quotients of Banach algebras

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

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Proof: Denote the quotient norm by $\|\cdot\|_{q}$.

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

 $\|ab+\mathcal{I}\|_{q}\leq\|a+\mathcal{I}\|_{q}\|b+\mathcal{I}\|_{q}$

for every $a,b\in\mathcal{A}$.

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

 $\displaystyle\|ab+\mathcal{I}\|_{q}$ $\displaystyle=$ $\displaystyle\inf_{z\,\in\,ab+\mathcal{I}}\|z\|=\inf_{u\,\in\,a+\mathcal{I}% \atop v\,\in\,b+\mathcal{I}}\|uv\|$ $\displaystyle\leq$ $\displaystyle\inf_{u\,\in\,a+\mathcal{I}\atop v\,\in\,b+\mathcal{I}}\|u\|\|v\|% \leq\inf_{u\,\in\,a+\mathcal{I}}\|u\|\inf_{v\,\in\,b+\mathcal{I}}\|v\|$ $\displaystyle=$ $\displaystyle\|a+\mathcal{I}\|_{q}\|b+\mathcal{I}\|_{q}$

$\square$

Title quotients of Banach algebras QuotientsOfBanachAlgebras 2013-03-22 17:41:54 2013-03-22 17:41:54 asteroid (17536) asteroid (17536) 6 asteroid (17536) Theorem msc 46H10 msc 46H05 msc 46B99