Unital strongly harmonic commutative Banach algebras
Volume 149 / 2002
Abstract
A unital commutative Banach algebra ${\cal A}$ is spectrally separable if for any two distinct non-zero multiplicative linear functionals $ \varphi $ and $ \psi $ on it there exist $ a $ and $ b $ in $ {\cal A}$ such that $ ab=0 $ and $ \varphi (a)\psi (b)\not =0. $ Spectrally separable algebras are a special subclass of strongly harmonic algebras. We prove that a unital commutative Banach algebra ${\cal A}$ is spectrally separable if there are enough elements in $ {\cal A}$ such that the corresponding multiplication operators on $ {\cal A}$ have the decomposition property $ (\delta ). $ On the other hand, if ${\cal A} $ is spectrally separable, then for each $ a\in {\cal A}$ and each Banach left $ {\cal A}$-module $ {\cal X} $ the corresponding multiplication operator $ L_a $ on $ {\cal X} $ is super-decomposable. These two statements improve an earlier result of Baskakov.
