Toeplitz operators and the Roe-Higson type index theorem

2014/05/19 by Seto, Tatsuki
#Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.1405.4852

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_w(M) on noncompact manifolds, which is slightly larger than the algebra of Higson functions. Out of ϕ that belongs to C_w(M) we construct an index class Ind(ϕ, D) in K₁-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 Ind(ϕ, 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.

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