Gelfand-Mazur theorem

Theorem - Let 𝒜 be a unital Banach algebraMathworldPlanetmath over that is also a division algebraMathworldPlanetmath (i.e. every non-zero element is invertible). Then 𝒜 is isometrically isomorphic to .

Proof : Let e denote the unit of 𝒜.

Let x𝒜 and σ(x) be its spectrum. It is known that the spectrum is a non-empty set ( in .

Let λσ(x). Since x-λe is not invertible and 𝒜 is a division algebra, we must have x-λe=0 and so x=λe

Let ϕ:𝒜 be defined by ϕ(λ)=λe.

It is clear that ϕ is an injective algebra homomorphism.

By the above discussion, ϕ is also surjective.

It is isometric because λe=|λ|e=|λ|

Therefore, 𝒜 is isometrically isomorphic to .

Title Gelfand-Mazur theorem
Canonical name GelfandMazurTheorem
Date of creation 2013-03-22 17:29:03
Last modified on 2013-03-22 17:29:03
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Theorem
Classification msc 46H05