Representable E-Theory for C0(X)-Algebras

Let X be a locally compact space, and let A and B be C0(X)-algebras. We define the notion of an asymptotic C0(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 C0(X)-algebras that are C0(X)-stable, C0(X)-homotopy-invariant, and half-exact. If A is RKK(X)-nuclear, these groups are naturally isomorphic to Kasparovʹs representable KK-theory groups RKK(X; A, B). Applications and examples are also discussed.