Homology and residues of adiabatic pseudodifferential operators

Abstract

We compute the Hochschild homology groups of the adiabatic algebra $\Psi_\alpha(X)$, a deformation of the algebra of pseudodifferential operators $\Psi(X)$ when $X$ is the total space of a fibration of closed manifolds. We deduce the existence and uniqueness of traces on $\Psi_\alpha(X)$ and some of its ideals and quotients, in the spirit of the noncommutative residue of Wodzicki and Guillemin. We introduce certain higher homological versions of the residue trace. When the base of the fibration is $S^1$, these functionals are related to the $\eta$ function of Atiyah-Patodi-Singer.