Proposition 3

Let A be a unital involutive Banach algebra. Let $a\in A$. The spectrum of $a^*$ is the image Sp$_A(a)^*$ of Sp$_A(a)$ under complex conjugation. For every $f\in \mathscr{O}$(Sp$_A(a)$), one has $f(a)^*=f^*(a^*)$.

The first assertion follows from I, p. 97. From what precedes, the mapping $\varphi$ of $\mathscr{O}$(Sp$_A(a)$) into A defined by $f\mapsto (f^*(a^*))^*$ is a continuous unital morphism of $\mathscr{O}$(Sp$_A(a)$) into A such that the image of the germ in a neighbourhood of Sp$_A(a)$ of the identity function of $\mathbf{C}$ is equal to $a$. Consequently, $\varphi$ is the mapping $f\mapsto f(a)$ of the holomorphic functional calculus (I, p. 74, th. 5).