Lemma 9

Let A be a commutative unital stellar subalgebra of $\mathscr{L}(E)$. For every $x$ and $y$ in E, the mapping $\mu$ of $\mathsf{X}(A)$ into $\mathbf{C}$ defined by $\mu(f) =\langle x|\mathscr{H}_A(f)y\rangle$ for every $f\in \mathscr{C}(\mathsf{X}(A))$ is a bounded measure on the compact space $\mathsf{X}(A)$. It is positive if $x=y$. Its norm is $\leqslant \|x\| \|y\|$, with equality if $x=y$.

The mapping $\mu$ is linear. For every function $f\in \mathscr{C}(\mathsf{X}$(A)), one has

hence $\mu$ is a bounded measure on $\mathsf{X}(A)$ of norm $\leqslant \|x\| \|y\|$.

Suppose that $x=y$. For every positive function $f\in \mathscr{C}(\mathsf{X}$(A)), the element $\mathscr{H}_A(f)$ is a positive element of A (example 1 of I, p. 115), hence a positive element of $\mathscr{L}$ (E), whence $\mu(f) =\langle x|\mathscr{H}_A(f)(x)\rangle \geqslant 0$ (prop. 8 of I, p. 138). This shows that $\mu$ is a positive measure. Since $\mu(1) =\|x\|^2$, the total mass of $\mu$ is $\|x\|^2$ (INT, III, p. 58, § 1, No.$^o8$, cor. 2).