C ^∗-Algebras of Two-Dimensional Singular Integral Operators with Shifts

Abstract

We construct a Fredholm symbol calculus for the C ^∗-algebra ${\mathfrak{B}}$ generated by the C ^∗-algebra ${\mathfrak{A}}$ of two-dimensional singular integral operators with continuous coefficients on a bounded closed simply connected domain $\overline{U}\subset\mathbb{R}^{2}$ with Liapunov boundary and by all unitary shift operators W _g where g runs through a discrete solvable group G=F⋊H of diffeomorphisms of $\overline{U}$ onto itself, where F is a commutative group of conformal mappings, H={e,γ} and γ is similar to the shift $z\mapsto \overline{z}$. As a result, we establish a Fredholm criterion for the operators $B\in{\mathfrak{B}}$.

The first author was partially supported by the SEP-CONACYT Project No. 168104 and by PROMEP via “Proyecto de Redes” (México).