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We formulate and prove an equivariant Bott periodicity theorem for infinite dimensional Euclidean vector spaces. The main features of our argument are (i) the construction of a non-commutative C*-algebra to play the role of the algebra of functions on infinite dimensional Euclidean space; and (ii) the construction of a certain index one elliptic partial differential operator which provides the basis for an inverse to the Bott periodicity map. These tools have applications to index theory and the Novikov conjecture, notably a proof of the Novikov conjecture for amenable groups (the applications will be considered elsewhere). 1998Academic Press

Let A be a graded C * -algebra. We characterize Kasparov's K-theory groupK 0 (A) in terms of graded * -homomorphisms by proving a general converse to the functional calculus theorem for self-adjoint regular operators on graded Hilbert modules. An application to the index theory of elliptic differential operators on smooth closed manifolds and asymptotic morphisms is discussed.

Let X be a locally compact space, and let A and B be C 0 (X)-algebras. We define 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 on 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. Academic Press

This paper provides an E-theoretic proof of an exact form, due to E. Troitsky, of the Mischenko-Fomenko Index Theorem for elliptic pseudodifferential operators over a unital C * -algebra. The main ingredients in the proof are the use of asymptotic morphisms of Connes and Higson, vector bundle modification, a Baum-Douglas-type group, and a KK-argument of Kasparov.