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We compute the Hochschild homology of the algebra of double-edge pseudo-differential operators. The double-edge algebra is naturally associated to a compact manifold whose boundary is the total space of a fibration of closed manifolds. We introduce residue-type traces on this algebra and on various of its ideals and quotients. As an application, we show that these traces are unique up to a constant. We use these traces to compute the index of Fredholm

We consider the calculus ?*,*de(X, de?½) of double-edge pseudodifferential operators naturally associated to a compact manifold X whose boundary is the total space of a fibration. This fits into the setting of boundary fibration structures, and we discuss the corresponding geometric objects. We construct a scale of weighted double-edge Sobolev spaces on which double-edge pseudodifferential operators act as bounded operators, characterize the Fredholm

We recall a Fredholm criterion for fully elliptic cusp (pseudo)differential operators on a compact manifold with corners of arbitrary codimension, acting on suitable Sobolev spaces. Then we give a formula for the index in terms of regularized `trace' functionals similar to the residue trace of Wodzicki and Guillemin.