somewhere near the beginning.

The Gelfand representation theorem

Filed under: Mathematics — Alex @ 5:35 pm 7/4/2008

I finally read up to the Gelfand representation theorem in Murphy’s book. This is such a beautiful mathematical result that I couldn’t resist the urge to share it. I tried to write this synopsis so anyone who understands the Banach-Alaoglu theorem should have no problem following the gist of it.

Recall that a (complex) Banach algebra is a (complex) Banach space with a multiplication operation which is continuous w.r.t. the norm. The Gelfand representation theorem says that we can represent an abelian Banach algebra A as the algebra of continuous functions on a locally compact Hausdorff space S. The important case to consider is that of a unital Banach algebra, one with an identity element; in this case, we can take the space to be compact. For the remainder, A will be a unital abelian complex Banach algebra.

The obvious question is, what is this space S? Can it be intrinsically connected to A, or is the proof nonconstructive? It turns out that S is a very natural space; in order to construct it, we need the concept of the spectrum of an algebra.

Let a \in A. We call the set \sigma(a) = \{ \lambda \in \C : \lambda 1 - a \text{ is not invertible} \} the spectrum of a. One can think of the spectrum as a generalization of the eigenvalues of a square matrix or the range of a bounded function. Recall that a (continuous) homomorphism between two algebras is a linear map which preserves the multiplication operation. A character on A is a non-zero homomorphism between A and \C. Let \Omega(A) denote the set of characters on A; it turns out, magically, that \sigma(a) = \{ \tau(a) : \tau \in \Omega(A) \} for all a\in A.

One can show that \Omega(A) is a weak* closed subset of the unit ball of A*, and in fact if we give \Omega(A) the relative weak* topology, it is compact. From here it’s easy to guess that \Omega(A), the character space or spectrum of A, is the appropriate S. Given a \in A, define its Gelfand transform \hat{a} : \Omega(A) \rightarrow \C by \tau \mapsto \tau(a). By the choice of topology on the character space, each such functional is continuous (and in fact, vanishes at infinity).

The Gelfand representation theorem specialized to unital abelian complex Banach algebras then says

The map A \rightarrow C_0(\Omega(A)), \quad a \mapsto \hat{a} is a norm-decreasing homomorphism, and r(a) = \|\hat{a}\|_\infty for all a\in A. Furthermore, \sigma(a) = \hat{a}(\Omega(A)).

Possibly relevant posts:

No Comments »

No comments yet.

RSS feed for comments on this post. TrackBack URL

Leave a comment