Subsection 2.4.2. (00MY) · ScalingStacks

2.4.2. Holomorphic functional calculus

It is easy to localize the homomorphism to holomorphic convex neighbourhood of its spectrum. For a spectrum of homomorphism which is not holomorphically convex, one uses Proposition 2.79 to localize the homomorphism to any neighbourhood of it.

00JK

Lemma 2.80. Let $\phi:\mathcal{A}\to\mathcal{B}$ be a Banach algebra homomorphism from an affinoid algebra $\mathcal{A}$ to a Banach algebra $\mathcal{B}$ . Then for any Laurent domain neighbourhood $V$ of $\Sigma_{\phi}$ , $\phi$ extends to a unique Banach algebra homomorphism $\phi_{V}:\mathcal{A}_{V}\to\mathcal{B}$ . ([Ber, Corollary 2.5.16])

00JL

Theorem 2.81. Let $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach algebras from an affinoid algebra to a Banach algebra. Let $V\subseteq\mathfrak{M}(\mathcal{A})$ be any special domain containing $\Sigma_{\phi}$ . Then there exists a Banach algebra homomorphism

\theta_{\phi}:\Gamma(V,\mathscr{O}_{\mathfrak{M}(\mathcal{A})})\to\mathcal{B}

satisfying $\phi=\theta_{\phi}\circ\iota_{V}$ , where $\iota_{V}:\mathcal{A}\to\mathcal{A}_{V}=\Gamma(V,\mathscr{O}_{\mathfrak{M}(\mathcal{A})})$ is the Banach algebra homomorphism corresponding to the inclusion $V\subseteq\mathfrak{M}(\mathcal{A})$ . ([Ber, Theorem 7.3.4])

00JM

Remark 2.82. One can verify that the resulting Banach algebra homomorphism does not depend on the choice of $\widetilde{\phi}$ .