Multiplication operators on Banach modules over spectrally separable algebras

Document Type : Research Paper


Department of Materials and Metallurgy‎, ‎Faculty of Natural Sciences and Engineering‎, ‎University of Ljubljana‎, ‎Aškerčeva c‎. ‎12‎, ‎SI-1000 Ljubljana‎, ‎Slovenia.


‎Let $\mathcal{A}$ be a commutative Banach algebra and $\mathscr{X}$ be a left Banach $\mathcal{A}$-module‎. ‎We study the set‎ ‎${\rm Dec}_{\mathcal{A}}(\mathscr{X})$ of all elements in $\mathcal{A}$ which induce a decomposable multiplication operator on $\mathscr{X}$‎. ‎In the case $\mathscr{X}=\mathcal{A}$‎, ‎${\rm Dec}_{\mathcal{A}}(\mathcal{A})$ is the well-known Apostol algebra of $\mathcal{A}$‎. ‎We show that ${\rm Dec}_{\mathcal{A}}(\mathscr{X})$ is intimately related with the largest spectrally separable subalgebra of $\mathcal{A}$ and in this context‎ ‎we give some results which are related to an open question if Apostol algebra is regular for any commutative algebra $\mathcal{A}$‎.‎


Main Subjects

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