Gelfand representation

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.

For any locally compact Hausdorff 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).

The spectrum of a commutative C*-algebra A, denoted <math>\hat{A}<math> consists of the set of non-zero complex-valued *-homomorphisms 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 iff 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 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.

The Banach-Alaoglu theorem of functional analysis asserts that the unit ball 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 compact Hausdorff space.

Statement of the theorem

The Gelfand map on the commutative C*-algebra A is defined as follows:

<math> [\gamma(x)] (f) = f(x). \quad<math>

Theorem. The Gelfand map γ is an isometric *-isomorphism from A onto C₀(X), where X is the spectrum of A.

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 iff x - λ 1 is not invertible iff 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 Gelfand map give rise to a contravariant functor from the category of C*-algebras and morphisms into the category of locally compact Hausdorff spaces and continuous maps.

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.

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 iff x commutes with its adjoint x*, or equivalently iff 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. The there is a *-morphism f → f(x) from the algebra of continuous functions on the spectrum 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

Reference

• William Arveson, An Invitation to C*-Algebra, Springer-Verlag, 1981.