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

Representations on Hilbert spaces for a nonlocal C*-algebra 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 is associated with an amenable discrete group of homeomorphisms g:TT that have piecewise continuous derivatives and the same nonempty set of periodic points. An isometric C*-algebra homomorphism of the quotient C*-algebra , where is the ideal of compact operators, into an n× n matrix algebra associated to a C*-algebra 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 as well as a Fredholm criterion for the operators in are obtained.