Abstract
The self-adjoint algebra of operators generated by the semigroup of translation operators acting on the Hilbert space of functions supported on the half-line is studied. A real-valued index is introduced and is used to determine the spectrum of the Wiener-Hopf integral operators with distribution kernel having an almost periodic Fourier transform. Further, the algebra is shown to contain no nonzero compact operators, and the quotient of the algebra by its commutator ideal is shown to be isometrically isomorphic to the Banach algebra of almost periodic functions on the line.
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