The Quotient Algebra of an Algebra X Modulo a Two-Sided Ideal J

The Quotient Algebra of an Algebra X Modulo a Two-Sided Ideal J

Definition: Let $X$ be an algebra and let $J \subseteq X$ be a two-sided ideal of $X$. Define an equivalence relation $\sim$ on $X$ where for all $x, y \in X$ we have $x \sim y$ if and only if $x - y \in J$. For each $x \in J$ let $x'$ denote the equivalence class containing $x$, that is, $x' = \{ y \in X : x - y \in L \}$. Let $X / J = \{ x' : x \in X \}$ denote this set of equivalence classes. Then the Quotient Algebra of $X$ Modulo $J$ is the set $X / J$ with the operations of addition, scalar multiplication, and multiplication defined respectively for all $z, w \in X / J$ by:
1) $z + w = (x + y)'$ where $x \in z$ and $y \in w$.
2) $az = (ax)'$ where $x \in z$.
3) $zw = (xy)'$ where $x \in z$ and $y \in w$.

If instead $X$ is only a linear space and $J$ is a linear subspace of $X$, we can still define the equivalence relation $\sim$ on $X$ as above and consider the set $X / J$ with just the operations of addition and scalar multiplication as defined by (1) and (2) above. In this case, $X / J$ becomes a linear space called the **Quotient Linear Space of $X$ Modulo $L$, as we have seen before.

Since a linear space $X$ and a linear space $J$ that is closed, we can define a norm $\| \cdot \| : X/J \to [0, \infty)$ defined for all $z \in X/J$ by:

(1)
\begin{align} \quad \| z \| &= \inf \{ \| x \| : x \in z \} \\ &= \inf \{ \| z - j \| : j \in J \} \end{align}

We are now ready to define (again) the canonical mapping $q : X \to X /J$.

Definition: Let $X$ be a normed linear space and let $J \subseteq X$ be a closed linear subspace of $X$. The Canonical Map is the map $q : X \to X / J$ defined for all $x \in X$ by $q(x) = x'$.
Proposition 1: Let $X$ be a normed linear space and let $J \subseteq X$ be a closed linear subspace. Let $q : X \to X / J$ be the canonical map. Then:
a) If $J$ is a proper subspace of $X$ then $\| q \| = 1$.
b) If $J = X$ then $\| q \| = 0$.
  • Proof of a) Suppose that $J$ is a proper subspace of $X$. For each $x \in X$ we have that:
(2)
\begin{align} \quad \| q(x) \| = \| x' \| = \inf \{ \| y \| : y \in x' \} \leq \| x \| \end{align}
  • (Where the inequality comes from the fact that $x \in x'$). So $\| q \| \leq 1$.
  • On the other hand, since $J$ is a proper subspace of $X$, we can take $z \in X \setminus J$.
  • Since $J$ is closed we have that $\| q(z) \| > 0$. Let $\epsilon > 0$ be such that $\epsilon < 1$. Then:
(3)
\begin{align} \quad \frac{\| q(z) \|}{1 - \epsilon} > \| q(z) \| = \inf \{ \| y \| : y \in z' \} \end{align}
  • By the definition of infimum, there must exist a point $y \in z'$ such that:
(4)
\begin{align} \quad \frac{\| q(z) \|}{1 - \epsilon} &> \| y \| \\ \quad \| q(z) \| &> \| y \| (1 - \epsilon) \end{align}
  • But since $y \in z'$ we have that $\| q(y) \| = \| q(z) \|$, and from above we have that:
(5)
\begin{align} \quad \| q(y) \| & > \| y \| (1 - \epsilon) \end{align}
  • Since $z - y \in J$ we see that $y \neq 0$ otherwise $z$ would be in $J$ (which it cannot be since $z \in X / J$). Thus $\| y \| \neq 0$. Since for all $0 < \epsilon < 1$ there exists such a $y$ which makes the above inequality hold, we conclude that $\| q \| \geq 1$.
  • Thus $\| q \| = 1$. $\blacksquare$
  • Proof of b) Suppose that $J = X$. Let $x \in X$. Then for every $y \in X$ we have that $x - y \in X = L$. So the only $J$-coset is $0'$.
  • If $q : X \to X / J$ is the canonical map then $q(x) = 0$ for every $x \in X$ and so clearly $\| q \| = 0$. $\blacksquare$
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