Search

If $A$ is a commutative $C^{\star }$ -algebra and if $\unicode[STIX]{x1D719}:A\rightarrow \mathbb{C}$ is a continuous multiplicative functional such that $\unicode[STIX]{x1D719}(x)$ belongs to the spectrum of $x$ for each $x\in A$ , then $\unicode[STIX]{x1D719}$ is linear and hence a character of $A$. This establishes a multiplicative Gleason–Kahane–Żelazko theorem for $C(X)$. [ABSTRACT FROM AUTHOR]

Abstract 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 (x y) 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. [ABSTRACT FROM AUTHOR]