Generalizations of the Gleason-Kahane-Żelazko theorem
Let \(A\) be a commutative, complex Banach algebra with a unit and let \(M\) be a one codimensional subspace of \(A\). A. M. Gleason and, independently, J. P. Kahane and W. . Zelazko proved that (*) \(M\) is an ideal if and only if \(M\) consists only of noninvertible elements. Equivalently, if each element \(f\) of \(M\) belongs to a proper ideal \(I_ f\), which may depend on \(f\), then \(M\) is actually an ideal. There is one- to-one correspondence between one codimensional subspaces (ideals) of any unital Banach algebra \(A\) and one dimensional subspaces of \(A^ \#\), the space of all linear functionals on \(A\) (linear-multiplicative functionals), hence the Gleason-Kahane-. Zelazko theorem can be formulated: (**) Let \(F\in A^ \#\). Then \(F\) is multiplicative if and only if for any \(f\in A\) we have \(F(f)\in\sigma(f)\), where \(\sigma(f)\) denotes the spectrum of \(f\). The aim of this note is to give the history of various extensions and generalizations of the above result and the present some open problems.
scientific article; zbMATH DE number 150387
scientific article