We study the C*-algebra generated in $ {L^2}\left( \mathbb{R} \right) $ by operators of multiplication by functions with finitely many discontinuities of the first kind and by convolution operators with the Fourier transforms of such functions. The algebra is represented as the restricted direct sum . We express the spectrum of the restricted direct sum in terms of the spectra of its summands. This result is used to express the spectrum of the algebra in terms of the spectra of and . We describe all equivalence classes of irreducible representations of the algebra , the topology on the spectrum of this algebra, and solving composition series. We discuss the abstract index group of the quotient algebra by the ideal of compact operators and by the ideal com generated by the commutators of elements of the algebra . Bibliography: 14 titles.