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 ${\mathbb{K}_m}$ .