) and \(K\), respectively. The author shows that if \(\varphi: {\mathcal B}(H)\to {\mathcal B}(K)\) is an additive mapping satisfying \(\varphi(|A|^2)=|\varphi(A)|^2\) and \(\psi(A)=\varphi(I)\varphi(A)\) for each \(A\in{\mathcal B}(H)\), then \(\psi\) is the sum of two *-homomorphisms with that one of which is \(C\)-linear and another is \(C\)-antilinear, where a map \(\varphi\) is a \(^*\)-homomorphism means that \(\varphi:{\mathcal B}(H)\to {\mathcal B}(K)\), \(\varphi\) preserves the ring structure and \(\varphi(A^*)=\varphi(A)^*\) for every \(A\in {\mathcal B}(H)\). The author also gives some conditions which imply the injectivity and the rank-preserving of \(\psi\).