C*-Algebras of Singular Integral Operators with Shifts Having the Same Nonempty Set of Fixed Points

Abstract

The C*-subalgebra \({\mathfrak{B}}\) of \({\mathcal{B}}(L^2({\mathbb{T}}))\) generated by all multiplication operators by slowly oscillating and piecewise continuous functions, by the Cauchy singular integral operator and by the range of a unitary representation of an amenable group of diffeomorphisms \(g : {\mathbb{T}} \rightarrow {\mathbb{T}}\) with any nonempty set of common fixed points is studied. A symbol calculus for the C*-algebra \({\mathfrak{B}}\) and a Fredholm criterion for its elements are obtained. For the C*-algebra \(\mathcal{A}\) composed by all functional operators in \({\mathfrak{B}}\), an invertibility criterion for its elements is also established. Both the C*-algebras \({\mathfrak{B}}\) and \({\mathcal{A}}\) are investigated by using a generalization of the local-trajectory method for C*-algebras associated with C*-dynamical systems which is based on the notion of spectral measure.