2.2.1. Basic constructions [00ML]

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2.2.1. Basic constructions

Definition 2.16.

Let $A$ be a $k$ -algebra (the unit of which is denoted by $\mathbf{1}$ ) and $\lVert\mathord{\cdot}\rVert$ be a seminorm on $A$ (viewed as a vector space over $k$ ).

(1)

The seminorm $\lVert\mathord{\cdot}\rVert$ is said to be sub-multiplicative if for any $(a,b)\in A\times A$ one has $\lVert ab\rVert\leq\lVert a\rVert\cdot\lVert b\rVert$ .
(2)

The seminorm $\lVert\mathord{\cdot}\rVert$ is called power-multiplicative if $\lVert a^{n}\rVert=\lVert a\rVert^{n}$ for any $a\in A$ and any $n\in\mathbb{N}\setminus\{0\}$ .
(3)

The seminorm $\lVert\mathord{\cdot}\rVert$ is called multiplicative if $\lVert ab\rVert=\lVert a\rVert\cdot\lVert b\rVert$ for any $(a,b)\in A^{2}$ .

A $k$ -algebra seminorm (resp. $k$ -algebra norm) on $A$ is defined to be a sub-multiplicative seminorm (resp. sub-multiplicative norm) $\lVert\mathord{\cdot}\rVert$ on $A$ such that $\lVert\mathbf{1}\rVert=1$ . We denote by $\vvvert\mathord{\cdot}\vvvert$ an algebra seminorm. Any $k$ -algebra equipped with a complete $k$ -algebra norm is called a Banach $k$ -algebra.

We use calligraphic letters to denote Banach algebras and Banach modules (defined below) and use the corresponding capital letters to denote the underlying $k$ -algebra or the underlying module of a $k$ -algebra. For example, a Banach $k$ -algebra $(A,\vvvert\mathord{\cdot}\vvvert)$ is denoted by $\mathcal{A}$ . If $A^{\prime}$ is a sub- $k$ -algebra of $A$ , then the restriction of $\vvvert\mathord{\cdot}\vvvert$ on $A^{\prime}$ is a $k$ -algebra norm. If this norm is complete, we say that $\mathcal{A}^{\prime}$ ( $A^{\prime}$ equipped with the restricted norm) is a Banach $k$ -sub-algebra of $\mathcal{A}$ . Similarly, if $Q$ is a quotient $k$ -algebra of $A$ , then the quotient of the norm $\vvvert\mathord{\cdot}\vvvert$ on $Q$ is a sub-multiplicative seminorm. If it is a complete norm, we say that $\mathcal{Q}$ ( $Q$ equipped with the quotient norm) is a Banach quotient $k$ -algebra of $\mathcal{A}$ .

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 $\boldsymbol{r}$ .

Definition 2.18.

Let $\mathcal{A}_{1},\mathcal{A}_{2}$ be two Banach $k$ -algebras, and $\phi:A_{1}\to A_{2}$ be a homomorphism of $k$ -algebras. We say that $\phi$ is a homomorphism of Banach $k$ -algebras if it is bounded as a $k$ -linear map. A homomorphism of Banach $k$ -algebra $\phi$ is often denoted by $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ . A homomorphism of Banach $k$ -algebra $\phi$ is called an isomorphism of Banach $k$ -algebras if there exists a homomorphism of Banach $k$ -algebras $\psi:\mathcal{A}_{2}\to\mathcal{A}_{1}$ such that $\phi\circ\psi=\mathrm{Id}_{\mathcal{A}_{2}}$ and $\psi\circ\phi=\mathrm{Id}_{\mathcal{A}_{1}}$ .