Multiplication operators on Banach modules over spectrally separable algebras

Let \pA be a commutative Banach algebra and \eX be a left Banach \pA-moduleý. ýWe study the setý ý\Dec\pA(\eX) of all elements in \pA which induce a decomposable multiplication operator on \eXý. ýIn the case \eX=\pAý, ý\Dec\pA(\pA) is the well-known Apostol algebra of \pAý. ýWe show that \Dec\pA(\eX) is intimately related with the largest spectrally separable subalgebra of \pA and in this contextý ýwe give some results which are related to an open question if Apostol algebra is regular for any commutative algebra \pA
ý.