$l^1$-higher index, $l^1$-higher rho invariant and cyclic cohomology

Abstract

In this paper, we study l 1 -higher index theory and its pairing with cyclic cohomology for both closed manifolds and compact manifolds with boundary. We first give a sufficient geometric condition for the vanishing of the l 1 -higher indices of Dirac-type operators on closed manifolds. This leads us to define an l 1 -version of higher rho invariants. We prove a product formula for these l 1 -higher rho invariants. A main novelty of our product formula is that it works in the general Banach algebra setting, in particular, the l 1 -setting.On compact spin manifolds with boundary, we also give a sufficient geometric condition for Dirac operators to have well-defined l 1 -higher indices. More precisely, we show that, on a compact spin manifold M with boundary equipped with a Riemannian metric which has product structure near the boundary, if the scalar curvature on the boundary is sufficiently large, then the l 1 -higher index of its Dirac operator D M is well-defined and lies in the Ktheory of the l 1 -algebra of the fundamental group. As an immediate corollary, we see that if the Bost conjecture holds for the fundamental group of M , then the C * -algebraic higher index of D M lies in the image of the Baum-Connes assembly map.By pairing the above K-theoretic l 1 -index results with cyclic cocycles, we prove an l 1 -version of the higher Atiyah-Patodi-Singer index theorem for manifolds with boundary. A key ingredient of its proof is the product formula for l 1 -higher rho invariants mentioned above.