Gleason-Kahane-Żelazko theorem

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