On the algebra generated by the Bergman projection and a shift operator. II (Q1885513)

[For part 1 of this series, see Integral Equations Oper. Theory 46, No.4, 455-471 (2003; Zbl 1031.30021).] Let \(G\subset\mathbb C\) be a bounded domain whose boundary is a finite union of non-intersecting simple closed curves of class \(C^1\). Let \(\alpha\) be a \(C^2\)-diffeomorphism of \(\overline G\) satisfying the Carleman condition \(\alpha\cdot\alpha =\text{id}_{\overline G}\), and let \(J_\alpha\) stand for the Jacobian matrix of \(\alpha\) with respect to the real variables \(x\) and \(y\), where \(z = x+iy\). The mapping \(\alpha\) induces the unitary operator \(W\) on \(L_2(G)\) defined by \((W_\varphi)(z) =\varphi(\alpha(z))\sqrt{| \det J_\alpha(z)| }\). Let \(K\) stand for the orthogonal projection of \(L_2(G)\) onto the Bergman space \(\mathcal A^2(G)\), which consists of all analytic functions of \(L_2(G)\). Let \(\mathcal R \) be the \(C^\ast\)-algebra generated by \(K\), \(W\) and \(C(\overline G)I\). When \(\det J_\alpha\gtrless0\), the symbol algebra of \(\mathcal R\), abstract index groups and index formulae for Fredholm operators in \(\mathcal R\) are obtained. Some interesting examples are considered.