Abstract
A unital commutative Banach algebraA is spectrally separable if for any two distinct non-zero multiplicative linear functionals ' and on it there exist a and b in A such that ab = 0 and '(a) (b)6 0: Spectrally separable algebras are a special subclass of strongly harmonic algebras. We prove that a unital commutative Banach algebra A is spectrally separable if there are enough elements in A such that the corresponding multiplication operators onA have the decomposition property ( ): On the other hand, if A is spectrally separable, then for each a 2 A and each Banach left A-module X the corresponding multiplication operator La on X is super-decomposable. These two statements improve an earlier result of Baskakov.
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