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\begin{document}
\begin{eqnarray*}
|\phi(ab) - \phi(a)\phi(b)|^2=  |\phi((a - \phi(a)) b)|^2\\
\leq \phi ((a - \phi(a))^2)\phi (b^*b)\\
=\phi(a^2 - 2\phi(a)a + \phi(a)^2)\phi(b^*b)\\
=[\phi(a^2) - 2\phi(a)^2 + \phi(a)^2\phi(1)] \phi(b^*b)\\
=0
\end{eqnarray*}
\end{document}