Unital strongly harmonic commutative Banach algebras

Janko Bračič¹

¹ IMFM University of Ljubljana Jadranska 19 1111 Ljubljana, Slovenia

Studia Mathematica, Tome 149 (2002) no. 3, pp. 253-266

Résumé

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.

Détail

DOI : 10.4064/sm149-3-3

Keywords: unital commutative banach algebra cal spectrally separable distinct non zero multiplicative linear functionals varphi psi there exist cal varphi psi spectrally separable algebras special subclass strongly harmonic algebras prove unital commutative banach algebra cal spectrally separable there enough elements cal corresponding multiplication operators cal have decomposition property delta other cal spectrally separable each cal each banach cal module cal corresponding multiplication operator cal super decomposable these statements improve earlier result baskakov