On the analyticity of the bivariant JLO cocycle

Abstract

The goal of this note is to outline a proof that, for any ℓ ≥ 0, the JLO bivariant cocycle associated with a family of Dirac type operators along a smooth fibration M → B over the pair of algebras (C∞(M), C∞(B)), is entire when we endow C∞(M) with the Cℓ+1 topology and C∞(B) with the Cℓ topology. As a corollary, we deduce that this cocycle is analytic when we consider the Fréchet smooth topologies on C∞(M) and C∞(B).