Abstract: I will introduce the K1Delta group of a C*-algebra. This topological abelian group is functorial and invariant under stabilisation. K1Delta can distinguish *-homomorphisms indistinguishable by (K)K-theory and traces -- we illustrate this on the irrational rotation algebra, showing that certain integer actions are not (stably) cocycle conjugate.
The Hausdorffisation of K1Delta forms the missing part of the total stable invariant, for which we prove a classification theorem for *-homomorphisms between stable C*-algebras containing projections.
This is joint work with Jamie Gabe.