C*-ALGEBRAS OF POLY-BERGMAN TYPE OPERATORS WITH PIECEWISE SLOWLY OSCILLATING COEFFICIENTS

Abstract

Given $n,m\in \mathbb {N}$ and a simply connected uniform domain $U\subset \mathbb {C}$ with a sufficiently smooth boundary $\partial U$, we study the $C^{\ast}$-algebra

$$\begin{aligned} {\mathfrak B}_{U,n,m}({\mathfrak L}):=\mathrm{alg}\{aI,B_{U,1},\ldots ,B_{U,n},\widetilde{B}_{U,1},\ldots ,\widetilde{B}_{U,m}: a\in {\mathfrak X}({\mathfrak L})\}, \end{aligned}$$

generated by the operators of multiplication by functions in ${\mathfrak X}({\mathfrak L})$, by the poly-Bergman projections $B_{U,1},\ldots ,B_{U,n}$ and by the anti-poly-Bergman projections $\widetilde{B}_{U,1},\ldots ,\widetilde{B}_{U,m}$ acting on the Lebesgue space $L^2(U)$. The $C^{\ast}$-algebra ${\mathfrak X}({\mathfrak L})$ is generated by the set $SO_{\partial} (U)$ of all bounded continuous functions on U that slowly oscillate at points of $\partial U$ and by the set $PC({\mathfrak L})$ of all piecewise continuous functions on the closure $\overline{U}$ of U with discontinuities on a finite union ${\mathfrak L}$ of piecewise Dini-smooth curves that have one-sided tangents at every point $z\in {\mathfrak L}$, possess a finite set $Y={\mathfrak L}\cap \partial U$, do not form cusps, and are not tangent to $\partial U$ at the points $z\in Y$. Making use of the Allan-Douglas local principle, the limit operators techniques, quasicontinuous maps, and properties of $SO_{\partial} (U)$ functions, a Fredholm symbol calculus for the $C^{\ast}$-algebra ${\mathfrak B}_{U,n,m}({\mathfrak L})$ is constructed and a Fredholm criterion for its operators is obtained.