  Let A denote the smallest C*-subalgebra of the algebra of all bounded
operators on L^2(R) containing: (i) all multiplications a(M) by functions a in
C[-\infty,+\infty], (ii) all multiplications e^{ijM}, j in Z, and (iii) all
operators of the form F^{-1}b(M)F, where F denotes the Fourier transform and b
is in C[-\infty,+\infty]. It is known that the principal symbol mapping extends
to a surjective C*-homomorphism \sigma from A into C(M), where M is a certain
compactification of two copies of R. It is also known that E, the kernel of
\sigma, contains the compact ideal K and that the quotient of E by K, is
isomorphic to the direct sum of two copies of C(S^1,K). Using the explicit form
of these two isomorphisms, we are able to compute the connecting mappings in
the cyclic exact sequence in K-theory associated to the homomorphism \sigma and
to proof that K_0(A) is isomorphic to Z and that K_1(A) is isomorphic to Z^2.
  The isomorphism from E/K into C(S^1,K) can be to extended to a
C*-homomorphism \gamma from A into the direct sum of two copies of C(S^1,B),
where B denotes the algebra of all bounded operators on L^2(Z). 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 the translation-by-one automorphism.
Using the Pimsner-Voiculescu exact sequence, we then compute the K-theory of
the image of \gamma. That leads to a second proof that K_0(A) is isomorphic to
Z and that K_1(A) is isomorphic to Z^2.

Operator Algebras;K-Theory and Homology