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

Representations on Hilbert spaces for a nonlocal C⁎C⁎-algebra BB 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 UgUg in the C⁎C⁎-algebra BB is associated with a discrete amenable group G   of orientation-preserving piecewise smooth homeomorphisms g:T→Tg:T→T that acts topologically freely on TT and admits distinct fixed points for different shifts. A C⁎C⁎-algebra isomorphism of the quotient C⁎C⁎-algebra B/KB/K, where KK is the ideal of compact operators, onto a C⁎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⁎C⁎-algebra BB or, equivalently, a faithful representation of the quotient C⁎C⁎-algebra B/KB/K on a suitable Hilbert space is constructed and a Fredholm criterion for the operators B∈BB∈B is established.