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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g1300701.png" /> be a non-zero linear and multiplicative [[Functional|functional]] on a complex [[Banach algebra|Banach algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g1300702.png" /> with a unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g1300703.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g1300704.png" /> denote the set of all invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g1300705.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g1300706.png" />, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g1300707.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g1300708.png" />. A.M. Gleason [[#References|[a1]]] and, independently, J.P. Kahane and W. Żelazko [[#References|[a5]]], [[#References|[a6]]] proved that the property characterizes multiplicative functionals: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g1300709.png" /> is a linear functional on a complex unital Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007012.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007014.png" /> is multiplicative. Equivalently: a [[Linear functional|linear functional]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007015.png" /> on a commutative complex unital Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007016.png" /> is multiplicative if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007019.png" /> stands for the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007020.png" /> (cf. also [[Spectrum of an element|Spectrum of an element]]). As there is a one-to-one correspondence between linear multiplicative functionals and maximal ideals, the theorem can also be phrased in the following way: A codimension-one subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007021.png" /> of a commutative complex unital Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007022.png" /> is an ideal if and only if each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007023.png" /> is contained in a non-trivial ideal. The theorem is not valid for real Banach algebras.
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Let $F$ be a non-zero linear and multiplicative [[Functional|functional]] on a complex [[Banach algebra|Banach algebra]] $\mathcal{A}$ with a unit $e$, and let $\mathcal{A} ^ { - 1 }$ denote the set of all invertible elements of $\mathcal{A}$. Then $F ( e ) = 1$, and for any $a \in \mathcal{A} ^ { - 1 }$ one has $F ( a ) \neq 0$. A.M. Gleason [[#References|[a1]]] and, independently, J.P. Kahane and W. Żelazko [[#References|[a5]]], [[#References|[a6]]] proved that the property characterizes multiplicative functionals: If $F$ is a linear functional on a complex unital Banach algebra $\mathcal{A}$ such that $F ( e ) = 1$ and $F ( a ) \neq 0$ for $a \in \mathcal{A} ^ { - 1 }$, then $F$ is multiplicative. Equivalently: a [[Linear functional|linear functional]] $F$ on a commutative complex unital Banach algebra $\mathcal{A}$ is multiplicative if and only if $F ( a ) \in \sigma ( a )$ for all $a \in \mathcal{A}$, where $\sigma ( a )$ stands for the spectrum of $a$ (cf. also [[Spectrum of an element|Spectrum of an element]]). As there is a one-to-one correspondence between linear multiplicative functionals and maximal ideals, the theorem can also be phrased in the following way: A codimension-one subspace $M$ of a commutative complex unital Banach algebra $\mathcal{A}$ is an ideal if and only if each element of $M$ is contained in a non-trivial ideal. The theorem is not valid for real Banach algebras.
  
 
The Gleason–Kahane–Żelazko theorem has been extended into several directions:
 
The Gleason–Kahane–Żelazko theorem has been extended into several directions:
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007024.png" /> is non-constant [[Entire function|entire function]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007025.png" /> is a linear functional on a complex unital Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007026.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007028.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007030.png" /> is multiplicative [[#References|[a3]]].
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1) If $\varphi$ is non-constant [[Entire function|entire function]] and $F$ is a linear functional on a complex unital Banach algebra $\mathcal{A}$, such that $F ( e ) = 1$ and $F ( a ) \neq 0$ for $a \in \varphi ( \mathcal{A} )$, then $F$ is multiplicative [[#References|[a3]]].
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007031.png" /> be a finite-codimensional subspace of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007032.png" /> of all continuous complex-valued functions on a compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007033.png" />. If each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007034.png" /> is equal to zero at some point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007035.png" />, then the functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007036.png" /> have a common zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007037.png" /> [[#References|[a2]]]. It is not known if the analogous result is valid for all commutative unital Banach algebras.
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2) Let $M$ be a finite-codimensional subspace of the algebra $C ( X )$ of all continuous complex-valued functions on a compact space $X$. If each element of $M$ is equal to zero at some point of $X$, then the functions from $M$ have a common zero in $X$ [[#References|[a2]]]. It is not known if the analogous result is valid for all commutative unital Banach algebras.
  
3) The assumption of linearity of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g130/g130070/g13007038.png" /> has been weakened, and the result has been extended to mappings between Banach and topological algebras.
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3) The assumption of linearity of the functional $F$ has been weakened, and the result has been extended to mappings between Banach and topological algebras.
  
 
See [[#References|[a4]]] for more information about the history, related problems, and further references.
 
See [[#References|[a4]]] for more information about the history, related problems, and further references.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Gleason,  "A characterization of maximal ideals"  ''J. d'Anal. Math.'' , '''19'''  (1967)  pp. 171–172</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Jarosz,  "Finite codimensional ideals in function algebras"  ''Trans. Amer. Math. Soc.'' , '''287''' :  2  (1985)  pp. 779–785</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Jarosz,  "Multiplicative functionals and entire functions II"  ''Studia Math.'' , '''124''' :  2  (1997)  pp. 193–198</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Jarosz,  "When is a linear functional multiplicative?" , ''Function Spaces: Proc. 3rd Conf. Function Spaces'' , ''Contemp. Math.'' , '''232''' , Amer. Math. Soc.  (1999)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.-P. Kahane,  W. Żelazko,  "A characterization of maximal ideals in commutative Banach algebras"  ''Studia Math.'' , '''29'''  (1968)  pp. 339–343</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W. Żelazko,  "A characterization of multiplicative linear functionals in complex Banach algebras"  ''Studia Math.'' , '''30'''  (1968)  pp. 83–85</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  A.M. Gleason,  "A characterization of maximal ideals"  ''J. d'Anal. Math.'' , '''19'''  (1967)  pp. 171–172</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  K. Jarosz,  "Finite codimensional ideals in function algebras"  ''Trans. Amer. Math. Soc.'' , '''287''' :  2  (1985)  pp. 779–785</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  K. Jarosz,  "Multiplicative functionals and entire functions II"  ''Studia Math.'' , '''124''' :  2  (1997)  pp. 193–198</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  K. Jarosz,  "When is a linear functional multiplicative?" , ''Function Spaces: Proc. 3rd Conf. Function Spaces'' , ''Contemp. Math.'' , '''232''' , Amer. Math. Soc.  (1999)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  J.-P. Kahane,  W. Żelazko,  "A characterization of maximal ideals in commutative Banach algebras"  ''Studia Math.'' , '''29'''  (1968)  pp. 339–343</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  W. Żelazko,  "A characterization of multiplicative linear functionals in complex Banach algebras"  ''Studia Math.'' , '''30'''  (1968)  pp. 83–85</td></tr></table>

Latest revision as of 16:59, 1 July 2020

Let $F$ be a non-zero linear and multiplicative functional on a complex Banach algebra $\mathcal{A}$ with a unit $e$, and let $\mathcal{A} ^ { - 1 }$ denote the set of all invertible elements of $\mathcal{A}$. Then $F ( e ) = 1$, and for any $a \in \mathcal{A} ^ { - 1 }$ one has $F ( a ) \neq 0$. A.M. Gleason [a1] and, independently, J.P. Kahane and W. Żelazko [a5], [a6] proved that the property characterizes multiplicative functionals: If $F$ is a linear functional on a complex unital Banach algebra $\mathcal{A}$ such that $F ( e ) = 1$ and $F ( a ) \neq 0$ for $a \in \mathcal{A} ^ { - 1 }$, then $F$ is multiplicative. Equivalently: a linear functional $F$ on a commutative complex unital Banach algebra $\mathcal{A}$ is multiplicative if and only if $F ( a ) \in \sigma ( a )$ for all $a \in \mathcal{A}$, where $\sigma ( a )$ stands for the spectrum of $a$ (cf. also Spectrum of an element). As there is a one-to-one correspondence between linear multiplicative functionals and maximal ideals, the theorem can also be phrased in the following way: A codimension-one subspace $M$ of a commutative complex unital Banach algebra $\mathcal{A}$ is an ideal if and only if each element of $M$ is contained in a non-trivial ideal. The theorem is not valid for real Banach algebras.

The Gleason–Kahane–Żelazko theorem has been extended into several directions:

1) If $\varphi$ is non-constant entire function and $F$ is a linear functional on a complex unital Banach algebra $\mathcal{A}$, such that $F ( e ) = 1$ and $F ( a ) \neq 0$ for $a \in \varphi ( \mathcal{A} )$, then $F$ is multiplicative [a3].

2) Let $M$ be a finite-codimensional subspace of the algebra $C ( X )$ of all continuous complex-valued functions on a compact space $X$. If each element of $M$ is equal to zero at some point of $X$, then the functions from $M$ have a common zero in $X$ [a2]. It is not known if the analogous result is valid for all commutative unital Banach algebras.

3) The assumption of linearity of the functional $F$ has been weakened, and the result has been extended to mappings between Banach and topological algebras.

See [a4] for more information about the history, related problems, and further references.

References

[a1] A.M. Gleason, "A characterization of maximal ideals" J. d'Anal. Math. , 19 (1967) pp. 171–172
[a2] K. Jarosz, "Finite codimensional ideals in function algebras" Trans. Amer. Math. Soc. , 287 : 2 (1985) pp. 779–785
[a3] K. Jarosz, "Multiplicative functionals and entire functions II" Studia Math. , 124 : 2 (1997) pp. 193–198
[a4] K. Jarosz, "When is a linear functional multiplicative?" , Function Spaces: Proc. 3rd Conf. Function Spaces , Contemp. Math. , 232 , Amer. Math. Soc. (1999)
[a5] J.-P. Kahane, W. Żelazko, "A characterization of maximal ideals in commutative Banach algebras" Studia Math. , 29 (1968) pp. 339–343
[a6] W. Żelazko, "A characterization of multiplicative linear functionals in complex Banach algebras" Studia Math. , 30 (1968) pp. 83–85
How to Cite This Entry:
Gleason-Kahane-Żelazko theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gleason-Kahane-%C5%BBelazko_theorem&oldid=22509
This article was adapted from an original article by K. Jarosz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article