Approximate Identities in A for X when X is a Banach Left A-Module 4

Approximate Identities in A for X when X is a Banach Left A-Module 4

Recall from the Approximate Identities in A for X when X is a Banach Left A-Module 3 page the following theorem: Let $A$ be a Banach algebra and let $X$ be a Banach left $A$-module. If $A$ has a bounded left approximate identity for $X$ and $z \in X$, $\delta > 0$ then there exists $a \in A$ and $y \in X$ such that $z = ay$ and $\| z - y \| \leq \delta$.

We now prove some important corollaries to this result.

Corollary 1: Let $A$ be a Banach algebra. If $A$ has a bounded left approximate identity and $z \in A$ and $\delta > 0$ then there exists $x, y \in A$ where $y$ is contained in the least closed left ideal of $A$ containing $z$, $z = xy$, and $\| x - y \| \leq \delta$.
  • Proof: Let $X$ be the least closed left ideal containing $z$. Then $XA \subseteq X \subseteq A$. So $X$ can be regarded as a Banach left $A$-module.
  • By Theorem 1, for the point $z$ there exists $x \in A$, $y \in X$ such that $z = xy$ and $\| x - y \| \leq \delta$. And of course, $y \in X$ simply says $y$ is contained in the least closed left ideal of $A$. $\blacksquare$
Corollary 2: Let $A$ be a Banach algebra. If $A$ has a bounded left approximate identity $\{ e(\lambda) \}_{\lambda \in \Lambda}$ and $(z_n) \subset A$ is such that $\displaystyle{\lim_{n \to \infty} z_n = 0}$ then there exists $a \in A$ and $(y_n) \subset A$ such that $z_n = ay_n$ for every $n \in \mathbb{N}$ and $\displaystyle{\lim_{n \to \infty} y_n = 0}$.
  • Proof: Let
(1)
\begin{align} \quad X = \left \{ (a_n) \subset A : \lim_{n \to \infty} a_n = 0 \right \} \end{align}
  • Equip $X$ with the operations of sequence addition, scalar multiplication, and define a norm on $X$ for all $(a_n) \in X$ by:
(2)
\begin{align} \quad \| (a_n) \|_X = \sup_{n \in \mathbb{N}} \{ \| a_n \|_A \} \end{align}
  • (Note that the supremum exists since $(\| (a_n) \|_A)$ is a bounded numerical sequence.)
  • Then $X$ can be seen as a Banach left $A$-module with module multiplication defined for all $a \in A$ and all $(a_n) \in X$ by:
(3)
\begin{align} \quad a(a_n) = (aa_n) \end{align}
  • We aim to show that $e(\lambda)(a_n) - (a_n) \to 0$ for all $(a_n) \in X$. We prove this below.
  • First since $\{ e(\lambda) \}_{\lambda \in \Lambda}$ is bounded there exists a $K > 0$ such that $\| e(\lambda) \|_A \leq K$.
  • Let $\epsilon > 0$ be given and let $(a_n) \in X$. Then $\lim_{n \to \infty} a_n = 0$, that is, $\lim_{n \to \infty} \| a_n \|_A = 0$. So for $\frac{\epsilon}{K + 1} > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then $\| a_n \|_A < \frac{\epsilon}{K+1}$. So if $n \geq N$ then:
(4)
\begin{align} \quad \| e(\lambda)(a_n) - (a_n) \|_X &= \| (e(\lambda)a_n - a_n) \|_X \\ &= \sup_{n \in \mathbb{N}} \left \{ \| e(\lambda)a_n - a_n \|_A \right \} \\ & \leq \sup_{n \in \mathbb{N}} \left \{ (\| e(\lambda) \|_A + 1) \| a_n \|_A \right \} \\ & \leq [\| e(\lambda) \|_A + 1] \sup_{n \in \mathbb{N}} \left \{ \| a_n \|_A \right \} \\ & \leq [K + 1] \| (a_n) \|_X \\ & < [K + 1] \frac{\epsilon}{K + 1} \\ & < \epsilon \end{align}
  • So $e(\lambda)(a_n) - (a_n) \to 0$ for all $(a_n) \in X$, that is, $\{ e(\lambda) \}_{\lambda \in \Lambda}$ is a bounded left approximate identity in $A$ for $X$.
  • So by Theorem 1, for the sequence $(z_n) \in X$ there exists an $a \in A$ and a sequence $(y_n) \in X$ such that $(z_n) = a(y_n)$, that is, $z_n = ay_n$ for each $n \in \mathbb{N}$. Furthermore, since $(y_n) \in X$ by definition of $X$ we have that $\displaystyle{\lim_{n \to \infty} y_n = 0}$. $\blacksquare$
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