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The C * -algebra An,m generated by the n poly-Bergman and m antipoly-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L 2 (Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional convolution operators with symbols admitting homogeneous discontinuities we reduce the study to simpler C * -algebras associated with points z ∈ Π ∪ ∂Π and pairs (z, λ) ∈ ∂Π × R. Applying a symbol calculus for the abstract unital C * -algebras generated by N orthogonal projections sum of which equals the unit and by M = n + m one-dimensional orthogonal projections and using relations for the Gauss hypergeometric function, we study local algebras at points z ∈ ∂Π being the discontinuity points of coefficients. A symbol calculus for the C * -algebra An,m is constructed and a Fredholm criterion for the operators A ∈ An,m is obtained. Mathematics Subject Classification (2000). Primary 47A53, 47L15; Secondary 33C05, 47G10, 47L30.

The C * -algebra A generated by the Bergman and anti-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L 2 (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 ∈ Π ∪ ∂Π and pairs (z, λ) ∈ ∂Π × 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 A and a Fredholm criterion for the operators A ∈ A are obtained. Finally, a C * -algebra isomorphism between the quotient algebra A π = A/K, where K is the ideal of compact operators, and its analogue A π D for the unit disk is constructed. Mathematics Subject Classification (2000). Primary 47A53, 47L15; Secondary 47G10, 47L30.

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