Some character generating functions on Banach algebras

We consider a multiplicative variation on the classical Kowalski–Słodkowski Theorem which identifies the characters among the collection of all functionals on a Banach algebra A. In particular we show that, if A is a C^⁎-algebra, and if ϕ:A↦C is a continuous function satisfying ϕ(1)=1 and ϕ(x)ϕ(y)∈σ(xy) for all x,y∈A (where σ denotes the spectrum), then ϕ generates a corresponding character ψ_ϕ on A which coincides with ϕ on the principal component of the invertible group of A. We also show that, if A is any Banach algebra whose elements have totally disconnected spectra, then, under the aforementioned conditions, ϕ is always a character.