2009/05/19 by Patricia Hess, Hess, Patricia
Mathematics · #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #math.KT #math.OA
paper · pdf · doi:10.48550/arxiv.0905.3166
This is a PhD Thesis supervised by Severino T. Melo
arxiv created 2009/05/19 · arxiv updated 2009/12/01
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 σbe 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 γfrom 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 γis 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.