Invertibility Criteria in $${\varvec{C^*}}$$ C ∗ -algebras of Functional Operators with Shifts and $${\varvec{PQC}}$$ PQC Coefficients

Abstract

Let G be an amenable discrete group of orientation-preserving piecewise smooth homeomorphisms $$g:\mathbb {T}\rightarrow \mathbb {T}$$ , with finite sets of discontinuities for their derivatives $$g'$$ , which acts topologically freely on $$\mathbb {T}{\setminus }\Lambda ^\circ $$ , where $$\Lambda ^\circ $$ is the interior of a nonempty closed set $$\Lambda \subset \mathbb {T}$$ composed by all common fixed points for all shifts $$g\in G$$ . Invertibility criteria are established for the operators in the $$C^*$$ -algebra $$\begin{aligned} {\mathcal {A}}:=\mathrm{alg}\,(PQC,U_G)\subset {\mathcal {B}}(L^2(\mathbb {T})) \end{aligned}$$ generated by all functional operators of the form $$\sum _{g\in F}a_gU_g$$ , where $$a_gI$$ are multiplication operators by piecewise quasicontinuous functions $$a_g\in PQC$$ on $$\mathbb {T}$$ , $$U_g:\varphi \mapsto |g'|^{1/2} (\varphi \circ g)$$ are unitary weighted shift operators on $$L^2(\mathbb {T})$$ , and F is any finite subset of the group G.