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

Latest revision as of 19:41, 5 June 2020


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
How to Cite This Entry:
Gel'fand representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gel%27fand_representation&oldid=47058
This article was adapted from an original article by E.A. Gorin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article