A C*-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 is an element of B is established. (C) 2013 Elsevier Inc. All rights reserved.