= Solution
Let $M^+(K)$ be the positive linear functionals on $C(K)$: those $\varphi$ for which $f\geq0$ implies $\varphi(f)\geq0$. For real $f$,
$$
-\|f\|_\infty1_K\leq f\leq\|f\|_\infty1_K,
$$
so positivity gives $|\varphi(f)|\leq\varphi(1_K)\|f\|_\infty$; decomposition into real and imaginary parts, or the positive-functional Cauchy--Schwarz inequality, gives the same bound for complex $f$. Hence $\varphi$ is continuous and
$$
\boxed{\|\varphi\|=\varphi(1_K)}.
$$
The <Riesz-Markov-Kakutani representation theorem> says that there is a unique finite regular positive Borel measure $\mu$ with
$$
\varphi(f)=\int_Kf\,d\mu.
$$
More generally, $C(K)^*=M(K)$ is the Banach space of finite regular complex Borel measures with the total-variation norm.
Now let $A\subseteq\mathcal B(H)$ be commutative, unital, and C-star, and put $K=\Phi_A$. The <Gelfand transform> is an isometric star-isomorphism $A\cong C(K)$. For $\xi,\eta\in H$, the functional
$$
\widehat T\longmapsto\langle T\xi,\eta\rangle
$$
on $C(K)$ is represented by a regular complex measure $\mu_{\xi,\eta}$. The diagonal measures are positive. Polarization and the Riesz theorem assemble them into a projection-valued measure $E$ characterized by
$$
\langle E(S)\xi,\eta\rangle=\mu_{\xi,\eta}(S)
$$
for Borel sets $S\subseteq K$. The multiplication identities first hold for continuous functions and extend to bounded Borel functions by a monotone-class argument. Therefore
$$
\Psi(f)=\int_Kf\,dE,
\qquad f\in L^\infty(K),
$$
defines a unital star-homomorphism with $\|\Psi(f)\|\leq\|f\|_\infty$, and
$$
\Psi(\widehat T)=T
$$
for every $T\in A$.
For a normal $T\in\mathcal B(H)$, apply this construction to the commutative C-star algebra $C^*(1,T)$. Its character space identifies with $\sigma(T)$, and the <Gelfand transform> of $T$ is the coordinate function $z(\lambda)=\lambda$. We obtain the <Borel functional calculus for a normal operator>
$$
\Psi:L^\infty(\sigma(T))\to\mathcal B(H),
\qquad
\Psi(z)=T.
$$
If $\sigma(T)\subseteq\mathbb T$, then $\overline z,z=z\overline z=1$ on the spectrum. Since $\Psi$ preserves products and involution,
$$
T^*T=\Psi(\overline z)\Psi(z)=I,
\qquad
TT^*=I.
$$
Thus $T$ is a <Unitary element of a C-star algebra>[unitary operator].
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