Proof. Firstly, let $x, y \in H$, $\seqf{B_j}\subset \cb_{X}$ be disjoint Borel sets, and $B = \bigsqcup_{j = 1}^{n} B_{j}$, then for each $1 \le i < j \le n$, $E(B_{i})(H) \perp E(B_{j})(H)$, so by the Cauchy-Schwarz inequality and the Pythagorean Theorem,

As the above holds for all finite sequences of disjoint Borel sets, $\norm{E_{x, y}}_{C(X; \complex)^*}\le \norm{x}_{H} \norm{y}_{H}$. Thus for any $\phi \in C(X; \complex)^{**}$,

Since the above holds for all $x, y \in H$, $I_{E}(\phi) \in B(H)$ with $\norm{I_E(\phi)}_{B(H)}\le \norm{\phi}_{C(X; \complex)^{**}}$.

(1): For each $x, y \in H$, $E_{x, y}\in C(X; \complex)^{*}$. Since $\angles{\int \phi dE \cdot x, y}_{H}= \dpn{E_{x, y}, \phi}{C(X; \complex)^*}$ for every $\phi \in C(X; \complex)^{**}$, $I_{E}$ is continuous from the weak* topology on $C(X; \complex)^{**}$ to the weak operator topology on $B(H)$.

(2): By Lemma 26.6.4, the simple functions $\Sigma(X; \complex)$ are uniformly dense in the bounded Borel functions $B^{\infty}(X; \complex)$. Since

1. (i)

   $I_{E}$ restricted to $\Sigma(X; \complex)$ is a *-homomorphism.

2. (ii)

   Multiplication and conjugation are continuous in the uniform norm on $B^{\infty}(X; \complex)$

3. (iii)

   Composition and transposition are continuous in the operator norm on $B(H)$

the map $I_{E}$ restricted to $B^{\infty}(X; \complex)$ is a *-homomorphism by continuity. By Goldstine’s Theorem, $C(X; \complex) \subset B^{\infty}(X; \complex)$ is weak*-dense in $C(X; \complex)^{**}$. So as

1. (i)

   $I_{E}$ restricted to $B^{\infty}(X; \complex)$ is a *-homomorphism.

2. (ii)

   The involution $\phi \mapsto \ol \phi$ is weak*-continuous on $C(X; \complex)^{**}$.

3. (iii)

   The transpose $T \mapsto T^{*}$ is weak-operator continuous on $B(H)$.

4. (iv)

   The product $(\phi, \psi) \mapsto \phi \psi$ is separately weak*-continuous on $C(X; \complex)^{**}$.

5. (v)

   The composition $(S, T) \mapsto ST$ is separately weak-operator continuous on $B(H)$.

the map $I_{E}$ is a *-homomorphism by (1). Finally, since $E(X) = I_{B(H)}$, $I_{E}$ is a unital *-homomorphism.$\square$