A nonlocal \(C^*\)-algebra of singular integral operators with shifts having periodic points
Let \(\mathbb T\) be the unit circle. A function \(f \in L^{\infty} (\mathbb T)\) is said to be slowly oscillating at a point \(\lambda \in \mathbb T\) if \[ \lim_{\epsilon \to 0}\operatorname{ess}\operatorname{sup} \{\;| f(z_1)-f(z_2)|: z_1, z_2 \in {\mathbb T}_{\epsilon} (\lambda) \}=0, \] where \( {\mathbb T}_{\epsilon} (\lambda) = \{z \in \mathbb T: \epsilon / 2 \leq |z- \lambda| \leq \epsilon \}\). Let \(PSO(\mathbb T)\) stand for the \(C^*\)-subalgebra of \(L^{\infty}(\mathbb T)\) generated by the piecewise continuous functions on \(\mathbb T\) and the slowly oscillating functions on \(\mathbb T\). In the paper under review, the authors construct a symbol calculus for the \(C^*\)-subalgebra \(\mathcal B\) of \(B(L^2 (\mathbb T))\) generated by \(\mathcal U\) and \(U_G\), where \( \mathcal U\) is the \(C^*\)-algebra of singular integral operators on \(L^2 (\mathbb T)\) with \(PSO(\mathbb T)\) coefficients and \( U_G\) is the group of unitary weighted shift operators given by \[ (U_g \varphi)(t)= |g'(t)|^{\frac{1}{2}} \varphi (g(t)),\; t \in \mathbb T,\; \varphi \in L^2(\mathbb T),\; g \in G, \] for an amenable discrete group of orientation \(G\) preserving homeomorphisms, \(G= \{ g: \mathbb T \to \mathbb T\}\), such that: (i) all the homeomorphisms \(g \in G\) have piecewise continuous derivatives \(g'\) with at most finite sets of discontinuities; (ii) there exists \(t_0 \in \mathbb T\) whose \(G\)-orbit has a finite cardinality \(n>1\); (iii) all elements \(g \in G \setminus \{e\}\) have the same nonempty set of periodic points. \smallskip Using this symbol calculus, the authors obtain a Fredholm criterion for the operators in \(\mathcal B\). Their method is based on a generalization of the reduction scheme of Litvinchuc, Gohberg and Krupnik. Other papers by the authors directly connected to this topic are [Integral Equations Oper. Theory 55, No. 1, 19--67 (2006; Zbl 1124.47010)], [Complex Anal. Oper. Theory 2, No. 2, 241--272 (2008; Zbl 1158.47066)].
scientific article; zbMATH DE number 6048410
scientific article