K-teoria de operadores pseudodiferenciais com simbolos semi-periodicos no cilindro (in Portuguese)
Abstract
Let A denote the C*-algebra of bounded operators on L2(RxS1) generated by: (a) multiplications by smooth functions on S1; (b) multiplications by continuous functions on the two point compactification of R; (c) multiplications by 2π-periodic continuous functions; (d) the operator L given by the inverse of the square root of the identity operator minus the Laplacian operator on RxS1; and (e) operators of the form DL, where D is either the differencial operator on R or a first order differential operator on S1 with smooth coefficients. Let σ be the complex-valued symbol on A that arises from the Gelfand map of the C*-algebra A/E, where E is the commutator ideal of A. This is the continuous extension of the usual principal symbol of pseudodifferential operators. It is known that E contains the compact ideal K of A and E/K is isomorphic to C(S1, K) C(S1, K), where here K is the algebra of all compact operators on ZxS1. This isomorphism can be extended to a C*-homomorphism γ from A into C(S1, B) C(S1, B), where B denotes the algebra of all bounded operators on ZxS1. We compute the index map in the six-term exact sequence associated to σ, using a Fedosov-Atiyah-Singer index formula. Given A generated by classes of operators in (a), (d) and (e), we prove that the image of γ is isomorphic to the direct sum of two copies of the crossed product of A by an automorphism. We use the Pimsner-Voiculescu exact sequence to compute the K-theory of the crossed product. So, we can prove that K0(A) is isomorphic to Z5 and K1(A) is isomorphic to Z4.