Spectral measures in C∗-algebras of singular integral operators with shifts

The C∗-subalgebra B of B(L2(T)) generated by all multiplication operators by slowly oscillating and piecewise continuous functions, by the Cauchy singular integral operator and by the range of a unitary representation of an amenable group of diffeomorphisms which acts topologically freely on T is studied. A symbol calculus for the C∗-algebra B and a Fredholm criterion for the operators B∈B are established by using a generalization of the local-trajectory method for studying C∗-algebras associated with C∗-dynamical systems. This method is related to the Allan–Douglas local principle and its generalization is based on the notion of spectral measure.