Example

— 1) The identity mapping of $\mathscr{K}(T)$ is a vectorial measure on T with values in $\mathscr{K}(T)$.

2) Let H be a complex Hilbert space, $\mathscr{L}(H)$ the normed algebra of continuous endomorphisms of H. Let A be a subalgebra of $\mathscr{L}(H)$, commutative, closed, self-adjoint and containing the identity; one shows that there exist a compact space X and an isomorphism of the normed algebra A onto the algebra $\mathscr{K}_\mathbf{C}(X) = \mathscr{C}_\mathbf{C}(X)$ of continuous complex functions on X, equipped with the norm $\|f\| = \sup_{x \in X} |f(x)|$. The inverse isomorphism, restricted to $\mathscr{K}(X)$, is a vectorial measure m on X, with values in $\mathscr{L}(H)$, such that $m(fg) = m(f)m(g)$._