Invertibility Criteria in C ^∗ -algebras of Functional Operators with Shifts and PQC Coefficients

Let G be an amenable discrete group of orientation-preserving piecewise smooth homeomorphisms g: T→ T, with finite sets of discontinuities for their derivatives g ^′ , which acts topologically freely on T\ Λ ^∘ , where Λ ^∘ is the interior of a nonempty closed set Λ ⊂ T composed by all common fixed points for all shifts g∈ G. Invertibility criteria are established for the operators in the C ^∗ -algebra A:=alg(PQC,UG)⊂B(L2(T))generated by all functional operators of the form ∑ _{g ∈ F} a _g U _g , where a _g I are multiplication operators by piecewise quasicontinuous functions a _g ∈ PQC on T, Ug:φ↦|g′|1/2(φ∘g) are unitary weighted shift operators on L ² (T) , and F is any finite subset of the group G.