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.
- Amenable group
- C -algebra
- Functional operator
- Piecewise quasicontinuous function