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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:

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 $M$ 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 $F$ 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.

====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>