Proposition 8

Let E be a complex Hilbert space. Let A be a commutative unital stellar subalgebra of $\mathscr{L}(E)$. Let $x$ be an element of E. Put $E_x= A\cdot x$ and let $\mu_x$ denote the spectral measure of $x$ relative to A.

There exists a unique isometric isomorphism $\theta_x$ of $L^2(\mathsf{X}(A), \mu_x)$ onto $E_x$ such that $\theta_x(f) =\mathscr{H}_A(f)(x)$ for every function $f\in \mathscr{C}(\mathsf{X}(A))$. For every element $a\in A$, the space $E_x$ is stable under $a$, and if $a_x$ denotes the endomorphism of $E_x$ deduced from $a$ by passing to subspaces, then one has $a_x\circ \theta_x=\theta_x\circ m_{\mathscr{G}_A(a)}$.

Let $\widetilde{\theta}_x$ be the linear mapping of $\mathscr{C}(\mathsf{X}(A))$ into $E_x$ defined by

For every function $f\in \mathscr{C}(\mathsf{X}$(A)), one has

Since $\mathscr{C}(\mathsf{X}(A))$ is dense in $\mathscr{L}^2(\mathsf{X}(A), \mu_x)$, there exists a unique isometric linear mapping of $L^2(\mathsf{X}(A), \mu_x)$ into $E_x$ extending $\widetilde{\theta}_x$; let $\theta_x$ denote it. The mapping $\theta_x$ is surjective since its image is closed (lemma 8 of I, p. 107) and contains $A\cdot x$. It is therefore an isometric isomorphism.

Let $a\in A$. Put $g=\mathscr{G}_A(a)\in \mathscr{C}(\mathsf{X}(A))$. The space $E_x$ is stable under $a$. For $f\in \mathscr{C}(\mathsf{X}$(A)), one has then

and it follows that $\theta_x\circ m_g=a_x\circ \theta_x$.