Corollary

Let E be a complex Hilbert space. Let A be a commutative unital stellar subalgebra of $\mathscr{L}(E)$ admitting a cyclic vector $x$. Let $\mu_x$ be the spectral measure of $x$ relative to A. There exists a unique isometric isomorphism

such that $\theta_x(f) =\mathscr{H}_A(f)$ for every function $f\in \mathscr{C}(\mathsf{X}(A))$. For every $a$ in A, one has $a\circ \theta_x=\theta_x\circ m_{\mathscr{G}_A(a)}$.

In fact, one has then $E_x= E$ and $a_x=a$ for every $a\in A$.

Let E be a complex Hilbert space. Let $x\in E$ and let A be a commutative unital stellar subalgebra of $\mathscr{L}(E)$. Let $E_x$ denote the closed subspace $A\cdot x$ of E. For every $a\in A$, one has then $a(E_x)\subset E_x$. Let $A_x$ denote the commutative unital stellar subalgebra of $\mathscr{L}(E_x)$ formed by the endomorphisms of $E_x$ deduced from the elements of A by passing to subspaces. The vector $x$ is a cyclic vector for $A_x\subset \mathscr{L}(E_x)$.