Representable E-theory for Co(X)-algebras

Let X be a locally compact space, and let A and B be Co(X)-algebras. We define the notion of an asymptotic Co(X)-morphism from A to B and construct representable E-theory groups RE(X;A,B). These are the universal groups on the category of separable Co(X)-algebras that are Co(X)-stable, Co(X)-homotopy-invariant, and half-exact. If A is RKK(X)-nuclear, these groups are naturally isomorphic to Kasparovs representable KK-theory groups RKK(X;A,B). Applications and examples are also discussed.