Multiplicative maps into the spectrum

We consider the converse of a famous result of W. Żelazko et al. which characterizes multiplicative functionals amongst the dual space members of a complex unital Banach algebra A. Specifically, we investigate when a continuous multiplicative map φ: A → ℂ, with values φ(x) belonging to the spectrum of x, is automatically linear. Our main result states that if A is a C^∗-algebra, then φ always generates a corresponding character φ of A. It is then shown that φ shares many linear properties with its induced character. Moreover, if A is a von Neumann algebra, then it turns out that φ itself is linear, and that it corresponds to its induced character.