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We prove that if $A$ and $B$ are $C^*$-algebras, then $A\\subset _{\\sigma \\Delta } B$ if and only if there exists an onto $*$-homomorphism $\\theta :B\\otimes \\mathcal K \\rightarrow A\\otimes \\mathcal K,$ where $\\mathcal K$ is the set of compact operators acting on an infinite dimensional separable Hilbert space. 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