Proposition 6.3.5: Quotient Banach Algebra

Proposition 6.3.5 (Quotient Banach Algebra).label Let $\frak{m}$ be a closed ideal of a Banach algebra $A$ (not necessarily abelian.) The quotient Banach space $A/\frak{m}$ is a Banach algebra with respect to the quotient algebra structure.

Proof. $A/\frak{m}$ is a vector space under the standard quotient construction and multiplication is well-defined since $\frak{m}$ is a 2-sided ideal. Since $\frak{m}$ is closed, $A/\frak{m}$ is a Banach space. It remains to show that $A/\frak{m}$ is a Banach algebra.
Let $\pi:A\to A/\frak{m}$ be the quotient map. The norm in $A/\frak{m}$ is given by $\norm{\pi(x)}{}=\INF{m\in\frak{m}}\norm{x+m}{}$. Suppose $\e>0$ then $\exists m,n\in\frak{m}$ satisfying

\begin{align*}\norm{x+m}{}\leq \norm{\pi(x)}{}+\e&&\norm{y+n}{}\leq\norm{\pi(y)}{}+\e\end{align*}

Thus

\begin{align*}\norm{\pi(x)\pi(y)}{}&=\norm{\pi(x+m)\pi(y+n)}{}=\norm{\pi((x+m)(y+n))}{}\\&\leq \norm{(x+m)(y+n)}{}\leq\norm{x+m}{}\norm{y+n}{}\\&\leq \parens{\norm{\pi(x)}{}+\e}\parens{\norm{\pi(y)}{}+\e}\end{align*}

Since $\e$ is arbitrary we are done.$\square$

The Eigenspace

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The Eigenspace

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