= Solution
Use the usual unital-subalgebra convention $I_H\in A$. The <spectral theorem for a commutative operator algebra> states that the <character space> $\Phi_A$, equipped with the <Gelfand topology>, is a <compact Hausdorff space>, and there is a unique regular <projection-valued measure> $E$ on its <Borel sigma-algebra> satisfying
$$
\boxed{E(\Phi_A)=I_H,\qquad a=\int_{\Phi_A}\widehat a(\chi)\,dE(\chi)\quad(a\in A),}
$$
where $\widehat a(\chi)=\chi(a)$ is the <Gelfand transform>. The map $a\mapsto\widehat a$ is an isometric unital star-isomorphism from $A$ onto $C(\Phi_A)$ by the <Commutative Gelfand--Naimark theorem>.
More explicitly, every $E(B)$ is an <orthogonal projection>, $E(\varnothing)=0$, $E(B\cap C)=E(B)E(C)$, and for disjoint <Borel sets> $B_n$,
$$
E\left(\bigcup_nB_n\right)v=\sum_nE(B_n)v\quad(v\in H),
$$
with convergence in the norm of the <Hilbert space>. Regularity means that each <scalar spectral measure> $B\mapsto\langle E(B)v,w\rangle$ is a <regular Borel measure>. The integral identity then implies $\|a\|=\sup_{\chi\in\Phi_A}|\widehat a(\chi)|$. No separability assumption on $H$ is needed.
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