The Banach algebra \({\mathfrak B}\) generated by multiplication operators, Wiener-Hopf operators and Mellin convolution operators with piecewise slowly oscillating and continuous data are studied on weighted Lebesgue spaces. The compactness of commutators of operators in the algebra \({\mathfrak B}\) is established. Using the limit operators techniques, describing the maximal ideal space of a central subalgebra of the quotient Banach algebra \({\mathfrak B}^\pi \) with respect to the ideal of compact operators, and studying the local algebras associated with the Allan-Douglas local principle, we establish homomorphic images for considered local algebras and derive necessary Fredholm conditions for the operators \(A\in {\mathfrak B}\) in terms of the invertibility of such images.