Gelf‎and ‎representation ‎Theorem

Question

In ‎proof ‎of "‎‎Gelf‎and ‎representation ‎Theorem‎" ‎(see 1.3.6 Theorem of Murphy's book )‎, I ‎am ‎understanding ‎that ‎why ‎the ‎map $$ A ‎‎\rightarrow‎ ‎C_{0}(‎\Omega(A)‎)~ , ‎~‎‎a‎‎\rightarrow‎ ‎\widehat{a}$$ ‎‎is a‎ ‎norm-decreasing ‎homomorphism. ‎‎

Answer 1

$$ \widehat{ab}(\tau) = \tau(ab) = \tau(a)\tau(b) = \hat{a}(\tau)\hat{b}(\tau), $$ $$\lVert \hat{a} \rVert_\infty = r(a) = \inf_{n \in \mathbb{N}} \lVert a^n\rVert^{1/n} \leq \lVert a \rVert.$$