invertible elements in a Banach algebra form an open set


Theorem - Let 𝒜 be a Banach algebra with identity elementMathworldPlanetmath e and G(𝒜) be the set of invertible elements in 𝒜. Let Br(x) denote the open ball of radius r centered in x.

Then, for all xG(𝒜) we have that

Bx-1-1(x)G(𝒜)

and therefore G(𝒜) is open in 𝒜.

Proof : Let xG(𝒜) and yBx-1-1(x). We have that

e-x-1y=x-1x-x-1y=x-1(x-y)x-1x-y<x-1x-1-1=1

So, by the Neumann series (http://planetmath.org/NeumannSeriesInBanachAlgebras) we conclude that e-(e-x-1y) is invertiblePlanetmathPlanetmath, i.e. x-1yG(𝒜).

As G(𝒜) is a group we must have yG(𝒜).

So Bx-1-1(x)G(𝒜) and the theorem follows.

Title invertible elements in a Banach algebra form an open set
Canonical name InvertibleElementsInABanachAlgebraFormAnOpenSet
Date of creation 2013-03-22 17:23:22
Last modified on 2013-03-22 17:23:22
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 6
Author asteroid (17536)
Entry type Theorem
Classification msc 46H05