Gelfand representation

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In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) has two related meanings:

In the former case, one may regard the Gelfand representation as a far-reaching generalization of the Fourier transform of an integrable function. In the latter case, the Gelfand-Naimark representation theorem is one avenue in the development of spectral theory for normal operators, and generalizes the notion of diagonalizing a normal matrix.

Historical remarks[edit]

One of Gelfand's original applications (and one which historically motivated much of the study of Banach algebras[citation needed]) was to give a much shorter and more conceptual proof of a celebrated lemma of Norbert Wiener (see the citation below), characterizing the elements of the group algebras L1(R) and \ell^1({\mathbf Z}) whose translates span dense subspaces in the respective algebras.

The model algebra[edit]

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.

Note that A is unital if and only if X is compact, in which case C0(X) is equal to C(X), the algebra of all continuous complex-valued functions on X.

Gelfand representation of a commutative Banach algebra[edit]

Let A be a commutative Banach algebra, defined over the field ℂ of complex numbers. A non-zero algebra homomorphism φ: A → ℂ is called a character of A; the set of all characters of A is denoted by ΦA.

It can be shown that every character on A is automatically continuous, and hence ΦA is a subset of the space A* of continuous linear functionals on A; moreover, when equipped with the relative weak-* topology, ΦA turns out to be locally compact and Hausdorff. (This follows from the Banach–Alaoglu theorem.) The space ΦA is compact (in the topology just defined) if[citation needed] and only if the algebra A has an identity element.

Given aA, one defines the function \widehat{a}:\Phi_A\to{\mathbb C} by \widehat{a}(\phi)=\phi(a). The definition of ΦA and the topology on it ensure that \widehat{a} is continuous and vanishes at infinity[citation needed], and that the map a\mapsto \widehat{a} defines a norm-decreasing, unit-preserving algebra homomorphism from A to C0A). This homomorphism is the Gelfand representation of A, and \widehat{a} is the Gelfand transform of the element a. In general, the representation is neither injective nor surjective.

In the case where A has an identity element, there is a bijection between ΦA and the set of maximal proper ideals in A (this relies on the Gelfand–Mazur theorem). As a consequence, the kernel of the Gelfand representation AC0A) may be identified with the Jacobson radical of A. Thus the Gelfand representation is injective if and only if A is (Jacobson) semisimple.

Examples[edit]

In the case where A = L1(R), the group algebra of R, then ΦA is homeomorphic to R and the Gelfand transform of fL1(R) is the Fourier transform \tilde{f}.

In the case where A = L1(R+), the L1-convolution algebra of the real half-line, then ΦA is homeomorphic to {zC: Re(z) ≥ 0}, and the Gelfand transform of an element fL1(R+) is the Laplace transform {\mathcal L}f.

The C*-algebra case[edit]

As motivation, consider the special case A = C0(X). Given x in X, let \varphi_x \in A^* be pointwise evaluation at x, i.e. \varphi_x(f) = f(x). Then \varphi_x is a character on A, and it can be shown that all characters of A are of this form; a more precise analysis shows that we may identify ΦA with X, not just as sets but as topological spaces. The Gelfand representation is then an isomorphism

C_0(X)\to C_0(\Phi_A).\

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

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. (It can be shown that every algebra homomorphism from A to the complex numbers is automatically a *-homomorphism, so that this definition of the term 'character' agrees with the one above.)

In particular, the spectrum of a commutative C*-algebra is a locally compact Hausdorff space: In the unital case, i.e. where 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 the zero homomorphism, and the removal of a single point from a compact Hausdorff space yields a locally compact Hausdorff 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 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 net {fk}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 {fk(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.

Statement of the commutative Gelfand-Naimark theorem[edit]

Let A be a commutative C*-algebra and let X be the spectrum of A. Let

\gamma:A \to C_0(X)

be the Gelfand representation defined above.

Theorem. The Gelfand map γ is an isometric *-isomorphism from A onto C0(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. (See the earlier remarks for the general, commutative Banach algebra case.) For any such m the quotient algebra A/m is one-dimensional (by the Gelfand-Mazur theorem), and therefore any a in A gives rise to a complex-valued function on Y.

In the case of C*-algebras with unit, the spectrum map gives rise to a contravariant functor from the category of C*-algebras with unit and unit-preserving continuous *-homomorphisms, to the category of compact Hausdorff spaces and continuous maps. This functor is one half of a contravariant equivalence between these two categories (its adjoint being the functor that assigns to each compact Hausdorff space X the C*-algebra C0(X)). In particular, given compact Hausdorff spaces X and Y, then C(X) is isomorphic to C(Y) (as a C*-algebra) if and only if X is homeomorphic to Y.

The 'full' 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[edit]

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 ff(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.

References[edit]