Author/Authors :
Bračič J. نويسنده Department of Materials and Metallurgy, Faculty of Natural Sciences and Engineering, University of Ljubljana, A?ker?eva c. 12, SI-1000 Ljubljana, Slovenia.
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 $mathcal{A}$.
