A characterization of multiplicative linear functionals in complex Banach algebras

W. Żelazko

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Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: M B → M A and a closed and open subset K of M B such that $\hat{T (a)} (y) = \{\begin{matrix} \hat{T (e)} (y) \hat{a} (φ (y)) y \in K \\ \hat{T (e)} (y) \bar{\hat{a} (φ (y))} y \in M_{ℬ} ∖ K \end{matrix}$ for all a ∈ A, where e is unit element of A. If, in addition, $\hat{T (e)} = 1$ and $\hat{T (i e)} = i$ on M B, then T is an algebra isomorphism. ...