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. Alternatively, it is a way of representing commutative Banach algebras as continuous functions. The Gelfand representation theorem is one avenue in the development of spectral theory for normal operators.

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

[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.

Given x in X, let φ_x ∈ A* be the pointwise evaluation at x, i.e. φ_x(f) = f(x). There is a natural bijection between space X and the collection of all such functionals, called "characters" of A. In fact, in the relative weak-* topology, this family of functionals is locally compact and Hausdorff. We can then embed an element f from A into A** as the map that sends φ_x to f(x). In fact, given an abstract commutative C*-algebra, these identifications 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. This is the idea behind the Gelfand representation. Details are given below.

[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 the complex numbers. Elements of the spectrum are called characters on A.

Note that spectrum is an overloaded word. It also refers to the spectrum σ(x) of an element x of an algebra with unit 1, that is the set of complex numbers r for which x - r 1 is not invertible in A. For unital C*-algebras, 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. Together with the spectral radius formula, this shows that  is a subset of the unit ball of A* and as such can be given the relative weak-* topology. This is the topology of pointwise convergence. A nets {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).

If A is a separable C*-algebra, the weak-* topology is metrizable on bounded subsets. Thus the spectrum of a separable commutative C*-algebra A can be regarded as a metric space. So the topology can be characterized via convergence of sequences.

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 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 1, all characters f must be unital, i.e. f(1) is the complex number one. This excludes the zero homomorphism. So  is closed under weak-* convergence and the spectrum is actually compact. In the non-unital case, the weak-* closure of  is  ∪ {0}, where 0 is zero the homomorphism, and the 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:

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. The spectrum is called the Structure space in this context. The Gelfand transform is injective if and only if A is semisimple if and only if A is isomorphic to a Banach function algebra.

[edit] References

• W. Arveson (1981). An Invitation to C*-Algebras. Springer-Verlag. ISBN 0-387-90176.