Proposition 20

For every $f\in \mathscr{O}_{\mathbf{R}}$(Sp$_{A_{(\mathbf{C})}}(x)$), there exists a unique element $f(x)$ of A such that $f(1\otimes x) = 1\otimes f(x)$ in $A_{(\mathbf{C})}$. The mapping $f\mapsto f(x)$ from $\mathscr{O}_{\mathbf{R}}$(Sp$_{A_{(\mathbf{C})}}(x)$) into A is the unique continuous unital homomorphism of $\mathbf{R}$-algebras which maps to $x$ the germ in $\mathscr{O}_{\mathbf{R}}$(Sp$_{A_{(\mathbf{C})}}(x)$) of the identical mapping of $\mathbf{C}$.

Let us put S = Sp$_{A_{(\mathbf{C})}}(x)$. By the above lemma, for every germ $f\in \mathscr{O}_{\mathbf{R}}$(Sp($x$)), one has $f(1\otimes x) =\overline{f(1 \otimes x)}$. The first assertion follows from this. Let us denote by $z$ the germ in $\mathscr{O}_{\mathbf{R}}(S)$ of the identical mapping of $\mathbf{C}$. The mapping $f\mapsto f(x)$ is a continuous unital homomorphism of the $\mathbf{R}$-algebra $\mathscr{O}_{\mathbf{R}}$(Sp($x$)) into A, which maps $z$ to $x$. It is the only one by prop. 19, since every morphism having these properties is uniquely determined on every full closed sub-$\mathbf{R}$-algebra of $\mathscr{O}(S)$ containing $z$.

Let $f\in \mathscr{O}_{\mathbf{R}}$(Sp$_{A_{(\mathbf{C})}}(x)$). The element $f(x)$ belongs to every full closed subalgebra of A containing $x$ (prop. 19), and therefore belongs to the bicommutant of $x$ in A. The complex spectrum of $f(x)$ is equal to $f$(Sp($x$)) (I, p. 75, prop. 8). For every $g\in \mathscr{O}_{\mathbf{R}}(f$(Sp$_{A_{(\mathbf{C})}}(x)$)), one has $g\circ f\in \mathscr{O}_{\mathbf{R}}$(Sp$_{A_{(\mathbf{C})}}(x)$) and (loc. cit.) $(g\circ f)(x) =g(f(x))$.

Let U be an open subset of $\mathbf{C}$, stable under complex conjugation. The set Ω of elements $x$ of A whose complex spectrum is contained in U is open in A (I, p. 76, Prop. 10). Let $f$ be a holomorphic function on U such that $f^*=f$. The mapping $x\mapsto f(x)$ of Ω into A is analytic (loc. cit.).

Let A, B be complete normable unital associative algebras over $\mathbf{R}$ and let $\varphi : A\rightarrow$ B be a continuous unital algebra morphism. Let $x\in A$. The complex spectrum of $\varphi (x)$ is contained in that of $x$ and, for every $f\in \mathscr{O}_{\mathbf{R}}$(Sp$_{A_{(\mathbf{C})}}(x)$), one has $f(\varphi (x)) =\varphi (f(x))$. This results at once from the analogous statement in the complex case (I, p. 75, Prop. 8).