C⁎-algebra of nonlocal convolution type operators

Abstract

The C⁎-subalgebra B of all bounded linear operators on the space L2(R), which is generated by all multiplication operators by piecewise slowly oscillating functions, by all convolution operators with piecewise slowly oscillating symbols and by the range of a unitary representation of the group of all affine mappings on R, is studied. A faithful representation of the quotient C⁎-algebra Bπ=B/K in a Hilbert space, where K is the ideal of compact operators on the space L2(R), is constructed by applying an appropriate spectral measure decompositions, a local-trajectory method and the Fredholm symbol calculus for the C⁎-algebra of convolution type operators without shifts. This gives a Fredholm symbol calculus for the C⁎-algebra B and a Fredholm criterion for the operators B∈B.