# Gelfand representation

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

- a way of representing commutative Banach algebras as algebras of continuous functions;
- the fact that for commutative C*-algebras, this representation is an isometric isomorphism.

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}(**R**) and whose translates span dense subspaces in the respective algebras.

## The model algebra

For any locally compact Hausdorff topological space *X*, the space *C*_{0}(*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 *C*_{0}(*X*) is equal to *C*(*X*), the algebra of all continuous complex-valued functions on *X*.

## Gelfand representation of a commutative Banach algebra

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 Φ_{A}.

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

Given *a* ∈ *A*, one defines the function by . The definition of Φ_{A} and the topology on it ensure that is continuous and vanishes at infinity^{[]}, and that the map defines a norm-decreasing, unit-preserving algebra homomorphism from *A* to *C*_{0}(Φ_{A}). This homomorphism is the *Gelfand representation of A*, and 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 Φ_{A} and the set of maximal ideals in *A* (this relies on the Gelfand–Mazur theorem). As a consequence, the kernel of the Gelfand representation *A* → *C*_{0}(Φ_{A}) may be identified with the Jacobson radical of *A*. Thus the Gelfand representation is injective if and only if *A* is (Jacobson) semisimple.

### Examples

In the case where *A* = *L*^{1}(**R**), the group algebra of **R**, then Φ_{A} is homeomorphic to **R** and the Gelfand transform of *f* ∈ *L*^{1}(**R**) is the Fourier transform .

In the case where *A* = *L*^{1}(**R**_{+}), the L^{1}-convolution algebra of the real half-line, then Φ_{A} is homeomorphic to {*z* ∈ **C**: Re(*z*) ≥ 0}, and the Gelfand transform of an element *f* ∈ *L*^{1}(**R**_{+}) is the Laplace transform .

## The C*-algebra case

As motivation, consider the special case *A* = *C*_{0}(*X*). Given *x* in *X*, let be pointwise evaluation at *x*, i.e. . Then 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

### 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 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

be the Gelfand representation defined above.

**Theorem**. The Gelfand map γ is an isometric *-isomorphism from *A* onto *C*_{0}(*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}(*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* 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.

## References

- W. Arveson (1981).
*An Invitation to C*-Algebras*. Springer-Verlag. ISBN 0-387-90176-0. - Frank F. Bonsall, John Duncan (1973).
*Complete Normed Algebras*. Springer-Verlag, New York. ISBN 0-387-06386-2. - N. Wiener (1932). "Tauberian theorems".
*Ann. Of Math. (2)*(Annals of Mathematics)**33**(1): 1–100. doi:10.2307/1968102. JSTOR 1968102.