Proposition 3

Let B be a commutative unital Banach algebra and $\varphi$ a continuous unital morphism of A into B. Let $\boldsymbol{a}= (a_1, . . . , a_n)\in A^n$. Put $\boldsymbol{b}= (\varphi (a_1), . . . , \varphi (a_n))$, so that Sp$^n_B(\boldsymbol{b})\subset$ Sp$^n_A(\boldsymbol{a})$. For every $f\in \mathscr{O}$(Sp$^n_A(\boldsymbol{a}); A$), one has

where $\varphi_*(f)$ denotes the germ of $\varphi \circ f$ in a neighbourhood of Sp$^n_B(\boldsymbol{b})$.

It is enough to prove that for every open neighbourhood U of Sp$^n_A(\boldsymbol{a})$ and every $f\in \mathscr{O}(U; A)$, one has $\varphi (\Theta_{\boldsymbol{a}}(f)) = \Theta_{\boldsymbol{b}}(\varphi \circ f)$, where $\varphi \circ f\in \mathscr{O}(U; B)$. Let $(h, u_1, . . . , u_n)$ be a sequence adapted to $\boldsymbol{a}$, where the support of $h$ is contained in U (lemma 1 of I, p. 52 and lemma 3 of I, p. 54). Let $\omega$ denote the associated differential form. For every $\boldsymbol{z}\in \mathbf{C}^n$, one has

so that the sequence $(h, \varphi \circ u_1, . . . , \varphi \circ u_n)$ is adapted to $\boldsymbol{b}$. Let $\omega '$ denote the associated differential form. Let $\mu$ denote Lebesgue measure on $\mathbf{C}^n$. Let $f\in \mathscr{O}(U; A)$. Let us write $\psi \cdot \mu$ for the vector measure associated with the differential form $f \omega$. The vector measure associated with the differential form

is equal to $(\varphi \circ \psi )\cdot \mu$. Therefore, by formula (5), and INT, VI, §2, n$^o2$, prop. 2, one has

$\Theta_{\boldsymbol{b}}(\varphi \circ f) =(2i\pi n$!$)^n\int_U(\varphi \circ f)\mu=(2ni\pi$!$)^n\varphi (\int_U\psi \mu)=\varphi (\Theta_{\boldsymbol{a}}(f))$,

as was required.