Point multipliers and the Gleason-Kahane-Żelazko theorem (Q1697691)

Let \(A\) be a unital Banach algebra with non-empty character space \(\sigma(A)\), and let \(X\) be a left Banach \(A\)-module. A linear functional \(\xi\) on \( X\) is said a point multiplier at \(\varphi\in\sigma(A)\) if \(\langle \xi,a\cdot x\rangle = \varphi(a)\langle \xi,x\rangle\), \(a\in A\), \(x\in X\). Denote by \(\sigma_A(X)\) the set of all such multipliers. Main results: Theorem 3.1. For \(A\) and \(X\) as above, with non-empty \(\sigma_A(X)\), we have (i) for a not necessarily continuous linear functional \(\xi\) on \(X\), its kernel is a submodule of \(X\) if and only if there is a \(\varphi\) in \(\sigma(A)\) with \(\langle \xi,a\cdot x \rangle=\varphi(a) \langle\xi,x\rangle\), \(a\in A\), \(x\in X\); (ii) a non-zero \(\xi\) in \(X^*\) is a point multiplier on \(X\) if and only if \(\langle \xi,a\cdot x\rangle \neq 0\) for all \(a\) in \(A^{-1}\), \(x\notin \ker (\xi)\). Theorem 3.14. For \(X\) as above, suppose that \(X=\sum_1^n X(\varphi_i)\), for distinct \(\varphi_i\) in \(\sigma(A)\) (\(X(\varphi)=\{x\in X: a\cdot x=\varphi(a)x\) for all \(a\in A\}\). Then the set of point multipliers on \(X\) can be identified with the disjoint union \(X^*_1\cup\dots\cup X^*_n\).