2.2.5. Banach module [00MQ]

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2.2.5. Banach module

One can also consider seminorms on modules over Banach algebra. Let $\mathcal{A}$ be a Banach $k$ -algebra. A (semi)normed $\mathcal{A}$ -module is defined to be an $A$ -module $M$ with a (semi)norm $\lVert\mathord{\cdot}\rVert$ such that $(M,\lVert\mathord{\cdot}\rVert)$ is a (semi)normed vector space over $k$ (denoted by $\mathcal{M}$ ), and that the multiplication is bounded, in the sense that there exists $C>0$ such that

\forall a\in\mathcal{A},\ \forall m\in M,\quad\lVert a\cdot m\rVert\leq C\vvvert a\vvvert\cdot\lVert m\rVert

One calls a Banach $\mathcal{A}$ -module a normed $\mathcal{A}$ -module $(M,\lVert\mathord{\cdot}\rVert)$ whose norm is complete.

Let $\mathcal{M}_{1}=(M_{1},\lVert\mathord{\cdot}\rVert_{1})$ , $\mathcal{M}_{2}=(M_{2},\lVert\mathord{\cdot}\rVert_{2})$ be Banach $\mathcal{A}$ -modules and $\phi:M_{1}\to M_{2}$ be a homomorphism of $A$ -modules. It is called bounded if there exists $C>0$ such that $\lVert\phi(m_{1})\rVert_{2}\leq C\lVert m_{1}\rVert_{1}$ for any $m_{1}\in M_{1}$ . In this case $\phi$ is said to be a homomorphism of Banach $\mathcal{A}$ -modules, and is denoted by $\phi:\mathcal{M}_{1}\rightarrow\mathcal{M}_{2}$ . In addition, the homomorphism $\phi$ of Banach $\mathcal{A}$ -modules is called admissible if it is admissible as linear map between normed-vector spaces over $k$ .

Definition 2.34.

Let $\mathcal{M}$ be a Banach $\mathcal{A}$ -module. It is called a Banach finite $\mathcal{A}$ -module if there exists $l\in\mathbb{N}_{+}$ and a surjective homomorphism of Banach $\mathcal{A}$ -modules $\mathcal{A}^{\oplus l}\to\mathcal{M}$ where $\mathcal{A}^{\oplus l}$ is the Banach $\mathcal{A}$ -module corresponding to the $A$ -module $A^{\oplus l}$ equipped with the norm $(a_{1},\dots,a_{l})\mapsto\max\vvvert a_{i}\vvvert$ . (Note that such a homomorphism is necessarily admissible.)

Proposition 2.35.

Let $\mathcal{A}$ be a Banach $k$ -algebra and $\mathcal{M}$ be a Banach $\mathcal{A}$ -module. If $A$ is Noetherian as a $k$ -algebra and $M$ is finitely generated as $A$ -module, then any $\mathcal{A}$ -sub-module of $\mathcal{M}$ is closed, and $\mathcal{M}$ is a Banach finite $\mathcal{A}$ -module. ([FvdP, Lemma 1.2.3]

Definition 2.36.

Let $\phi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ be a homomorphism between Banach $k$ -algebras. It is called Banach finite if $\mathcal{A}_{2}$ is a Banach finite $\mathcal{A}_{1}$ -module. In this case $\mathcal{A}_{2}$ is called a Banach finite $\mathcal{A}_{1}$ -algebra.

Remark 2.37.

If a $k$ -Banach algebra homomorphism $\phi$ is finite as homomorphism of $k$ -algebra, and $\mathcal{A}_{1}$ is Noetherian, then $\phi$ is automatically Banach finite: there is a surjective $\mathcal{A}_{1}$ -module homomorphism $p:\mathcal{A}_{1}^{\oplus n}\to\mathcal{A}_{2}$ , by Proposition 2.35 $\ker(p)$ is closed. Then $p$ is continuous hence is admissible by Corollary 2.5. So $\mathcal{A}_{2}$ is a Banach finite $\mathcal{A}_{1}$ -module.