Left X Modules Right X Modules And X Bimodules

Left X-Modules, Right X-Modules, and X-Bimodules

Left X-Modules, Right X-Modules, and X-Bimodules

Definition: Let $X$ be an algebra over $\mathbf{F}$ and let $M$ be a linear space over $\mathbf{F}$. We say that $M$ is a Left $X$-Module if with the mapping of left module multiplication $(x, m) \to xm$ of $X \times M$ into $M$ is such that the following axioms are satisfied:
LM 1) For each fixed $x \in X$ the map $m \to xm$ is linear on $M$.
LM 2) For each fixed $m \in M$ the map $x \to xm$ is linear on $X$.
LM 3) For all $x_1, x_2 \in X$ and all $m \in M$ we have that $x_1(x_2m) = (x_1x_2)m$.
Similarly, we say that $M$ is a Right $X$-Module if with the mapping of right module multiplication $(x, m) \to mx$ of $X \times M$ into $M$ is such that the following axioms are satisfied:
RM 1) For each fixed $x \in X$ the map $m \to mx$ is linear on $M$.
RM 2) For each fixed $m \in M$ the map $x \to mx$ is linear on $X$.
RM 3) For all $x_1, x_2 \in X$ and all $m \in M$ we have that $(mx_1)x_2 = m(x_1x_2)$.
We say that $M$ is an $X$-Bimodule if it is both a left $X$-module and a right $X$-module, as well as the additional axiom holds:
BM 1) For all $x_1, x_2 \in X$ and all $m \in M$ we have that $x_1(mx_2) = (x_1m)x_2$.
Definition: Let $X$ be an algebra with unit $e$.
1) If $M$ is a left $X$-module then $M$ is said to be Unit Linked if for every $m \in M$ we have that $em = m$.
2) If $M$ is a right $X$-module then $M$ is said to be Unit Linked if for every $m \in M$ we have that $me = m$.
3) If $M$ is an $X$-bimodule then $M$ is said to be Unit Linked if for every $m \in M$ we have that $em = me = m$.

Example 1

Let $X = \mathbb{R}$ and let $M$ be any linear space. Then $M$ is an $\mathbb{R}$-bimodule with module multiplication $f : \mathbb{R} \times M \to M$ defined for all $(t, m) \in \mathbb{R} \times M$ by $f(t, m) = tm$

Clearly for each fixed $t \in \mathbb{R}$ the map $f_t : M \to M$ defined by $f_t(m) = tm$ is linear since $t(m_1 + m_2) = tm_1 + tm_2$ for all $m_1, m_2 \in M$. Also it is clear that for each fixed $m \in M$ the map $f_m : \mathbb{R} \to M$ defined by $f_m(t) = tm$ is linear since $(t_1 + t_2)m = t_1m + t_2m$ for all $t_1, t_2 \in \mathbb{R}$. Lastly, we have that $s(tm) = (st)m$ for all $s, t \in \mathbb{R}$ and $m \in M$.

So every linear space $M$ is a left $\mathbb{R}$-module. It is easy to check that every linear space $M$ is also a right $\mathbb{R}$-module and that it is actually an $\mathbb{R}$-bimodule that is unit-linked by the unit number 1 in $\mathbb{R}$.

Example 2

Let:

(1)
\begin{align} \quad X = \left \{ \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix} : x_{11}, x_{12}, x_{21}, x_{22} \in \mathbb{R} \right \} \end{align}

That is, $X$ consists of all $2 \times 2$ matrices with real entries. With the operations of matrix addition and scalar multiplication, and matrix multiplication, it is clear that $X$ is an algebra over $\mathbb{R}$. Let $M = \mathbb{R}^2$. Then $M$ can be viewed as a left $X$-module with module multiplication $X \times M \to M$ by:

(2)
\begin{align} \quad \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix} (a, b) = (x_{11}a + x_{12}b, x_{21}a + x_{22}b) \end{align}

It is also unit linked where the unit in $X$ is the $2 \times 2$ identity matrix $I_2$ since clearly for any $(a, b) \in \mathbb{R}^2$ we have that $I_2(a, b) = (a, b)$.

Banach Left X-Modules, Banach Right X-Modules, and Banach X-Bimodules

Definition: Let $X$ be a normed algebra and let $M$ be a normed linear space, both over $\mathbf{F}$.
1) $M$ is said to be a Normed Left $X$-Module if $M$ is a left $X$-module and there exists a $K > 0$ such that for all $x \in X$ and all $m \in M$ we have that $\| xm \| \leq K \| x \| \| m \|$.
2) $M$ is said to be a Normed Right $X$-Module if $M$ is a right $X$-module and there exists a $K > 0$ such that for all $x \in X$ and all $m \in M$ we have that $\| mx \| \leq K \| x \| \| m \|$.
3) $M$ is said to be a Normed $X$-Bimodule if $M$ is both a normed left $X$-module and a normed right $X$-module.
A normed right/left/ $X$-(bi)module is said to be a Banach Left/Right $X$-(Bi)Module if $M$ is also a Banach space.
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