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A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the-algebragenerated by all multiplication operators by piecewise quasicontinuous ( PQC ) functions, by the Cauchy singular integral operatorand by the unitary weighted shift operators,acting on the spaceover the unit circle. Here  G  denotes a discrete amenable group of orientation-preserving piecewise smooth homeomorphismswith finite sets of discontinuities for their derivatives, which acts topologically freely on, whereis the interior of the nonempty closed setcomposed by all common fixed points for all shifts, with boundaryof zero Lebesgue measure. A Fredholm symbol calculus for the-algebrais constructed and a Fredholm criterion for the operatorsis established.

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