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We consider a multiplicative variation on the classical Kowalski-S lodkowski 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.2010 Mathematics Subject Classification. 15A60, 46H05, 46H10, 46H15, 47B10.

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