Theorem 1

Let A be a commutative unital stellar subalgebra of $\mathscr{L}(E)$. There exist a locally compact topological space X, a positive measure $\mu$ on X, an isometric isomorphism $\theta$ of $L^2(X, \mu)$ onto E, and an isometric morphism of stellar algebras $\pi$ of A into $\mathscr{C}_b(X)$ such that for every $a\in A$, one has

By Prop. 9, there exists a subset C of E such that E is the Hilbert sum of the subspaces $E_x$ for $x\in C$. Let X be the locally compact topological space $\mathsf{X}(A)\times C$, where C is endowed with the discrete topology. There exists a unique measure $\mu$ on X such that $\mu|(\mathsf{X}(A)\times \{x\})$ is identified with the spectral measure $\mu_x$ of $x$ for every $x\in C$ (INT, III, p. 65, § 2, n$^o1$, Prop. 1); this measure is positive (cf. loc. cit.). One then has a decomposition as Hilbert sum

(Prop. 1 of IV, p. 179, c) applied to the sets $\mathsf{X}(A)\times \{x\}$ for $x\in C)$. There therefore exists a unique isometry $\theta : L^2(X, \mu)\rightarrow E$ which coincides with $\theta_x: L^2(\mathsf{X}(A), \mu_x)\rightarrow E_x$ on $L^2(\mathsf{X}(A), \mu_x)$ for every $x\in C$.

Let $p: X\rightarrow \mathsf{X}(A)$ be the canonical projection; it is surjective, hence the linear mapping $p^*:\mathscr{C}_b(\mathsf{X}(A))\rightarrow \mathscr{C}(X)$ defined by $f\mapsto f\circ p$ is injective. Let us write $\pi =p^*\circ \mathscr{G}_A$; it is an injective morphism of stellar algebras of A into $\mathscr{C}_b(X)$; it is therefore isometric (Prop. 9 of I, p. 112).

Let $a\in A$. For every $x\in C$, we have $a_x\circ \theta_x=\theta_x\circ m_{\mathscr{G}_A(a)}$. The continuous linear mappings $a\circ \theta$ and $\theta \circ m_{\pi(a)}$ therefore coincide on the subspaces $L^2(\mathsf{X}(A), \mu_x)$, and are consequently equal.