Proposition 13

We have the following properties:

a) For every $f\in \mathscr{C}'$(Sp$'_A(x)$), we have Sp$'_A(f(x)) =f$(Sp$'_A(x)$) ;

b) For every $f\in \mathscr{C}'$(Sp$'_A(x)$) and for every $g\in \mathscr{C}'$(Sp$'_A(f(x))$), we have $(g\circ f)(x) =g(f(x))$ ;

c) Let $A'$ be a stellar algebra and let $\pi$ be a morphism from A into $A'$; then $\pi (x)$ is normal in $A'$, we have Sp$'_{A'}(\pi (x))\subset$ Sp$'_A(x)$ and $\pi (f(x)) =f(\pi (x))$ for every $f\in \mathscr{C}'$(Sp$'_A(x)$);

d) If A is commutative, and if $f\in \mathscr{C}'$(Sp$'_A(x)$), then $\mathscr{G}'_A(f(x)) =$ $f\circ \mathscr{G}'_A(x)$.