Multiplication operators on Banach modules over spectrally separable algebras

Let A be a commutative Banach algebra and X be a left Banach A-module. We study the set DecA(X ) of all elements in A which induce a decomposable multiplication operator on X . In the case X = A, DecA(A) is the well-known Apostol algebra of A. We show that DecA(X ) is intimately related with the largest spectrally separable subalgebra of 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 A.