A Nonlocal C*-Algebra of Singular Integral Operators with Shifts Having Periodic Points

Abstract

Representations on Hilbert spaces for a nonlocal C*-algebra ${{\mathfrak {B}}}$ of singular integral operators with piecewise slowly oscillating coefficients and unitary shift operators are constructed. The group of unitary shift operators U _g of the C*-algebra ${{\mathfrak {B}}}$ is associated with an amenable discrete group of homeomorphisms ${g:{\mathbb{T}}\to{\mathbb{T}}}$ that have piecewise continuous derivatives and the same nonempty set of periodic points. An isometric C*-algebra homomorphism of the quotient C*-algebra ${{\mathfrak {B}}^\pi={\mathfrak {B}}/{\mathcal {K}}}$, where ${{\mathcal {K}}}$ is the ideal of compact operators, into an n × n matrix algebra associated to a C*-algebra ${{\mathfrak {B}}_0}$ of singular integral operators with shifts having only fixed points is established making use of a spectral measure. Based on this generalization of the Litvinchuk–Gohberg–Krupnik reduction scheme, a symbol calculus for the C*-algebra ${{\mathfrak {B}}}$ as well as a Fredholm criterion for the operators in ${{\mathfrak {B}}}$ are obtained.