On the problem of classification of Banach algebras of singular integral operators withPC-coefficients inL p spaces on composite contours (original) (raw)
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Representations on Hilbert spaces for a nonlocal C * -algebra B of singular integral operators with piecewise slowly oscillating coefficients extended by a group of unitary shift operators are constructed. The group of unitary shift operators U g in the C * -algebra B is associated with a discrete amenable group G of orientation-preserving piecewise smooth homeomorphisms g : T → T that acts topologically freely on T and admits distinct fixed points for different shifts. A C * -algebra isomorphism of the quotient C * -algebra B/K, where K is the ideal of compact operators, onto a C * -algebra of Fredholm symbols is constructed by applying the local-trajectory method, spectral measures and a lifting theorem. As a result, a Fredholm symbol calculus for the C * -algebra B or, equivalently, a faithful representation of the quotient C * -algebra B/K on a suitable Hilbert space is constructed and a Fredholm criterion for the operators B ∈ B is established.