K-Theory for group C^*-algebras

These notes are based on a lecture course given by the first author in the Sedano Winter School on K-theory held in Sedano, Spain, on January 22-27th of 2007. They aim at introducing K-theory of C^*-algebras, equivariant K-homology and KK-theory in the context of the Baum-Connes conjecture.

💡 Research Summary

The manuscript is a set of lecture notes that aim to introduce the K‑theory of C*‑algebras, equivariant K‑homology, and Kasparov’s bivariant KK‑theory in the specific context of the Baum‑Connes conjecture. It begins with a concise historical motivation, explaining why the K‑theory of reduced group C*‑algebras is a central object in non‑commutative geometry and how it connects to classical conjectures such as Novikov’s.

The first technical chapter develops the basic K‑theory for a C*‑algebra A. The groups K₀(A) and K₁(A) are defined via projections in matrix algebras over A and unitary elements, respectively. Bott periodicity is proved, and the long exact sequence associated with a short exact sequence 0 → I → A → A/I → 0 is derived, showing how K‑theory behaves under extensions. Concrete calculations are carried out for commutative algebras C(X), matrix algebras Mₙ(ℂ), and reduced group algebras C*_r(G). In particular, for abelian groups the Fourier transform identifies Kₙ(C*_r(G)) with the topological K‑theory of the Pontryagin dual.

The second chapter introduces equivariant K‑homology Kⁿ_G(X) for a G‑space X. Using Kasparov cycles (H,π,F) that are G‑equivariant and satisfy the usual compactness conditions, the authors define K‑homology groups and discuss their functorial properties. A version of Poincaré duality is proved, showing that for a compact G‑manifold the equivariant K‑homology is naturally isomorphic to equivariant K‑theory shifted by the dimension. This sets the stage for the assembly map, which will later identify these groups with the K‑theory of the group C*‑algebra.

The third chapter is devoted to KK‑theory. A Kasparov (A,B)‑module (E,π,T) is defined, where E is a countably generated Hilbert B‑module, π is a ‑representation of A, and T is an operator satisfying the usual “almost‑commutation” and “compactness” conditions. The group KK(A,B) is the set of homotopy classes of such modules. The authors develop the internal product ⊗_B : KK(A,B) × KK(B,C) → KK(A,C), showing that it is associative and provides KK with a triangulated category structure. They also discuss the external product, the Künneth formula, and the Universal Coefficient Theorem in the setting of nuclear C‑algebras. The crucial observation is that ordinary K‑theory and equivariant K‑homology can be realized as special cases: Kₙ(A) ≅ KK(ℂ,A) and Kⁿ_G(X) ≅ KK_G(C₀(X),ℂ).

The fourth and final chapter formulates the Baum‑Connes conjecture. For a second countable locally compact group G, one considers a universal proper G‑space ĒG (a contractible G‑CW complex with proper action). The conjecture asserts that the assembly map
μ : Kⁿ_G(ĒG) → Kₙ(C*_r(G))
induced by the Kasparov product with the “γ‑element” is an isomorphism. The notes explain how to construct μ using the descent homomorphism in KK‑theory and the canonical class in KK_G(C₀(ĒG),ℂ). Known cases where the conjecture holds are surveyed: amenable groups (including all abelian groups), groups acting properly on trees, hyperbolic groups, many linear groups such as SL₂(ℝ), and groups with the Haagerup property. Recent progress on groups with finite asymptotic dimension and on certain lattices in higher rank Lie groups is also mentioned. The authors highlight the remaining challenges, especially for non‑amenable groups lacking a γ‑element equal to 1, and discuss how counter‑examples would impact related conjectures in topology and geometry.

An appendix supplies detailed computations for C*_r(ℤ), C*_r(F_n) (free groups), and C*_r(SL₂(ℝ)), illustrating how the long exact sequence in K‑theory and the Mayer‑Vietoris principle in KK‑theory can be applied. The notes conclude with a set of exercises designed to reinforce the material and to encourage readers to explore current research directions. Overall, the manuscript offers a coherent, self‑contained exposition that bridges elementary K‑theory and the sophisticated machinery required for the Baum‑Connes conjecture, making it a valuable resource for graduate students and researchers entering the field.