Gelfand–Mazur theorem

The Gelfand - Mazur (after Israel Gelfand and Stanisław Mazur ) is one of the starting points of the theory of Banach algebras. He says that the only - Banach algebra is, which is a division ring.

Lemma on the range

Let A be a Banach algebra with unit element 1 - Then there is a for each, so it is not invertible.

We call the set of all for which is not invertible, the spectrum of. This makes it possible to formulate this statement more concise so that the spectrum of an element of a Banach algebra with identity - is not empty.

Evidence

The proof consists of a combination of functional analysis ( Hahn- Banach ) and function theory ( Liouville's theorem ):

We assume that is invertible for each. Then for distinct

Turning now to an arbitrary and share above equation by. It follows

.

The right hand side exists, by continuity, for, since the algebraic operations including inversion in A are continuous and f is continuous. Therefore, the function is holomorphic at all. You vanishes at infinity, and since f is continuous. Therefore, this feature is limited and constant by the theorem of Liouville, so she must be on very equal to 0. Since was arbitrary, it follows from the Hahn- Banach theorem that, but that may not be for an invertible element. This contradiction completes the proof.

Gelfand - Mazur

If the - Banach algebra A is a division ring, then.

If that is so, there is a by the above lemma, so it is not invertible. Since 0 is the only non - invertible element in a skew field, has to be. So every element of A is a multiple of the one, and the assertion follows.

Swell

• R.V. Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras, 1983
• R. Meise, D. Vogt: Introduction to Functional Analysis, Vieweg (1992 )

• Functional Analysis
• Set ( mathematics)