Difference between revisions of "Gel'fand representation"
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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> | ||
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− | |||
====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 | + | 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 | + | 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 |
Gel'fand representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gel%27fand_representation&oldid=47058