View source for Gleason-Kahane-Żelazko theorem

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.

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]]].

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.

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.

See [[#References|[a4]]] for more information about the history, related problems, and further 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>