quotients of Banach algebras

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

Proof: Denote the quotient norm by q.

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


for every a,b𝒜.

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

ab+q = infzab+z=infua+vb+uv
= a+qb+q

Title quotients of Banach algebras
Canonical name QuotientsOfBanachAlgebras
Date of creation 2013-03-22 17:41:54
Last modified on 2013-03-22 17:41:54
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 6
Author asteroid (17536)
Entry type Theorem
Classification msc 46H10
Classification msc 46H05
Classification msc 46B99