Examples of Modules - For Normed Left A-Modules M, M is a Left (A + F)-Module
Let $\mathfrak{A}$ be a normed algebra and let $M$ be a normed left $\mathfrak{A}$-module. Recall that the unitization $\mathfrak{A} + \mathbf{F}$ is normed algebra with unit $(0, 1)$ with norm defined on $\mathfrak{A} + \mathbf{F}$ for all $(a, \alpha) \in \mathfrak{A} + \mathbf{F}$ by:
(1)As we will see, $M$ will become an left ($\mathfrak{A} + \mathbf{F}$)-module with module multiplication defined for all $m \in M$ and for all $(a, \alpha) \in \mathfrak{A} + \mathbf{F}$ by:
(2)Note that indeed this module multiplication takes elements of $(\mathfrak{A} + \mathbf{F}) \times M$ and outputs elements of $M$. To see this, observe that since $M$ is a left $\mathfrak{A}$-module we have that $am \in M$. Furthermore, since $M$ is by definition also a linear space we have that $\alpha m \in M$. Thus $am + \alpha m \in M$.
- 1. For each fixed $(a, \alpha) \in \mathfrak{A} + \mathbf{F}$ we have that $((a, \alpha), m) \to (a, \alpha)m$ is linear since for all $m_1, m_2 \in M$ we have that:
- 2. And for each fixed $m \in M$ we have that $((a, \alpha), m) \to (a, \alpha)m$ is linear since for all $(a, \alpha), (b, \beta) \in \mathfrak{A} + \mathbf{F}$ we have that:
- 3. We have that for all $(a, \alpha), (b, \beta) \in \mathfrak{A} + \mathbf{F}$ and all $m \in M$:
- While:
- So $(a, \alpha)[(b, \beta)m] = [(a, \alpha)(b, \beta)]m$. Thus $M$ is a normed left ($\mathfrak{A} + \mathbf{F}$)-module.
- 4. Lastly, for all $(a, \alpha) \in \mathfrak{A} + \mathbf{F}$ and for all $m \in M$ we have that:
So we conclude that $M$ is a normed left $(\mathfrak{A} + \mathbf{F})$-module.
A similar argument shows that if $M$ is a normed right $\mathfrak{A}$-module then $M$ is also a normed right $(\mathfrak{A} + \mathbf{F})$-module.