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I need some help understanding a proof of the following claim: every cyclic representation is isomorphic to some $L^2$ space. First, formal definitions:
[Equivalent representations] Let $\mathcal{A}$ be a (group) a *-algebra, and let $(\mathcal{H}, \pi)$, and $(K, \rho)$ are two *-representations of $\mathcal{A}$, then these representations are equivalent, if there is a unitary operator $U: H\to K$, $(U^{-1}: K\to H)$, such that \begin{equation*} U\pi(a)=\rho(a)U \end{equation*} $U$ is the representation that equivalates $\pi$ and $\rho$. ``U intertwines $\pi$ and $\rho$''
[Proposition]: Let $\mathcal{A}$ be a unital *-normed algebra, and let $(\mathcal{H}, \pi, \xi)$ and $(K, \rho, \eta)$, where $\xi, \eta$ are cyclic vectors, and they each determine a positive linear functional on $\mathcal{A}$. Let $\mu_\xi$, and $\mu_\eta$ be the correspondonig positive linear functionals, and if $\mu_\xi=\mu_\eta$, then $(\mathcal{H}, \pi)$ and $(K, \rho)$ are unitarily equiavalent via a unitary operator such that $U\xi=\eta$.
Now to the proposition that I need help with.
[Proposition] $(H_\lambda, \pi_\lambda, \xi_\mu)$ is isomorphic to some $L^2(X,\mu_\lambda, \xi_\lambda)$
Proof: Let $\mathcal{A}$ be a commutative unital *-normed algebra and let $(\mathcal{H}, \mu)$ be a cyclic *-representation with cyclic vector $\xi$. Then let $\mathcal{B}=\overline{\mu(a)}$ be the norm closure is a $C^*$-algebra. So $\mathcal{B}=C(X)$. Then $\mu_\xi$ is a positive linear functional on $\mathcal{B}=C(X)$. Hence $\mu_\xi$ gives a finite regular Borel measure on $X$, $\tilde{\mu}_\xi$, so you can form the $L^2(X,\tilde{\mu}_\xi)$ with $\mathcal{B}$ acting on $L^2(X, \tilde{\mu}_\xi)$ by pointwise multiplication. This is basically the GNS representation. Then you have the cyclic vector as the constant function 1. And find that $\mu_1=\mu_\xi$, so \begin{equation*} (\mathcal{H}, \pi, \xi)\cong (L^2(X, \tilde{\mu}_\xi), 1) \end{equation*} via the unitary $U$ such that $U\xi=1$.
Question: the GNS representation that started with the positive linear functional $\mu_{\xi}$ that takes its values in $\mathcal{B}$, is a representation of algebra $\mathcal{B}$, not algebra $\mathcal{A}$, as representation $(H,\mu)$. How these two can be equivalent if they represent different algebras?
