Super-connections and non-commutative geometry

We show that Quillen's formalism for computing the Chern character of the index using superconnections extends to arbitrary operators with functional calculus. We thus remove the condition that the operators have, up to homotopy, a gap in the spectrum. This is proved using differential graded algebras and non-commutative differential forms. Our results also give a new proof of the coincidence of the Chern character of a difference bundle defined using super-connections with the classical definition.