On the Algebra Generated by the Bergman Projection and a Shift Operator I

Let G (subset of) C be a domain with smooth boundary and let (alpha) be a C^2- diffeomorphism on G satisfying the Carleman condition (alpha) (ring operator) (alpha) = id(G).We denote by R the C*-algebra generated by the Bergman projection of G, all multiplication operators aI (a (element of) C(G) and the operator W (phi)= (radical)(|det J(alpha)|) (phi) (ring operator) (alpha) , where det J(alpha) is the Jacobian of (alpha). A symbol algebra of R is determined and Fredholm conditions are given. We prove that the C*-algebra generated by the Bergman projection of the upper half-plane and the operator (W(phi))(z) = (phi)(-z) is isomorphic and isometric to C^2 * M2(C).