quotients of Banach algebras

Theorem - Let $𝒜$ be a Banach algebra and $ℐ\subseteq 𝒜$ a closed (http://planetmath.org/ClosedSet) ideal. Then $𝒜/ℐ$ is Banach algebra under the quotient norm.

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Proof: Denote the quotient norm by $\parallel \cdot {\parallel }_q$.

By the parent entry (http://planetmath.org/QuotientsOfBanachSpacesByClosedSubspacesAreBanachSpacesUnderTheQuotientNorm) we know that $𝒜/ℐ$ is a Banach space under the quotient norm. Thus, we only need to show the normed algebra inequality:

$${\parallel ab+ℐ\parallel }_q\le {\parallel a+ℐ\parallel }_q{\parallel b+ℐ\parallel }_q$$

for every $a,b\in 𝒜$.

Using the fact that $𝒜$ is a Banach algebra and the definition of quotient norm we have that:

${\parallel ab+ℐ\parallel }_q$$=$$\underset{z\in ab+ℐ}{inf}\parallel z\parallel =\underset{\genfrac{}{}{0pt}{}{u\in a+ℐ}{v\in b+ℐ}}{inf}\parallel uv\parallel$

$\le$$\underset{\genfrac{}{}{0pt}{}{u\in a+ℐ}{v\in b+ℐ}}{inf}\parallel u\parallel \parallel v\parallel \le \underset{u\in a+ℐ}{inf}\parallel u\parallel \underset{v\in b+ℐ}{inf}\parallel v\parallel$

$=$${\parallel a+ℐ\parallel }_q{\parallel b+ℐ\parallel }_q$

$\mathrm{\square }$