Toeplitz Operators and the Roe-Higson Type Index Theorem in Riemannian Surfaces

Abstract

Let $M$ be a non-compact complete Riemannian manifold of dimension two and $N$ a circle in $M$. We assume that $M$ is partitioned by $N$. We define a unital $C^{\ast}$-algebra $C_{b}^{\ast}(M)$, which is slightly larger than the Roe algebra of $M$. We also construct $[u_{\phi}]$ in $K_{1}(C_{b}^{\ast}(M))$, which is a counter part of Roe's odd index class. We prove that Connes' pairing of Roe's cyclic one-cocycle with $[u_{\phi}]$ is equal to the Fredholm index of a Toeplitz operator on $N$. It is a part of an extension of the Roe-Higson index theorem to even-dimensional partitioned manifolds.