C*-Algebras of Bergman Type Operators with Piecewise Constant Coefficients over Sectors

Abstract

Given \({m \in \mathbb{N}}\), we study the C*-algebra \({\mathfrak{A}_m(\mathfrak{L})}\) generated by the operators of multiplication by piecewise constant functions with discontinuities on a system \({\mathfrak{L}}\) of rays starting from the origin and by the Bergman and anti-Bergman projections acting on the Lebesgue space \({L^2(\mathbb{K}_m)}\) over the sector $$\begin{array}{ll}\mathbb{K}_m = \{z = re^{i\theta}\,: \, r > 0, \theta \in (0, \pi /m) \}.\end{array}$$ A symbol calculus for the C*-algebra \({\mathfrak{A}_m(\mathfrak{L})}\) is constructed and an invertibility criterion for operators \({A \in \mathfrak{A}_m(\mathfrak{L})}\) in terms of their symbols is established. The C*-algebras of Bergman type operators are studied for the first time in domains with non-smooth boundaries, and obtained results essentially depend on the angle of the sector \({\mathbb{K}_m}\) .