ScalingStacks

Proof. By Proposition 3.5, there are homomorphisms of Banach algebras induced by the identity map on the dense $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$:

$$\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon/2)_{X|Y}^{\mathrm{aff}})\xrightarrow[\sigma(\epsilon/2)|_{Y}]{}\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon/2)_{X|Y})\xrightarrow[\iota(\epsilon/2)]{}\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}).$$

Let $\tau(\epsilon/2)$ denote the composed homomorphism of Banach $k$-algebras. It is a homomorphism from an affinoid algebra to a Banach algebra. It has dense image, so by Proposition 2.27 the induced continuous map

$$\tau(\epsilon/2)^{*}:\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))\rightarrow\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon/2)_{X|Y}^{\mathrm{aff}}))$$

is injective and is closed. As both spaces are compact and Hausdorff, this map is a homeomorphism from its domain to its image. So the homomorphism spectrum $\Sigma_{\tau(\epsilon/2)}$ is homeomorphic to $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))$, and is contained in $W_{\epsilon/2}$.

One performs spectral calculus for the homomorphism $\tau(\epsilon/2)$ and the special domain $W_{\epsilon/2}$: by Theorem 2.81, there exist a homomorphism of Banach $k$-algebras

$$\theta(\epsilon/2)_{W_{\epsilon/2}}:\mathcal{W}_{\epsilon/2}\rightarrow\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y})$$

which extends the identity map on the dense sub-$k$-algebra $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$. ∎