The principal-symbol index map for an algebra of pseudodifferential operators

Abstract

Let B be a compact Riemannian manifold, let Ω denote the cylinder R × B, ∆ Ω its Laplace operator and Λ = (1 -∆ Ω ) -1/2 .Let A denote the C*-algebra of bounded operators on L 2 (R × B) generated by all the classical pseudodifferential operators on R × B of the form LΛ N , N a nonnegative integer and L an N -th order differential operator whose (local) coefficients approach 2π-periodic functions at +∞ and -∞.Let E denote the kernel of the continuous extension of the principal symbol to A.The problem of computing the K-theory index map) is reduced to the problem of computing the Fredholm indices of two elliptic operators on the compact manifold S 1 × B.For B = S 1 , Hess went further and proved in her thesis that K 0 (A) ∼ = Z 5 and K 1 (A) ∼ = Z 4 .

Locations

• Contemporary mathematics - American Mathematical Society - View
• arXiv (Cornell University) - View - PDF