The paper is devoted to studying the Fredholmness of operators in the Banach algebra \({\mathfrak B}\) of singular integral operators with complex conjugation and slowly oscillating coefficients on a weighted Lebesgue space over a star-like curve \(\Gamma \) with arcs forming angles and cusps at their common point. Applying a reduction of \({\mathfrak B}\) to a Banach algebra generated by the operators of multiplication by slowly oscillating matrix functions, by Wiener–Hopf operators and by Mellin convolution operators on a weighted Lebesgue space of vector functions over half-line \(\mathbb {R}_+\) , we construct a Fredholm symbol calculus for the Banach algebra \({\mathfrak B}\) and establish a Fredholm criterion for the operators \(N\in {\mathfrak B}\) .
