A unital commutative Banach algebra is spectrally separable if for any two distinct non-zero multiplicative linear functionals φ and ψ on it there exist a and b in such that ab = 0 and φ(a)ψ(b) ≠ 0. Spectrally separable algebras are a special subclass of strongly harmonic algebras. We prove that a unital commutative Banach algebra is spectrally separable if there are enough elements in such that the corresponding multiplication operators on have the decomposition property (δ). On the other hand,...