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In this paper, we study $l^1$-higher index theory and its pairing with cycliccohomology for both closed manifolds and compact manifolds with boundary. Wefirst give a sufficient geometric condition for the vanishing of the$l^1$-higher indices of Dirac-type operators on closed manifolds. This leads usto define an $l^1$-version of higher rho invariants. We prove a product formulafor these $l^1$-higher rho invariants. A main novelty of our product formula isthat 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 geometriccondition for Dirac operators to have well-defined $l^1$-higher indices. Moreprecisely, we show that, on a compact spin manifold $M$ with boundary equippedwith a Riemannian metric which has product structure near the boundary, if thescalar curvature on the boundary is sufficiently large, then the $l^1$-higherindex of its Dirac operator $D_M$ is well-defined and lies in the $K$-theory ofthe $l^1$-algebra of the fundamental group. As an immediate corollary, we seethat if the Bost conjecture holds for the fundamental group of $M$, then the$C^\ast$-algebraic higher index of $D_M$ lies in the image of the Baum-Connesassembly 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 formanifolds with boundary. A key ingredient of its proof is the product formulafor $l^1$-higher rho invariants mentioned above.

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