K-teoria de operadores pseudodiferenciais com simbolos semi-periodicos no cilindro (in Portuguese)

TL;DR

This paper studies a class of pseudodifferential operators with semi-periodic symbols on a cylindrical domain, computes their index, and determines their K-theory groups using advanced operator algebra techniques.

Contribution

It introduces a new algebra of pseudodifferential operators with semi-periodic symbols and computes its K-theory and index map using the Fedosov-Atiyah-Singer formula and crossed product methods.

Findings

K0 group is isomorphic to Z^5

K1 group is isomorphic to Z^4

Index map computed via Fedosov-Atiyah-Singer formula

Abstract

Let A denote the C*-algebra of bounded operators on L^2(RxS^1) generated by: (a) multiplications by smooth functions on S^1; (b) multiplications by continuous functions on the two point compactification of R; (c) multiplications by 2\pi-periodic continuous functions; (d) the operator L given by the inverse of the square root of the identity operator minus the Laplacian operator on RxS^1; and (e) operators of the form DL, where D is either the differencial operator on R or a first order differential operator on S^1 with smooth coefficients. Let \sigma 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(S^1, K)\oplus C(S^1, K), where here K is the algebra of all compact operators on ZxS^1. This isomorphism can be extended to a C*-homomorphism \gamma from A into C(S^1, B)\oplus C(S^1, B), where B denotes the algebra of all bounded operators on ZxS^1. We compute the index map in the six-term exact sequence associated to \sigma, using a Fedosov-Atiyah-Singer index formula. Given A\dagger generated by classes of operators in (a), (d) and (e), we prove that the image of \gamma is isomorphic to the direct sum of two copies of the crossed product of A\dagger 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 Z^5 and K1(A) is isomorphic to Z^4.