\(C^*\)-algebras of singular integral operators with shifts having the same nonempty set of fixed points
Let \(\mathbb{T}\) be the unit circle and \(B(L^2(\mathbb{T}))\) the \(C^*\)-algebra of all bounded linear operators acting on \(L^2(\mathbb{T})\). Consider the \(C^*\)-algebra \(U\subset B(L^2(\mathbb{T}))\) generated by all multiplication operators by piecewise slowly oscillating functions, and by Cauchy singular integral operators. Let \(G\) be an amenable discrete group of orientation-preserving diffeomorphisms of \(\mathbb{T}\) onto intself. The authors consider the \(C^*\)-algebra \(V= \text{alg}(U,G)\). The case when all the shifts \(g\in G\), \(g\neq \text{id}\), have the same finite set of fixed points was studied by the authors in [J.~Funct.\ Anal.\ 242, No.\,1, 86--126 (2007; Zbl 1108.47012)]. Here, the authors allow arbitrary common set of fixed points, with possibly nonempty interior. A symbolic calculus is given with an explicit characterization of Fredholm operators.
scientific article; zbMATH DE number 5320875