Multiplication operators on Banach modules over spectrally separable algebras

Abstract

‎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 $mathcal{A}$‎.‎