Gleason-Kahane-Żelazko theorem
Let be a non-zero linear and multiplicative functional on a complex Banach algebra
with a unit
, and let
denote the set of all invertible elements of
. Then
, and for any
one has
. A.M. Gleason [a1] and, independently, J.P. Kahane and W. Żelazko [a5], [a6] proved that the property characterizes multiplicative functionals: If
is a linear functional on a complex unital Banach algebra
such that
and
for
, then
is multiplicative. Equivalently: a linear functional
on a commutative complex unital Banach algebra
is multiplicative if and only if
for all
, where
stands for the spectrum of
(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
of a commutative complex unital Banach algebra
is an ideal if and only if each element of
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 is non-constant entire function and
is a linear functional on a complex unital Banach algebra
, such that
and
for
, then
is multiplicative [a3].
2) Let be a finite-codimensional subspace of the algebra
of all continuous complex-valued functions on a compact space
. If each element of
is equal to zero at some point of
, then the functions from
have a common zero in
[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 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 |
Gleason-Kahane-Żelazko theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gleason-Kahane-%C5%BBelazko_theorem&oldid=11915