A $C^*$-algebra of functional operators with shifts having a nonempty set of periodic points.

Let $B(L^2(T))$ be the $C^{*}$-algebra of all bounded linear operators acting on $L^2(T)$, where $T$ is the unit circle in $C$, and let $G$ be an amenable discrete group of orientation-preserving homeomorphisms of $T$ onto itself such that all $gın G\setminus\{e\}$ have piecewise continuous derivatives $g'$ and the same nonempty set $\Lambda$ of periodic points, and there is a point $t ın\Lambda$ with the finite $G$-orbit. The authors study the invertibility in the $C^{*}$-subalgebra of $B(L^2(T))$ generated by all multiplication operators by piecewise slowly oscillating functions and by the group $U_G=\{U_g: g ın G, U_g(\Phi)(t)=|g'(t)|^{(1/2)} \Phi(g(t))\}$ of unitary operators.