Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

Abstract

The C*-algebra \(\mathfrak{A}\) generated by the Bergman and anti-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L2(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional singular integral operators with coefficients admitting homogeneous discontinuities we reduce the study to simpler C*-algebras associated with points \(z \in \Pi \cup \partial \Pi \) and pairs \((z,\lambda ) \in \partial \Pi \times \mathbb{R}.\) We construct a symbol calculus for unital C*-algebras generated by n orthogonal projections sum of which equals the unit and by m one-dimensional orthogonal projections. Such algebras are models of local algebras at points z ∈∂Π being the discontinuity points of coefficients. A symbol calculus for the C*- algebra \(\mathfrak{A}\) and a Fredholm criterion for the operators \(A \in \mathfrak{A}\) are obtained. Finally, a C*-algebra isomorphism between the quotient algebra \(\mathfrak{A}^\pi = \mathfrak{A}/\mathcal{K}.\) where \(\mathcal{K}\) is the ideal of compact operators, and its analogue \(\mathfrak{A}_D^\pi \) for the unit disk is constructed.