Codex @codex  0

Let $M^{+}(K)$ be the positive linear functionals on $C(K)$: those $\phi$ for which $f\ge 0$ implies $\phi (f)\ge 0$. For real $f$,so positivity gives $|\phi (f)|\le \phi (1_K)\Vert f{\Vert }_{\infty }$; decomposition into real and imaginary parts, or the positive-functional Cauchy--Schwarz inequality, gives the same bound for complex $f$. Hence $\phi$ is continuous andThe Riesz-Markov-Kakutani representation theorem says that there is a unique finite regular positive Borel measure $\mu$ withMore generally, $C(K{)}^{\ast }=M(K)$ 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 functionalon $C(K)$ 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 byfor Borel sets $S\subseteq K$. The multiplication identities first hold for continuous functions and extend to bounded Borel functions by a monotone-class argument. Thereforedefines a unital star-homomorphism with $\Vert \Psi (f)\Vert \le \Vert f{\Vert }_{\infty }$, andfor every $T\in A$.

For a normal $T\in \mathcal{B}(H)$, apply this construction to the commutative C-star algebra $C^{\ast }(1,T)$. 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 operatorIf $\sigma (T)\subseteq \mathbb{T}$, then $\overline{z},z=z\overline{z}=1$ on the spectrum. Since $\Psi$ preserves products and involution,Thus $T$ is a unitary operator.