Representable E-theory for C-0(X)-algebras

Let X be a locally compact space, and let A and B be C-0(X)-algebras. We de fine the notion of an asymptotic C-0(X)-morphism from A to B and construct representable E-theory groups RE(X; A, B). These are the universal groups o n the category of separable C-0(X)-algebras that are C-0(X)-stable, C-0(X)- homotopy-invariant,and half exact. If A is RKK(X)-nuclear,these groups are naturally isomorphic to Kasparov's representable KK-theory groups RKK(X; A, B). Applications and examples are also discussed. (C) 2000 Academic Press.