K-Theory of pseudodifferential operators with semi-periodic symbols

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Let A denote the C*algebra of bounded operators on L2(R) generated by: (i) all multiplications a(M) by functions a∈ C[-∞,+∞], (ii) all multiplications by 2π-periodic continuous functions and (iii) all Fourier multipliers F⁻¹b(M)F, where F denotes the Fourier transform and b is in C[-∞,+∞]. The Fredholm property for operators in A is governed by two symbols, the principal symbol σand an operator-valued symbol γ. We give two proofs of the fact that K0(A) and K1(A) are isomorphic, respectively, to Z and Z⊕ Z: by computing the connecting mappings in the standard K-theory six-term exact sequences associated to σand to γ. For the second computation, we prove that the image of γis isomorphic to the direct sum of two copies of the crossed product of C[-∞,+∞] by an automorphism, and then use the Pimsner-Voiculescu exact sequence to compute its K-theory.