C*-Algebras of Integral Operators with Piecewise Slowly Oscillating Coefficients and Shifts Acting Freely

摘要

We establish a symbol calculus for the C*-subalgebra $mathcal{A}$ of $mathcal{B}left( {L^2 left( mathbb{T} right)} right)$ generated by the operators of multiplication by slowly oscillating and piecewise continuous functions and the operators $e_{h,lambda } S_mathbb{T} e_{h,lambda }^{ - 1} Ileft( {h in mathbb{R},lambda in mathbb{T}} right)$ where $S_{mathbb{T}} $ is the Cauchy singular integral operator and $e_{{h,lambda }} (t): = exp {left( {h(t + lambda )/(t - lambda )} right)},t in mathbb{T}backslash {left{ lambda right}}.$ The C*-algebra $mathcal{A}$ is invariant under the transformations $$A mapsto U_{z} A{user1{U}}^{{{user1{ - 1}}}}_{{user1{z}}} quad {text{and}}quad A mapsto e_{{h,lambda }} Ae^{{ - 1}}_{{h,lambda }} I{left( {z,lambda in mathbb{T},;h in mathbb{R}} right)},$$ where U z is the rotation operator ${left( {U_{z} varphi } right)}(t): = varphi (zt),;t in mathbb{T}.$ Using the localtrajectory method, which is a natural generalization of the Allan-Douglas local principle to nonlocal type operators, we construct symbol calculi and establish Fredholm criteria for the C*-algebra $mathfrak{B}$ generated by the operators $A in mathcal{A}$ and $U_{z} (z in mathbb{T}),$ for the C*-algebra $mathfrak{C}$ generated by the operators $A in mathcal{A}$ and $e_{{h,lambda }} I{left( {h in mathbb{R},;lambda in mathbb{T}} right)},$ and for the C*-algebra $mathfrak{D}$ generated by the algebras $mathfrak{B}$ and $mathfrak{C}.$ The C*-algebra $mathfrak{B}$ can be considered as an algebra of convolution type operators with piecewise slowly oscillating coefficients and shifts acting freely.