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Peralta, Mohsen Niazi","submitted_at":"2015-03-04T15:42:30Z","abstract_excerpt":"We introduce the notion of weak-2-local derivation (respectively, $^*$-derivation) on a C$^*$-algebra $A$ as a (non-necessarily linear) map $\\Delta : A\\to A$ satisfying that for every $a,b\\in A$ and $\\phi\\in A^*$ there exists a derivation (respectively, a $^*$-derivation) $D_{a,b,\\phi}: A\\to A$, depending on $a$, $b$ and $\\phi$, such that $\\phi \\Delta (a) = \\phi D_{a,b,\\phi} (a)$ and $\\phi \\Delta (b) = \\phi D_{a,b,\\phi} (b)$. We prove that every weak-2-local $^*$-derivation on $M_n$ is a linear derivation. 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Peralta, Mohsen Niazi","submitted_at":"2015-03-04T15:42:30Z","abstract_excerpt":"We introduce the notion of weak-2-local derivation (respectively, $^*$-derivation) on a C$^*$-algebra $A$ as a (non-necessarily linear) map $\\Delta : A\\to A$ satisfying that for every $a,b\\in A$ and $\\phi\\in A^*$ there exists a derivation (respectively, a $^*$-derivation) $D_{a,b,\\phi}: A\\to A$, depending on $a$, $b$ and $\\phi$, such that $\\phi \\Delta (a) = \\phi D_{a,b,\\phi} (a)$ and $\\phi \\Delta (b) = \\phi D_{a,b,\\phi} (b)$. We prove that every weak-2-local $^*$-derivation on $M_n$ is a linear derivation. 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