On weighted Lebesgue spaces over \(\mathbb {R}_+\) with power weights, the Banach algebra \({\mathfrak D}\) generated by multiplication operators, Wiener-Hopf operators and Mellin convolution operators with piecewise slowly oscillating data is studied. In contrast to [21], where such algebra was studied for piecewise continuous data, we consider piecewise slowly oscillating data for every generator of \({\mathfrak D}\) . Using the limit operators techniques, studying the compactness of commutators of operators in \({\mathfrak D}\) with slowly oscillating data and applying the Allan-Douglas local principle, we describe the maximal ideal space \({\mathfrak M}\) of a new central subalgebra of the quotient Banach algebra \({\mathfrak D}^\pi \) with respect to the ideal of compact operators. The set \({\mathfrak M}\) depends on three parameters \(\xi ,\eta ,\mu \) . Taking the closed two-sided ideals \(\mathcal {J}^\pi _{\xi ,\eta ,\mu }\) of \({\mathfrak D}^\pi \) for every point \((\xi ,\eta ,\mu )\in {\mathfrak M}\) , we describe the quotient Banach algebras \({\mathfrak D}^\pi _{\xi ,\eta ,\mu } ={\mathfrak D}^\pi /\mathcal {J}^\pi _{\xi ,\eta ,\mu }\) and reduce the study to investigating the local algebras \({\mathfrak D}^\pi _{\xi ,\eta ,\mu }\) for each of four subsets of \({\mathfrak M}\) .
