Functional calculus and topological K-theory

Abstract

The difference between algebraic and topological K-theory has its roots in analysis: K0 behaves like topological K-theory as long as certain tools of functional analysis apply. A central issue is the holomorphic functional calculus, which underlies several important approximation lemmas in K-theory. We shall do functional analysis using bornologies instead of topologies. We prefer bornologies because bornological algebras are very close to Banach algebras and hence provide a very natural setting for the functional calculus.