View source for Gel'fand representation

A mapping establishing a correspondence between an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g0436301.png" /> of a commutative Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g0436302.png" /> and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g0436303.png" /> on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g0436304.png" /> of maximal ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g0436305.png" />. There exists a one-to-one correspondence between the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g0436306.png" /> and the homomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g0436307.png" /> into the field of complex numbers. If the corresponding identification is made, the Gel'fand representation is realized by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g0436308.png" />. 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|Group algebra of a locally compact group]]) the Gel'fand representation coincides with the [[Fourier transform|Fourier transform]] (for more details see [[Banach algebra|Banach algebra]]). The Gel'fand transform was introduced by I.M. Gel'fand [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.M. Gel'fand, "Normierte Ringe" ''Mat. Sb.'' , '''9 (51)''' : 1 (1941) pp. 3–24 {{MR|}} {{ZBL|0134.32102}} {{ZBL|0031.03403}} </TD></TR></table>



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

Using the Gel'fand representations of specially chosen algebras one can prove various approximation theorems (cf., e.g., [[#References|[a2]]], Sect. 11.13). A well-known such theorem is Wiener's theorem (cf. also [[#References|[a1]]], Chapt. XI, Sect. 2): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g0436309.png" /> is a non-vanishing absolutely-convergent (Fourier) series on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363011.png" /> can be represented as an absolutely-convergent Fourier series on this interval.

In algebraic geometry a very similar representation/transform is used. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363012.png" /> be a commutative ring with unity. To an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363013.png" /> one associates the morphism of affine schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363014.png" /> (function) given by the ring homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363016.png" /> (cf. [[Affine scheme|Affine scheme]]). In the case of affine varieties over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363017.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363018.png" />, where now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363019.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363020.png" />-algebra, takes the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363021.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363022.png" /> at the closed point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363023.png" /> represented by the maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043630/g04363024.png" />, showing the relationship of this construction with the Gel'fand transform.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5 {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Rudin, "Functional analysis" , McGraw-Hill (1979) {{MR|1157815}} {{MR|0458106}} {{MR|0365062}} {{ZBL|0867.46001}} {{ZBL|0253.46001}} </TD></TR></table>