Algebras Generated by the Bergman Projection and Operators of Multiplication by Piecewise Continuous Functions

Abstract

Let D be the unit disk. For \( \mathcal{A}\subset L_{\infty}(D) \) containing piecewise continuous functions, we study the C*-algebras \( \mathcal{R_{A}} \) generated by the Bergman projection for D and operators of multiplication by functions of \( \mathcal{A} \). These algebras are related to the algebra generated by more than two projections depending on how many limits a function has at a boundary point. We find the description of the symbol algebra of \( \mathcal{R_{A}} \), denoted here by \( \mathcal{\widehat{R}_{A}} \). Interesting facts about representations of \( \mathcal{\widehat{R}_{A}} \) are found and we construct a special family of coefficients such that the algebra \( \mathcal{\widehat{R}_{A}} \) has irreducible representations of predefined dimensions.