Equivariant KK-Theory and the Continuous Rokhlin Property

The Universal Property of Equivariant Kk-theory

Let G be a locally compact, σ-compact group. We prove that the equivariant KK-theory, KK, is the universal category for functors from G-algebras to abelian groups which are stable, homotopy invariant and split-exact. This is a generalization of Higsons characterisation of (non-equivariant) KK-theory.

Equivariant KK-theory and noncommutative index theory

Duality, Correspondences and the Lefschetz Map in Equivariant Kk-theory: a Survey

We survey work by the author and Ralf Meyer on equivariant KKtheory. Duality plays a key role in our approach. We organize the survey around the objective of computing a certain homotopy-invariant of a space equipped with a proper action of a group or groupoid called the Lefschetz map. The latter associates an equivariant K-homology class to an equivariant Kasparov self-morphism of a space X ad...

The Rokhlin property and the tracial topological rank

Let A be a unital separable simple C∗-algebra with TR(A) ≤ 1 and α be an automorphism. We show that if α satisfies the tracially cyclic Rokhlin property then TR(A ⋊α Z) ≤ 1. We also show that whenever A has a unique tracial state and αm is uniformly outer for each m and αr is approximately inner for some r > 0, α satisfies the tracial cyclic Rokhlin property. By applying the classification theo...

Finite Cyclic Group Actions with the Tracial Rokhlin Property

We give examples of actions of Z/2Z on AF algebras and AT algebras which demonstrate the differences between the (strict) Rokhlin property and the tracial Rokhlin property, and between (strict) approximate representability and tracial approximate representability. Specific results include the following. We determine exactly when a product type action of Z/2Z on a UHF algebra has the tracial Rok...