On C*-algebras of singular integral operators with PQC coefficients and shifts with fixed points

A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the (Formula presented.) -algebra (Formula presented.) generated by all multiplication operators by piecewise quasicontinuous (PQC) functions, by the Cauchy singular integral operator (Formula presented.) and by the unitary weighted shift operators (Formula presented.), (Formula presented.) acting on the space (Formula presented.) over the unit circle (Formula presented.). Here G denotes a discrete amenable group of orientation-preserving piecewise smooth homeomorphisms (Formula presented.) with finite sets of discontinuities for their derivatives (Formula presented.), which acts topologically freely on (Formula presented.), where (Formula presented.) is the interior of the nonempty closed set (Formula presented.) composed by all common fixed points for all shifts (Formula presented.), with boundary (Formula presented.) of zero Lebesgue measure. A Fredholm symbol calculus for the (Formula presented.) -algebra (Formula presented.) is constructed and a Fredholm criterion for the operators (Formula presented.) is established.