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Let $f: M_n \\rightarrow [0,\\infty)$ be a continuous map such that $f(\\mu UAU^*)= f(A)$ for any complex unit $\\mu$, $A \\in M_n$ and unitary $U \\in M_n$, $f(X)=0$ if and only if $X=0$ and the induced map $t \\mapsto f(tX)$ is monotonic increasing on $[0,\\infty)$ for any rank 1 nilpotent $X \\in M_n$. Characterizations are given for surjective maps $\\phi$ on $M_n$ satisfying $f(AB-BA) = f(\\phi(A)\\phi(B)-\\phi(B)\\phi(A))$. 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