# Gel'fand representation

A mapping establishing a correspondence between an element $a$ of a commutative Banach algebra $A$ and a function $\widehat{a}$ on the space $X$ of maximal ideals of $A$. There exists a one-to-one correspondence between the points of $X$ and the homomorphisms of $A$ into the field of complex numbers. If the corresponding identification is made, the Gel'fand representation is realized by the formula $\widehat{a} ( x) = x( a)$. In the special case of the group algebra of a locally compact Abelian group (with convolution taken as multiplication in the algebra, cf. also Group algebra of a locally compact group) the Gel'fand representation coincides with the Fourier transform (for more details see Banach algebra). The Gel'fand transform was introduced by I.M. Gel'fand [1].

#### References

 [1] I.M. Gel'fand, "Normierte Ringe" Mat. Sb. , 9 (51) : 1 (1941) pp. 3–24 Zbl 0134.32102 Zbl 0031.03403

Using the Gel'fand representations of specially chosen algebras one can prove various approximation theorems (cf., e.g., [a2], Sect. 11.13). A well-known such theorem is Wiener's theorem (cf. also [a1], Chapt. XI, Sect. 2): If $f ( t) = \sum _ {n = - \infty } ^ \infty c _ {n} e ^ {2 \pi i t n }$ is a non-vanishing absolutely-convergent (Fourier) series on the interval $[ 0 , 1 ]$, then $1 / f ( t)$ can be represented as an absolutely-convergent Fourier series on this interval.
In algebraic geometry a very similar representation/transform is used. Let $A$ be a commutative ring with unity. To an element $a \in A$ one associates the morphism of affine schemes $\mathop{\rm Spec} ( A) \rightarrow \mathop{\rm Spec} ( \mathbf Z [ T] )$( function) given by the ring homomorphism $\mathbf Z [ T] \rightarrow A$, $T \mapsto a$( cf. Affine scheme). In the case of affine varieties over an algebraically closed field $k$, the function $\widehat{a} : \mathop{\rm Spec} ( A) \rightarrow \mathop{\rm Spec} ( k [ T] )$, where now $A$ is a $k$- algebra, takes the value $a$ $\mathop{\rm mod} \mathfrak m \in k$ at the closed point of $\mathop{\rm Spec} ( A)$ represented by the maximal ideal $\mathfrak m$, showing the relationship of this construction with the Gel'fand transform.