caa a22 4500 467931429 CHVBK 20180406152944.0 cr unu---uuuuu 170328e20060501xx s 000 0 eng 10.1007/s00020-005-1377-1 doi (NATIONALLICENCE)springer-10.1007/s00020-005-1377-1 C *-Algebras of Integral Operators with Piecewise Slowly Oscillating Coefficients and Shifts Acting Freely [Elektronische Daten] [M. Bastos, C. Fernandes, Yu. Karlovich] Abstract.: 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 $$e_{h,\lambda } S_\mathbb{T} e_{h,\lambda }^{ - 1} I\left( {h \in \mathbb{R},\lambda \in \mathbb{T}} \right)$$ 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 Uz 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. Birkhäuser Verlag, Basel, 2006 C *-algebra of operators nationallicence piecewise slowly oscillating coefficients nationallicence shifts nationallicence maximal ideal nationallicence local principle nationallicence local-trajectory method nationallicence two projections theorem nationallicence symbol calculus nationallicence Fredholmness nationallicence representation nationallicence Bastos M. Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049 - 001, Lisboa, Portugal aut Fernandes C. Departamento de Matemática Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2825, Monte de Caparica, Portugal aut Karlovich Yu Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, C.P. 62210 Cuernavaca, Morelos, México aut Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhauser.ch 55/1(2006-05-01), 19-67 0378-620X 55:1<19 2006 55 20 https://doi.org/10.1007/s00020-005-1377-1 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00020-005-1377-1 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bastos M. Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049 - 001, Lisboa, Portugal aut NATIONALLICENCE 700 1- Fernandes C. Departamento de Matemática Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2825, Monte de Caparica, Portugal aut NATIONALLICENCE 700 1- Karlovich Yu Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, C.P. 62210 Cuernavaca, Morelos, México aut NATIONALLICENCE 773 0- Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhauser.ch 55/1(2006-05-01), 19-67 0378-620X 55:1<19 2006 55 20 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer