Spectral theorem for a commutative operator algebra

If $A$ is a commutative C-star algebra of bounded operators on a Hilbert space containing its identity, there is a unique regular projection-valued measure $E$ on the character space ${\Phi }_A$ such that $E({\Phi }_A)=I$ and $a={\int }_{{\Phi }_A}\hat{a}dE$ for every $a\in A$, where $\hat{a}$ is the Gelfand transform.

For a faithful representation of $C(K)$ by integration against a regular projection-valued measure $P$, every nonempty open subset $U$ satisfies $P(U)\ne 0$. Choose a nonzero continuous function supported inside $U$, using compact Hausdorff normality. If $P(U)=0$, its spectral integral vanishes, contradicting faithfulness. Regularity is understood through the scalar spectral measures.