Gelfand representation

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 algebra structure 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.

The importance of X being locally compact and Hausdorff is that this turns X into a completely regular space. In such a space every closed subset of X is the common zero set of a family of continuous complex-valued functions on X, allowing one to recover the topology of X from C₀(X).

Note that C₀(X) 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

Let ${\displaystyle A}$ be a commutative Banach algebra, defined over the field ${\displaystyle \mathbb{C}}$ of complex numbers. A non-zero algebra homomorphism (a multiplicative linear functional) ${\displaystyle \mathrm{\Phi }:A\to \mathbb{C}}$ is called a character of ${\displaystyle A}$; the set of all characters of ${\displaystyle A}$ is denoted by ${\displaystyle {\mathrm{\Phi }}_A}$.

It can be shown that every character on ${\displaystyle A}$ is automatically continuous, and hence ${\displaystyle {\mathrm{\Phi }}_A}$ is a subset of the space ${\displaystyle A^{\ast }}$ of continuous linear functionals on ${\displaystyle A}$; moreover, when equipped with the relative weak-* topology, ${\displaystyle {\mathrm{\Phi }}_A}$ turns out to be locally compact and Hausdorff. (This follows from the Banach–Alaoglu theorem.) The space ${\displaystyle {\mathrm{\Phi }}_A}$ is compact (in the topology just defined) if and only if the algebra ${\displaystyle A}$ has an identity element.

Given ${\displaystyle a\in A}$, one defines the function ${\displaystyle \hat{a}:{\mathrm{\Phi }}_A\to \mathbb{C}}$ by ${\displaystyle \hat{a}(\varphi )=\varphi (a)}$. The definition of ${\displaystyle {\mathrm{\Phi }}_A}$and the topology on it ensure that ${\displaystyle \hat{a}}$ is continuous and vanishes at infinity, and that the map ${\displaystyle a\mapsto \hat{a}}$ defines a norm-decreasing, unit-preserving algebra homomorphism from ${\displaystyle A}$ to ${\displaystyle C_0({\mathrm{\Phi }}_A)}$. This homomorphism is the Gelfand representation of ${\displaystyle A}$, and ${\displaystyle \hat{a}}$ is the Gelfand transform of the element ${\displaystyle a}$. In general, the representation is neither injective nor surjective.

In the case where ${\displaystyle A}$ has an identity element, there is a bijection between ${\displaystyle {\mathrm{\Phi }}_A}$ and the set of maximal ideals in ${\displaystyle A}$ (this relies on the Gelfand–Mazur theorem). As a consequence, the kernel of the Gelfand representation ${\displaystyle A\to C_0({\mathrm{\Phi }}_A)}$ may be identified with the Jacobson radical of ${\displaystyle A}$. Thus the Gelfand representation is injective if and only if ${\displaystyle A}$ is (Jacobson) semisimple.

Examples

The Banach space ${\displaystyle A=L^1(\mathbb{R})}$ is a Banach algebra under the convolution, the group algebra of ${\displaystyle \mathbb{R}}$. Then ${\displaystyle {\mathrm{\Phi }}_A}$ is homeomorphic to ${\displaystyle \mathbb{R}}$ and the Gelfand transform of ${\displaystyle f\in L^1(\mathbb{R})}$ is the Fourier transform ${\displaystyle \tilde{f}}$. Similarly, with ${\displaystyle A=L^1({\mathbb{R}}_{+})}$, the group algebra of the multiplicative reals, the Gelfand transform is the Mellin transform.

For ${\displaystyle A={\ell }^{\mathrm{\infty }}}$, the representation space is the Stone–Čech compactification ${\displaystyle \beta \mathbb{N}}$. More generally, if ${\displaystyle X}$ is a completely regular Hausdorff space, then the representation space of the Banach algebra of bounded continuous functions is the Stone–Čech compactification of ${\displaystyle X}$.

The C*-algebra case

As motivation, consider the special case A = C₀(X). Given x in X, let ${\displaystyle {\phi }_x\in A^{\ast }}$ be pointwise evaluation at x, i.e. ${\displaystyle {\phi }_x(f)=f(x)}$. Then ${\displaystyle {\phi }_x}$ 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 Φ_A with X, not just as sets but as topological spaces. The Gelfand representation is then an isomorphism

${\displaystyle C_0(X)\to C_0({\mathrm{\Phi }}_A).}$

The spectrum of a commutative 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. (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 {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.

Statement of the commutative Gelfand–Naimark theorem

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

${\displaystyle \gamma :A\to C_0(X)}$

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 commutative 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

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 a normal 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.