Gelfand representation

In mathematics, the Gelfand representation in functional analysis (named after I. M. Gelfand) is either of two things:
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 formula_1 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 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 "C"0("X").
Note that "C"0("X") 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 formula_2 be a commutative Banach algebra, defined over the field formula_3 of complex numbers. A non-zero algebra homomorphism (a multiplicative linear functional) formula_4 is called a "character" of formula_2; the set of all characters of formula_2 is denoted by formula_7.
It can be shown that every character on formula_2 is automatically continuous, and hence formula_7 is a subset of the space formula_10 of continuous linear functionals on formula_2; moreover, when equipped with the relative weak-* topology, formula_7 turns out to be locally compact and Hausdorff. (This follows from the Banach–Alaoglu theorem.) The space formula_7 is compact (in the topology just defined) if and only if the algebra formula_2 has an identity element.
Given formula_15, one defines the function formula_16 by formula_17. The definition of formula_7and the topology on it ensure that formula_19 is continuous and vanishes at infinity, and that the map formula_20 defines a norm-decreasing, unit-preserving algebra homomorphism from formula_2 to formula_22. This homomorphism is the "Gelfand representation of formula_2", and formula_19 is the "Gelfand transform" of the element formula_25. In general, the representation is neither injective nor surjective.
In the case where formula_2 has an identity element, there is a bijection between formula_7 and the set of maximal ideals in formula_2 (this relies on the Gelfand–Mazur theorem). As a consequence, the kernel of the Gelfand representation formula_29 may be identified with the Jacobson radical of formula_2. Thus the Gelfand representation is injective if and only if formula_2 is (Jacobson) semisimple.
Examples.
In the case where formula_32, the group algebra of formula_33, then formula_7 is homeomorphic to formula_33 and the Gelfand transform of formula_36 is the Fourier transform formula_37.
In the case where formula_38, the formula_39-convolution algebra of the real half-line, then formula_7 is homeomorphic to formula_41, and the Gelfand transform of an element formula_42 is the Laplace transform formula_43.
The C*-algebra case.
As motivation, consider the special case "A" = "C"0("X"). Given "x" in "X", let formula_44 be pointwise evaluation at "x", i.e. formula_45. Then formula_46 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 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 "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" 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
This allows us to apply continuous functions to bounded normal operators on Hilbert space.