[proofplan] The map $\Psi_T$ factors as the composition $L^\infty(K, \mathcal{B}(K)) \to L^\infty(P) \xrightarrow{\Phi_P} \mathcal{L}(H)$, where the first arrow is the natural quotient — bounded Borel functions modulo $P$-null sets — and $\Phi_P$ is the integration map from the previous theorem **Integration Against $P$**. Each property reduces to a known property of $\Phi_P$ together with the previous theorem **Spectral Theorem for Normal Operators**: (i) the algebraic structure passes through $\Phi_P$, with the identity-on-spectrum function recovering $T$ by the spectral integration formula; (ii) the norm bound is the contraction property of the quotient $L^\infty \to L^\infty(P)$ followed by the isometry $\Phi_P$, with equality on $C(K)$ from the inverse Gelfand transform; (iii) reduces to the commutant identity in the previous theorem on commutative unital $C^*$-algebras once we use Fuglede–Putnam to pass between $T$-commutation and $f(T)$-commutation; (iv) follows from the fact that $*$-homomorphisms between unital $C^*$-algebras carry invertible elements to invertible elements. [/proofplan] [step:Define $\Psi_T$ as the composition $L^\infty(K, \mathcal{B}(K)) \to L^\infty(P) \xrightarrow{\Phi_P} \mathcal{L}(H)$] The natural map \begin{align*} \iota: L^\infty(K, \mathcal{B}(K)) \to L^\infty(P) \end{align*} sends a bounded Borel function $f$ on $K$ to its equivalence class modulo $P$-null sets (subsets $N$ with $P(N) = 0$). This is a unital $*$-homomorphism: addition, scalar multiplication, pointwise multiplication, and complex conjugation are all preserved by the quotient. Moreover $\iota$ is *norm-decreasing*: \begin{align*} \|\iota(f)\|_{L^\infty(P)} = \inf\{c \ge 0 : P(\{|f| > c\}) = 0\} \le \|f\|_{L^\infty(K)} = \sup_{\lambda \in K} |f(\lambda)|. \end{align*} The inequality holds because if $c = \|f\|_{L^\infty(K)}$, then $\{|f| > c\} = \varnothing$, hence $P(\{|f| > c\}) = 0$. By the previous theorem **Integration Against $P$**, there exists a unique isometric, unital $*$-homomorphism \begin{align*} \Phi_P: L^\infty(P) \to \mathcal{L}(H) \end{align*} satisfying $(\Phi_P(f) x, y)_H = \int_K f \, dP_{x,y}$. Define \begin{align*} \Psi_T := \Phi_P \circ \iota: L^\infty(K, \mathcal{B}(K)) \to \mathcal{L}(H), \qquad f \mapsto f(T) := \int_K f \, dP. \end{align*} [/step] [step:Establish property (i) — $\Psi_T$ is a unital $*$-homomorphism with $\mathrm{id}(T) = T$] *Algebraic structure.* Both $\iota$ and $\Phi_P$ are unital $*$-homomorphisms (Step 1 and theorem 2692), so the composition $\Psi_T$ is a unital $*$-homomorphism: for $f, g \in L^\infty(K)$ and $\alpha \in \mathbb{C}$, \begin{align*} \Psi_T(\alpha f + g) &= \alpha \Psi_T(f) + \Psi_T(g), \\ \Psi_T(fg) &= \Psi_T(f) \Psi_T(g), \\ \Psi_T(\bar{f}) &= \Psi_T(f)^*, \\ \Psi_T(1) &= I. \end{align*} *$\mathrm{id}(T) = T$.* The function $\mathrm{id}: K \to \mathbb{C}$, $\lambda \mapsto \lambda$, lies in $C(K) \subseteq L^\infty(K)$. By the previous theorem **Spectral Theorem for Normal Operators** applied to $T$, \begin{align*} T = \int_K \lambda \, dP(\lambda) = \int_K \mathrm{id}(\lambda) \, dP(\lambda) = \Phi_P(\iota(\mathrm{id})) = \Psi_T(\mathrm{id}) = \mathrm{id}(T). \end{align*} [/step] [step:Establish property (ii) — norm bound and equality on $C(K)$] *Bound for general $f \in L^\infty(K)$.* For $f \in L^\infty(K, \mathcal{B}(K))$, \begin{align*} \|f(T)\|_{\mathcal{L}(H)} = \|\Phi_P(\iota(f))\|_{\mathcal{L}(H)} = \|\iota(f)\|_{L^\infty(P)} \le \|f\|_{L^\infty(K)}, \end{align*} using that $\Phi_P$ is isometric (theorem 2692) and that $\iota$ is norm-decreasing (Step 1). *Equality on $C(K)$.* Let $f \in C(K)$. To show $\|f(T)\|_{\mathcal{L}(H)} = \|f\|_{L^\infty(K)}$, it suffices (given the upper bound just proved) to establish the lower bound $\|f(T)\|_{\mathcal{L}(H)} \ge \|f\|_{L^\infty(K)}$. Recall from the proof of the previous theorem **Spectral Theorem for Normal Operators** that the resolution of the identity $P$ on $\sigma(T) = K$ comes from pushing forward the resolution $P_A$ on the character space $K_A$ along the homeomorphism $\Theta: K_A \to K$, $\chi \mapsto \chi(T)$. Under this identification, the integration map $\Phi_P$ restricted to $C(K)$ equals the inverse Gelfand transform of the algebra $A := C^*(T, I)$: \begin{align*} \Phi_P|_{C(K)}: C(K) \to A \subseteq \mathcal{L}(H), \qquad f \mapsto \gamma(f \circ \Theta) = \Gamma^{-1}(f \circ \Theta), \end{align*} where $\Gamma: A \to C(K_A)$ is the Gelfand transform. By the [Commutative Gelfand-Naimark Theorem](/theorems/2689), $\Gamma$ is an isometric $*$-isomorphism, so $\Gamma^{-1}$ is also an isometric $*$-isomorphism. The pull-back $f \mapsto f \circ \Theta$ from $C(K)$ to $C(K_A)$ is an isometric $*$-isomorphism (being composition with a homeomorphism, which preserves the supremum norm). Composing two isometries gives an isometry. Hence \begin{align*} \|\Phi_P(f)\|_{\mathcal{L}(H)} = \|\Gamma^{-1}(f \circ \Theta)\|_{\mathcal{L}(H)} = \|f \circ \Theta\|_{C(K_A)} = \|f\|_{C(K)} = \|f\|_{L^\infty(K)}, \end{align*} since the supremum and the $L^\infty$ norm coincide for continuous functions on a compact space. (Strictly: $\|f\|_{L^\infty(K)} = \sup_{\lambda \in K} |f(\lambda)| = \|f\|_{C(K)}$ for $f \in C(K)$, where the $L^\infty$ norm is the supremum norm; there is no a.e.\ ambiguity for continuous functions.) Hence $\|f(T)\|_{\mathcal{L}(H)} = \|f\|_{L^\infty(K)}$ for $f \in C(K)$. [/step] [step:Establish property (iii) — commutant identity] Let $S \in \mathcal{L}(H)$. We prove the equivalence \begin{align*} ST = TS \iff S \circ f(T) = f(T) \circ S \text{ for all } f \in L^\infty(K). \end{align*} *$(\Leftarrow)$.* Apply the assumption to $f = \mathrm{id}$. By Step 2, $\mathrm{id}(T) = T$, so $ST = TS$. *$(\Rightarrow)$.* Suppose $ST = TS$. By the previous theorem **Spectral Theorem for Normal Operators**, the commutant identity for $T$ gives: $S$ commutes with $T$ if and only if $S$ commutes with $P(E)$ for all $E \in \mathcal{B}(K)$. Hence $S P(E) = P(E) S$ for all $E$. By property (iii) of the previous theorem **Integration Against $P$** applied to the integration map $\Phi_P: L^\infty(P) \to \mathcal{L}(H)$: $S$ commutes with $\Phi_P(g)$ for every $g \in L^\infty(P)$. Specialising $g = \iota(f)$ for $f \in L^\infty(K)$: $S \Phi_P(\iota(f)) = \Phi_P(\iota(f)) S$, i.e.\ $S f(T) = f(T) S$. This proves both directions. [/step] [step:Establish property (iv) — spectral inclusion] We show $\sigma(f(T)) \subseteq \overline{f(K)}$ for every $f \in L^\infty(K, \mathcal{B}(K))$. The closure $\overline{f(K)}$ in $\mathbb{C}$ is the *essential range* of $f$ when $f$ is continuous; for general bounded Borel $f$, the relevant closed set is the essential range with respect to $P$, which is contained in $\overline{f(K)}$. We work with the form $\sigma(f(T)) \subseteq \overline{f(K)}$ as stated. Suppose $\mu \in \mathbb{C} \setminus \overline{f(K)}$. Then there exists $\delta > 0$ such that $|\mu - f(\lambda)| \ge \delta$ for all $\lambda \in K$. Hence the function \begin{align*} g: K &\to \mathbb{C} \\ \lambda &\mapsto \frac{1}{f(\lambda) - \mu} \end{align*} is well-defined, bounded by $1/\delta$, and Borel measurable (the reciprocal of a non-vanishing bounded Borel function is bounded and Borel). Hence $g \in L^\infty(K, \mathcal{B}(K))$. Apply the unital $*$-homomorphism $\Psi_T$ from Step 2: \begin{align*} \Psi_T(g) \cdot \Psi_T(f - \mu) = \Psi_T(g \cdot (f - \mu)) = \Psi_T(1) = I. \end{align*} Similarly $\Psi_T(f - \mu) \cdot \Psi_T(g) = I$. Hence $\Psi_T(f - \mu) = f(T) - \mu I$ is invertible in $\mathcal{L}(H)$ with inverse $g(T)$, so $\mu \notin \sigma(f(T))$. By contrapositive, $\sigma(f(T)) \subseteq \overline{f(K)}$. *Spectral mapping for continuous $f$.* For $f \in C(K)$, $\overline{f(K)} = f(K)$ since $f(K)$ is compact (continuous image of a compact set), hence closed. Combined with the inclusion just proved, $\sigma(f(T)) \subseteq f(K)$. For the reverse inclusion in the continuous case: let $\mu = f(\lambda_0)$ for some $\lambda_0 \in K$. We show $\mu \in \sigma(f(T))$. By the previous theorem on commutative unital $C^*$-algebras applied to the algebra $A = C^*(T, I)$: $\mu \in \sigma_A(f(T)) = \sigma(f(T))$ (the second equality from [Spectral Inclusions and Permanence](/theorems/2688) part (3) for the normal element $f(T) \in A$ — note $f(T)$ is normal as a function of the normal $T$, since $\Psi_T$ is a $*$-homomorphism and $f, \bar{f}$ commute pointwise). The Gelfand transform identifies $\sigma_A(f(T))$ with the image of $f(T)$ under all characters of $A$, and the character space of $A$ is $K = \sigma(T)$, with the character $\chi_\lambda(T) = \lambda$ giving $\chi_\lambda(f(T)) = f(\lambda)$ (because $\Psi_T$ extends the inverse Gelfand transform and characters intertwine functional calculi for continuous functions). Hence $\sigma_A(f(T)) = \{f(\lambda) : \lambda \in K\} = f(K)$, giving $\mu \in f(K) \implies \mu \in \sigma(f(T))$. Combining, $\sigma(f(T)) = f(K)$ for $f \in C(K)$. [/step] [step:Wrap up] Properties (i)–(iv) have been established. The map \begin{align*} \Psi_T: L^\infty(K, \mathcal{B}(K)) \to \mathcal{L}(H), \quad f \mapsto f(T) = \int_K f \, dP, \end{align*} is a unital $*$-homomorphism (Step 2) extending the inverse Gelfand transform on $C(K)$ (Step 3 — equality of norms and the explicit identification with $\Gamma^{-1}$), bounded with $\|f(T)\| \le \|f\|_{L^\infty(K)}$ (Step 3), satisfying the commutant identity (Step 4), and respecting the spectrum via $\sigma(f(T)) \subseteq \overline{f(K)}$ (Step 5). This is the **Borel Functional Calculus** for the normal operator $T$. [/step]