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Peralta, Mohsen Niazi","submitted_at":"2014-12-05T08:40:22Z","abstract_excerpt":"We prove that, for a complex Hilbert space $H$ with dimension bigger or equal than three, every linear mapping $T: B(H)\\to B(H)$ satisfying the 3-local property is a $^*$-monomorphism, that is, every linear mapping $T: B(H) \\to B(H)$ satisfying that for every $a$ in $B(H)$ and every $\\xi,\\eta$ in $H$, there exists a $^*$-automorphism $\\pi_{a,\\xi,\\eta}: B(H)\\to B(H)$, depending on $a$, $\\xi$, and $\\eta$, such that $$T(a) (\\xi) = \\pi_{a,\\xi,\\eta} (a) (\\xi), \\hbox{ and } T(a) (\\eta) = \\pi_{a,\\xi,\\eta} (a) (\\eta),$$ is a $^*$-monomorphism. This solves a question posed by L. 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Peralta, Mohsen Niazi","submitted_at":"2014-12-05T08:40:22Z","abstract_excerpt":"We prove that, for a complex Hilbert space $H$ with dimension bigger or equal than three, every linear mapping $T: B(H)\\to B(H)$ satisfying the 3-local property is a $^*$-monomorphism, that is, every linear mapping $T: B(H) \\to B(H)$ satisfying that for every $a$ in $B(H)$ and every $\\xi,\\eta$ in $H$, there exists a $^*$-automorphism $\\pi_{a,\\xi,\\eta}: B(H)\\to B(H)$, depending on $a$, $\\xi$, and $\\eta$, such that $$T(a) (\\xi) = \\pi_{a,\\xi,\\eta} (a) (\\xi), \\hbox{ and } T(a) (\\eta) = \\pi_{a,\\xi,\\eta} (a) (\\eta),$$ is a $^*$-monomorphism. This solves a question posed by L. 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