ScalingStacks

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 $\Omega^{\mathrm{h}}=\Omega$.