C*-algebras of Bergman-type operators with piecewise continuous coefficients and shifts

Abstract

Let U be a bounded simply connected domain in ℂ with sufficiently smooth boundary Γ. Let G be a commutative group of conformal mappings of onto itself which is similar to the group of elliptic, hyperbolic or parabolic mappings of the closed unit disc onto itself, and let be a G-invariant set of simple Lyapunov curves in such that for every an at most finite number of curves in are intersecting at z and at every z ∈ Γ these curves form with Γ pairwise distinct angles lying in (0, π). Let be the C*-algebra generated by n poly-Bergman projections, m anti-poly-Bergman projections and by all multiplication operators aI acting on the space L 2(U), where a are piecewise continuous functions on with possible discontinuities on subsets of that are continuous at common fixed points of g ∈ G. For mentioned groups G, applying a local-trajectory method and a Fredholm symbol calculus for the C*-algebra , we establish Fredholm criteria for the operators B in the C*-algebras generated by all operators and all weighted shift operators W g (g ∈ G), where W g f = g′(f ○ g) for f ∈ L 2(U).