On the spectrum of the algebra of singular integral operators with discontinuities in symbols in momenta and coordinates

Victor Kasatkin

Abstract: We study the $C^*$-algebra B generated in $L^2(\mathbb{R})$ 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 B is represented as the restricted direct sum A1 ⊕_C A2. 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 B in terms of the spectra of A1 and A2. We describe all equivalence classes of irreducible representations of the algebra B, the topology on the spectrum of this algebra, and solving composition series. We discuss the abstract index group of the quotient algebra B by the ideal of compact operators and by the ideal com(B) generated by the commutators of elements of the algebra B.