K-teoria de operadores pseudodiferenciais na reta com simbolos semiperiodicos (in Portuguese)

Abstract

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 σfrom A into C(M), where M is a certain compactification of two copies of R. It is also known that E, the kernel of σ, 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 σand 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 γfrom 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 γis 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 γ. That leads to a second proof that K_0(A) is isomorphic to Z and that K_1(A) is isomorphic to Z^2.