A -algebra of singular integral operators with shifts admitting distinct fixed points

Representations on Hilbert spaces for a nonlocal C ⁎ -algebra B of singular integral operators with piecewise slowly oscillating coefficients extended by a group of unitary shift operators are constructed. The group of unitary shift operators U g in the C ⁎ -algebra B is associated with a discrete amenable group G of orientation-preserving piecewise smooth homeomorphisms g : T → T that acts topologically freely on T and admits distinct fixed points for different shifts. A C ⁎ -algebra isomorphism of the quotient C ⁎ -algebra B / K , where K is the ideal of compact operators, onto a C ⁎ -algebra of Fredholm symbols is constructed by applying the local-trajectory method, spectral measures and a lifting theorem. As a result, a Fredholm symbol calculus for the C ⁎ -algebra B or, equivalently, a faithful representation of the quotient C ⁎ -algebra B / K on a suitable Hilbert space is constructed and a Fredholm criterion for the operators B ∈ B is established.