Subsection 2.4. (00MW) · ScalingStacks

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2.4. Spectral calculus

Gelfand-Shilov theory allows one to do multi-variable spectral calculus for (commutative) Banach algebras over $\mathbb{C}$ . In particular, one can localize a homomorphism between Banach algebras onto a neighbourhood of its spectrum. Similar theory, as develloped in [Ber, Chapter 7], exists in the non-Archimedean base field setting.

2.4.1. Holomorphic envelop

The holomorphic convexity of spectrum of a homomorphism of Banach $k$ -algebra depends on the dense-ness of its image. In case where the spectrum of a homomorphism is not holomorphic convex, one can add variables to the source algebra so that spectrum of extended homomorphism is holomorphically convex.

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Definition 2.74. Let $\mathcal{A}$ and $\mathcal{B}$ be Banach $k$ -algebras, and $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach algebras. The spectrum of homomorphism $\phi$ is the image of $\mathfrak{M}(\mathcal{B})$ in $\mathfrak{M}(\mathcal{A})$ under $\phi^{\star}$ . Denote it by $\Sigma_{\phi}$

00JF

Definition 2.75. Let $\mathcal{A}$ be a Banach $k$ -algebra. Let $\Omega$ be a compact subset of $\mathfrak{M}(\mathcal{A})$ . The holomorphic convex envelop of $\Omega$ in $\mathfrak{M}(\mathcal{A})$ is the subset

\Omega^{\mathrm{h}}:=\{z\in\mathfrak{M}(\mathcal{A})\ |\ \forall f\in\mathcal{A},\lvert f\rvert_{z}\leq\sup_{z^{\prime}\in\Omega}\lvert f\rvert_{z^{\prime}}\}

The subset $\Omega$ is said to be holomorphically convex if $\Omega^{\mathrm{h}}=\Omega$ .

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Lemma 2.76. The intersection of all Weierstrass neighbourhoods of $\Omega$ in $\mathfrak{M}(\mathcal{A})$ coincide with $\Omega^{\mathrm{h}}$ . ([Ber, Proposition 2.6.1])

00JH

Proposition 2.77. Let $\mathcal{A}$ be a $k$ -affinoid algebra, $\mathcal{B}$ be a Banach $k$ -algebra. Let $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach $k$ -algebras. Let $\mathcal{B}^{\prime}$ be the closed sub-algebra generated by the image of $\phi$ of $\mathcal{A}$ in $\mathcal{B}$ and let $\phi^{\prime}:\mathcal{A}\to\mathcal{B}^{\prime}$ be the restricted homomorphism. Then $(\Sigma_{\phi})^{\mathrm{h}}=\Sigma_{\phi^{\prime}}$ . ([Ber, Proposition 7.3.1])

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Corollary 2.78. Let $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach algebras from an affinoid algebra to a Banach algebra with dense image. Then $\Sigma_{\phi}$ is holomorphically convex.

One has the following analogue of Arens-Calderon theorem, which holomorphically convexifies the spectrum of a homomorphism of Banach $k$ -algebras by adding variables on the source algebra.

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Proposition 2.79. Let $\mathcal{A}$ be a $k$ -affinoid algebra, $\mathcal{B}$ be a Banach $k$ -algebra. Let $\phi:\mathcal{A}\to\mathcal{B}$ be a homomorphism of Banach $k$ -algebras. Then for any open neighbourhood $U$ in $\mathfrak{M}(\mathcal{A})$ of the spectrum $\Sigma_{\phi}$ , there exists a homomorphism of Banach algebras extending $\phi$

\widetilde{\phi}:\widetilde{\mathcal{A}}:=\mathcal{A}\{r_{1}^{-1}T_{1},\dots,r_{n}^{-1}T_{n}\}\to\mathcal{B}

such that $\mathrm{pr}((\Sigma_{\phi})^{\mathrm{h}})\subseteq U$ , where $\mathrm{pr}:\mathfrak{M}(\widetilde{\mathcal{A}})\to\mathfrak{M}(\mathcal{A})$ is the canonical map of projection. ([Ber, Proposition 7.3.3])

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.

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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])

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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])

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