Toeplitz operators and the Roe–Higson type index theorem

Abstract

Let $M$ be a complete Riemannian manifold and assume that $M$ is partitioned by a hypersurface $N$. In this paper we introduce a novel class of functions $C_{\mathrm{w}}(M)$ on noncompact manifolds, which is slightly larger than the algebra of Higson functions. Out of $\varphi$ that belongs to $C_{\mathrm{w}}(M)$ we construct an index class Ind$(\varphi ,D)$ in $K_1$-group of the Roe algebra of $M$ by using the Kasparov product. It is supposed to be a counterpart of Roe's odd index class. We finally prove that Connes' pairing of $\mathrm{Ind}(\varphi ,D)$ and Roe's cyclic $1$-cocycle is equal to the Fredholm index of a Toeplitz operator on $N$. This is an extension of the Roe–Higson index theorem to even-dimensional partitioned manifolds.