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