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

Comments

The Gel'fand representation is also called the Gel'fand transform, cf. [a2] and Commutative Banach algebra.

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.

References

[a1]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5 MR0617913 Zbl 0435.46002
[a2]	W. Rudin, "Functional analysis" , McGraw-Hill (1979) MR1157815 MR0458106 MR0365062 Zbl 0867.46001 Zbl 0253.46001