Gelfand representation

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In mathematics, the Gelfand representation in functional analysis allows a complete characterisation of commutative C*-algebras as algebras of continuous complex-valued functions. The Gelfand representation theorem is one avenue in the development of spectral theory for normal operators.

1 The model algebra
2 The spectrum of a commutative C*-algebra
3 Statement of the theorem
4 Applications
5 Commutative Banach algebras
6 Reference

[edit] The model algebra

For any locally compact Hausdorff topological space X, the space C₀(X) of continuous complex-valued functions on X which vanish at infinity is in a natural way a commutative C*-algebra:

The structure of algebra over the complex numbers is obtained by considering the pointwise operations of addition and multiplication.
The involution is pointwise complex conjugation.
The norm is the uniform norm on functions.

Conversely given a commutative C*-algebra A, one can produce a locally compact Hausdorff space X so that A is *-isomorphic to C₀(X). Moreover, if A is unital, then X is compact, so C₀(X) is equal to C(X), the algebra of all continuous complex-valued functions on X.

In fact the space X can be described precisely: it is the so-called spectrum of A.

[edit] The spectrum of a commutative C*-algebra

See also: Spectrum of a C*-algebra

The spectrum or Gelfand space of a commutative C*-algebra A, denoted Â, consists of the set of non-zero *-homomorphisms from A to $\mathbb{C}$ (the complex numbers). Elements of the spectrum are called characters on A. The spectrum is a subset of the unit ball of A* and as such can be given the weak-* topology. In terms of convergence of nets, this topology can be described as follows: a net {f_k}_k of elements of the spectrum of A converges to f if and only if for each x in A, the net of complex numbers {f_k(x)}_k converges to f(x). Note that if A is a separable C*-algebra, the weak-* topology is metrizable. Thus the spectrum of a separable commutative C*-algebra A can be regarded as a metric space.

Note that spectrum is an overloaded word. It also refers to the spectrum σ(x) of an element x of an algebra with unit, that is the set of complex numbers r for which x - r 1 is not invertible in A. The two notions are connected in the following way: σ(x) is the set of complex numbers f(x) where f ranges over Gelfand space of A. Equivalently, σ(x) is the range of γ(x), where γ is the Gelfand representation defined below.

The Banach-Alaoglu theorem of functional analysis asserts that the unit ball of the dual of a Banach space is weak-* compact. It follows from the Banach-Alaoglu theorem that the spectrum of a commutative C*-algebra is a locally compact Hausdorff space. In the case the C*-algebra has a multiplicative unit element it is easy to see that the spectrum is actually compact, since the condition for a linear functional to be in the spectrum is closed under weak-* convergence. In the general case, removal of a single point from a compact Hausdorff space yields a locally compact Hausdorff space.

[edit] Statement of the theorem

Let A be a commutative C*-algebra and let X be the spectrum of A. The Gelfand map, or the Gelfand representation γ on A is defined as follows:

$[\gamma(x)] (f) = f(x), \quad x\in A,\ f\in X.$

Theorem. The Gelfand map γ is an isometric *-isomorphism from A onto C₀(X).

The idea of the proof is as follows. If A has an identity element, we claim that for any element x of A, the range of values of the function γ(x) is the same as the spectrum of the element of x. In fact λ is a spectral value of x if and only if x - λ 1 is not invertible if and only if x − λ 1 belongs to at least one maximal ideal m of A. Now by the Gelfand-Mazur theorem on Banach fields, the quotient A/m is naturally identified with the complex numbers C. It remains to show the resulting homomorphism is a *-homomorphism and that the spectral radius of x equals the norm of x. See the Arveson reference below.

The spectrum of a commutative C*-algebra can also be viewed as the set of all maximal ideals m of A, with the hull-kernel topology. For any such m it is shown that A/m is naturally identified to the field of complex numbers C. Therefore any a in A gives rise to a complex-valued function on Y.

The spectrum map give rise to a contravariant functor from the category of C*-algebras with unit and morphisms into the category of compact Hausdorff spaces and continuous maps. In particular, given compact Hausdorff spaces X and Y, then C(X) is isomorphic to C(Y) if and only if X is homeomorphic to Y. In the case of C*-algebras with unit, the Gelfand map is a natural transformation.

The Gelfand–Naimark theorem is a result for arbitrary (abstract) noncommutative C*-algebras A, which though not quite analogous to the Gelfand representation, does provide a concrete representation of A as an algebra of operators.

[edit] Applications

One of the most significant applications is the existence of a continuous functional calculus for normal elements in C*-algebra A: An element x is normal if and only if x commutes with its adjoint x*, or equivalently if and only if it generates a commutative C*-algebra C*(x). By the Gelfand isomorphism applied to C*(x) this is *-isomorphic to an algebra of continuous functions on a locally compact space. This observation leads almost immediately to:

Theorem. Let A be a C*-algebra with identity and x an element of A. Then there is a *-morphism f → f(x) from the algebra of continuous functions on the spectrum σ(x) into A such that

It maps 1 to the multiplicative identity of A;
It maps the identity function on the spectrum to x.

This allows us to apply continuous functions to bounded normal operators on Hilbert space.

[edit] Commutative Banach algebras

The same construction may be carried out for a commutative Banach algebra A. In this case, the representation one obtains is a continuous homomorphism into C₀(X), but it is not in general an isomorphism of Banach algebras.