[Part I, cf. Stud. Math. 119, No. 3, 289-297 (1996; Zbl 0868.46037).] The following result is proved in this very interesting paper: Let \(A\) be a complex Banach algebra with a unit \(e\), let \(F\) be a nonconstant entire function, and let \(T\) be a linear functional with \(T(e) = 1\) and such that \(T\circ F\) defined from \(A\) to the complex field is nonsurjective. Then \(T\) is multiplicative. This proves a conjecture of R. Arens. The particular case when \(F\) is the exponential function was proved independently in 1967 by A. Gleason and J. P. Kahane and W. Zelazko.