Gelfand-Mazur theorem
Theorem - Let be a unital Banach algebra over that is also a division algebra (i.e. every non-zero element is invertible). Then is isometrically isomorphic to .
Proof : Let denote the unit of .
Let and be its spectrum. It is known that the spectrum is a non-empty set (http://planetmath.org/SpectrumIsANonEmptyCompactSet) in .
Let . Since is not invertible and is a division algebra, we must have and so
Let be defined by .
It is clear that is an injective algebra homomorphism.
By the above discussion, is also surjective.
It is isometric because
Therefore, 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 |