Algebras generated by the Bergman projection and operators of multiplication by piecewise continuous functions (Q1402342)

Let \(D\) be the unit disk in \(\mathbb{C}\) and \(A^2(D)\) be the subspace of \(L_2(D)\) consisting of all analytic functions on \(D\). The Bergman projection \(B\) of \(D\) is the orthogonal projection from \(L_2(D)\) onto \(A^2(D)\). For \({\mathcal A}\subset L_\infty(D)\) consider the algebra \({\mathcal R}_{\mathcal A}\) generated by the Bergman projection and by operators of multiplication by functions of \({\mathcal A}\). Indeed \({\mathcal R}_{\mathcal A}\) is the \(C^*\)-algebra generated by all operators of the form \[ A=a(z)B+ b(z)(I-B)+K \] where \(a(z)\) and \(b(z)\) belongs to \(\mathcal A\) and \(K\) is a compact operator. The symbol algebra of \({\mathcal R}_{\mathcal A}\) is denoted by \(\widehat{\mathcal R}_{\mathcal A}\) and is defined by \(\widehat{\mathcal R}_{\mathcal A}/{\mathcal K}(L_2(D))\) where \({\mathcal K}(L_2(D))\) is the algebra of compact operators in \(L_2(D)\). The author finds in this paper the description of \(\widehat{\mathcal R}_{\mathcal A}\). Interesting facts about representations of \(\widehat{\mathcal R}_{\mathcal A}\) are found and the author constructs a special family of coefficients such that the algebra \(\widehat {\mathcal R}_{\mathcal A}\) has irreducible representations of predefined dimensions.