= Scalar-measure construction of a projection-valued measure
For a unital star-representation $\pi:C(K)\to\mathcal B(H)$, the <Riesz-Markov-Kakutani representation theorem> gives regular measures $\mu_{x,y}$ representing $f\mapsto\langle\pi(f)x,y\rangle$. Define $P(E)$ by $\langle P(E)x,y\rangle=\mu_{x,y}(E)$. Positivity and $\mu_{x,x}(K)=\|x\|^2$ make these positive contractions. The identities $\mu_{\pi(f)x,y}=f\mu_{x,y}$ imply commutation with $\pi(f)$ and $\mu_{P(E)x,y}=1_E\mu_{x,y}$; hence $P(F)P(E)=P(F\cap E)$. Thus each $P(E)$ is an <orthogonal projection>. Scalar countable additivity and orthogonality give countable additivity in the <strong operator topology>, constructing the normalized regular <projection-valued measure>.
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