Toeplitz operators and the Roe-Higson type index theorem

TL;DR

This paper extends the Roe-Higson index theorem to even-dimensional partitioned manifolds by introducing a new class of functions and relating an index class to Toeplitz operators via cyclic cocycles.

Contribution

It introduces a broader class of functions on noncompact manifolds and establishes a new index theorem connecting Toeplitz operators and Roe's index theory in even dimensions.

Findings

Defined a new class of functions $C_{w}(M)$ on noncompact manifolds.

Constructed an index class $ ext{Ind}(, D)$ in the $K_1$-group of the Roe algebra.

Proved the pairing of the index class with Roe's cyclic cocycle equals the Fredholm index of a Toeplitz operator.

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 $\phi$ that belongs to $C_{\mathrm{w}}(M)$ we construct an index class $\mathrm{Ind}(\phi , D)$ 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 $\mathrm{Ind}(\phi , D)$ 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 manifold.