Rank-preserving multiplicative maps on \(B(X)\) (Q1347933)

Let \(X\) (resp., \(H\)) be a Banach space (resp., Hilbert space). By \({\mathcal B}(X)\) (resp., \({\mathcal B}(H))\) denote the algebra of all bounded linear operators on \(X\) (resp., \(H\)). The authors describe the structure of multiplicative maps preserving rank on \({\mathcal B}(X)\). Furthermore, they prove that every multiplicative local approximate automorphism of \({\mathcal B}(X)\) the range of which contains all rank-1 idempotents is in fact an automorphism. They also characterize multiplicative maps preserving co-rank on \({\mathcal B}(H)\) and prove that a multiplicative map \(\Phi :{\mathcal B}(H)\rightarrow {\mathcal B}(K)\) the range of which contains all rank-1 projections preserves the property \(A^*B=0\Leftrightarrow \Phi (A)^*\Phi (B)=0\) for any \(A\), \(B\in {\mathcal B}(H)\) if and only if there exists a unitary or conjugate linear unitary operator \(U\in {\mathcal B}(H, K)\) such that \(\Phi (A)=UAU^*\) for all \(A\in {\mathcal B}(H)\).