Let \(PQC_\mathbb {R}\) be the \(C^*\) -algebra of all piecewise quasicontinuous functions on the real line \(\mathbb {R}\) that are equivalent to piecewise slowly oscillating functions at infinity, and let \(PQC_{\mathbb {R},p,w}\) be the corresponding Banach algebra of Fourier multipliers on the weighted Lebesgue space \(L^p(\mathbb {R},w)\) with \(p\in (1,\infty )\) and a Muckenhoupt weight w. Under some conditions, the maximal ideal space \(\Omega \) of a central subalgebra \(\mathcal {Z}^\pi _{\mathbb {R},p,w}\) of the quotient Banach algebra \(\mathfrak {A}^\pi _{\mathbb {R},p,w}=\mathfrak {A}_{\mathbb {R},p,w}/\mathcal {K}_{p,w}\) is described, and invertibility criteria for elements of \(\mathcal {Z}^\pi _{\mathbb {R},p,w}\) are obtained, where the Banach algebra \(\mathfrak {A}_{\mathbb {R},p,w}\) is generated by the multiplication operators aI with \(a\in PQC_\mathbb {R}\) and the convolution operators \(W^0(b)\) with \(b\in PQC_{\mathbb {R},p,w}\) , and \(\mathcal {K}_{p,w}\) is the ideal of all compact operators on the space \(L^p(\mathbb {R},w)\) . Applying the Allan-Douglas local principle, the Fredholmness of operators \(A\in \mathfrak {A}_{\mathbb {R},p,w}\) is established in terms of invertibility of cosets \(A_{\xi ,\eta }^\pi \) for all pairs \((\xi ,\eta )\in \Omega \) in certain explicitly given local algebras \(\Lambda _{\xi ,\eta }^\pi \) .
