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Suppose $F(\\cdot )$ is a unitary invariant norm, the pseudo spectra, the pseudo spectral radius, the $C$-numerical range, or the $C$-numerical radius for some finite rank operator $C$. The structure is determined for surjective maps $\\Phi :{\\mathcal A}\\rightarrow \\mathcal B$ satisfying $F(A^*B)=F(\\Phi (A)^*\\Phi (B))$ for all $A, B \\in {\\mathcal A}$. 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