Definitions

# Gelfand representation

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

One of Gelfand's original applications (and one which historically motivated much of the study of Banach algebras) 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 $L^1\left(\left\{mathbb R\right\}\right)$ and $ell^1\left(\left\{mathbb Z\right\}\right)$ whose translates span dense subspaces in the respective algebras.

## 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.

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.

## The Gelfand representation of a commutative Banach algebra

Let A be a commutative Banach algebra, defined over the field $\left\{mathbb C\right\}$ of complex numbers. A non-zero algebra homomorphism $phi:Ato \left\{mathbb C\right\}$ is called a character of $A$; the set of all characters of A is denoted by $Phi_A$.

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

Given $ain A$, one defines the function $widehat\left\{a\right\}:Phi_Ato\left\{mathbb C\right\}$ by $widehat\left\{a\right\}\left(phi\right)=phi\left(a\right)$. The definition of $Phi_A$ and the topology on it ensure that $widehat\left\{a\right\}$ is continuous and vanishes at infinity, and that the map $amapsto widehat\left\{a\right\}$ defines a norm-decreasing, unit-preserving algebra homomorphism from A to $C_0\left(Phi_A\right)$. This homomorphism is the Gelfand representation of A, and $widehat\left\{a\right\}$ 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 $Phi_A$ 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 $A to C_0\left(Phi_A\right)$ may be identified with the Jacobson radical of A. Thus the Gelfand representation is injective if and only if A is semisimple.

### Examples

In the case where $A=L^1\left(\left\{mathbb R\right\}\right)$, the group algebra $\left\{mathbb R\right\}$, then $Phi_A$ is homeomorphic to $\left\{mathbb R\right\}$ and the Gelfand transform of $fin L^1\left(\left\{mathbb R\right\}\right)$ is the Fourier transform $tilde\left\{f\right\}$.

In the case where $A=L^1\left(\left\{mathbb R\right\}_+\right)$, the L1-convolution algebra of the real half-line, then $Phi_A$ is homeomorphic to $\left\{ zin\left\{mathbb C\right\} : Re \left(z\right) geq 0 \right\}$, and the Gelfand transform of an element $fin L^1\left(\left\{mathbb R\right\}\right)$ is the Laplace transform $\left\{mathcal L\right\}f$.

## The C*-algebra case

As motivation, consider the special case $A=C_0\left(X\right)$. Given x in X, let $varphi_x in A^*$ be pointwise evaluation at x, i.e. $varphi_x\left(f\right) = f\left(x\right)$. 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 $Phi_A$ with X, not just as sets but as topological spaces. The Gelfand representation is then an isomorphism $C_0\left(X\right)to C_0\left(Phi_A\right)$.

### 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 Â is closed under weak-* convergence and the spectrum is actually compact. In the non-unital case, the weak-* closure of Â is Â ∪ {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 Â 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 {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

Let A be a commutative C*-algebra and let X be the spectrum of A. Let $gamma:A to C_0\left(X\right)$ 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 $C_0\left(X\right)$). 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 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

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