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, if is spectrally separable, then for each a ∈ and each Banach left -module the corresponding multiplication operator on is super-decomposable. These two statements improve an earlier result of Baskakov.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-3-3,
author = {Janko Bra\v ci\v c},
title = {Unital strongly harmonic commutative Banach algebras},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {253-266},
zbl = {1033.47025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-3-3}
}
Janko Bračič. Unital strongly harmonic commutative Banach algebras. Studia Mathematica, Tome 151 (2002) pp. 253-266. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-3-3/