ScalingStacks

Example 2.17. Let $\mathcal{A}$ be a Banach $k$-algebra. The Tate $k$-Banach algebra over $\mathcal{A}$ of multiradius $\boldsymbol{r}=(r_{1},\dots,r_{n})\in(\mathbb{R}_{+})^{N}$ is the algebra over $k$

$$\Big\{\sum_{J\in\mathbb{N}^{n}}a_{J}\boldsymbol{T}^{J},\text{ }a_{J}\in A\text{ and }\lim_{|J|\to\infty}\vvvert a_{J}\vvvert\cdot\boldsymbol{r}^{J}=0\Big\}$$

(for $J=(j_{1},\dots,j_{n})\in\mathbb{N}^{n}$, we denote $\prod_{i\in\{1,\dots,n\}}T_{i}^{j_{i}}$ by $\boldsymbol{T}^{J}$ and $\prod_{i\in\{1,\dots,n\}}r_{i}^{j_{i}}$ by $\boldsymbol{r}^{J}$) with a complete $k$-algebra norm defined by

$$\Big\vvvert\sum_{J\in\mathbb{N}^{n}}a_{J}\boldsymbol{T}^{J}\Big\vvvert_{\mathcal{T}_{\mathcal{A}}(\boldsymbol{r})}:=\sup_{J}\vvvert a_{J}\vvvert\cdot\boldsymbol{r}^{J}$$

This Banach algebra is denoted by $\mathcal{A}\{r_{1}^{-1}T_{1},\dots,r_{n}^{-1}T_{n}\}$, and is called an $\mathcal{A}$-Tate algebra of multiradius $\boldsymbol{r}$.