Definition 2

Let A be a commutative unital stellar subalgebra of $\mathscr{L}(E)$. For every $x$ and $y$ in E, the linear form $f\mapsto \langle x|\mathscr{H}_A(f)y\rangle$ on $\mathscr{C}(\mathsf{X}(A))$ is called the spectral measure of $(x, y)$ relative to A. If $x=y$, it is said to be the spectral measure of $x$ relative to A.