In Takesaki's 1967 article A Duality in the Representation Theory of C ∗ -Algebras, he defines $H$ as the infinitely-dimensional Hilbert space such that every cyclic representation of a $C^*$-algebra $A$ is a representation on a subspace of $A$ on $H$. He then goes on denote the set of all reps on $H$ as $\mathrm{Rep} (A: H)$. For an isometric operator $j: H \oplus H \to H$, he defines the $j$-direct sum of reps as $(\pi_1 \oplus_j \pi_2)(x) = j (\pi_1 (x) \oplus \pi_2(x)) j^{-1}$. For $\pi \in \mathrm{Rep} (A: H)$, define the orthogonal projection $p(\pi) : H \to \overline{\pi(A) H}$. Later on, the author defines the admissible operator fields as functions $x: \mathrm{Rep}(A: H) \to \mathcal{B}({H})$ such that:
$x(u \pi u^* ) = u x(\pi) u^* $ for every unitary $u \in \mathcal{b}(H)$ and $\pi \in \mathrm{Rep}(A : H)$;
$x(\pi_1 \oplus_j \pi_2) = x(\pi_1) \oplus_j x(\pi_2)$ for every pair $\pi_1, \pi_2 \in \mathrm{Rep}(A : H)$;
$x(\pi) = p(\pi) x(\pi) p(\pi)$ for every $\pi \in \mathrm{Rep} (A: H)$;
$x$ is bounded on $\mathrm{Rep}(A : H)$.
The author then claims that, due to "Gelfand's fundamental representation theorem of C*-algebras", every continuous admissible operator field on $\mathrm{Rep}(A: H)$ for an abelian $A$ is of the form $\pi \mapsto \pi(a)$ for some $a \in A$.
I'm curious which theorem the author is referencing? I'm familiar with the fact that for an abelian $C^*$-algebra we have a representation $A \to C_0 (\Sigma)$, where $\Sigma$ is a T2 locally compact space. How does that imply the bolded statement?
Every bit of help is appreciated. Thanks in advance.
EDIT: Forgot to mention, the topology on $\mathrm{Rep} (A: H)$ is the topology of pointwise convergence relative to *-strong topology on $\mathcal{B} (H)$.