Definition 3

For every function $f\in \mathscr{K}$ (Sp($u$)), one defines the endomorphism $f(u)$ of E by $f(u) =f_{\beta}(b(u))$.

The mapping $f\mapsto f_{\beta}$ is a complex algebra morphism of $\mathscr{K}$ (Sp($u$)) into $\mathscr{C}$(Sp($b(u)$)), hence the mapping $f\mapsto f(u)$ is a complex algebra morphism of $\mathscr{K}$ (Sp($u$)) into $\mathscr{L}(E)$. One has $\overline{f}(u) =f(u)^*$ since $\overline{f}_{\beta}=\overline{f_{\beta}}$. If $f\geqslant 0$ then $f_{\beta}\geqslant 0$, hence $f(u)$ is a positive endomorphism of E.