Introduction to the Baum-Connes Conjecture

Let us start with a countable group Γ. We linearize Γ by associating to it the complex group algebra CΓ, where CΓ is the C-vector space with basis Γ. It can also be viewed as the space of functions f: Γ → C with finite support. The product in CΓ is induced by the multiplication in Γ. Namely, for \( f = \sum\nolimits_{8 \in \Gamma } {{f_s}s} \) and \( g = \sum\nolimits_{t \in \Gamma } {{g_t}t} \) elements in CΓ, then which is the usual convolution of f and g, and thus for all t ∈Γ.

Let X be a finite simplicial complex, connected and aspherical (for each \( i \geqslant 2,\pi _i \left( X \right) = 0 \)) and \( \Gamma = {\pi _1}\left( X \right) \). Then X is a classifying space for Γ (or Eilenberg-Mac Lane K(Γ, 1) space). In particular such an X is unique up to homotopy. Note that, under these assumptions Γ is torsion free.

Let A be a unital algebra over C.

In a series of papers [45] [46] [47] [50] from 1980 to 1988, G. Kasparov defined an equivariant KK-theory for pairs of C*-algebras, a powerful machinery to deal both with K-theory and K-homology of C*-algebras.

To illustrate the difficulty of constructing an assembly map, consider the following situation. Let X be a proper Γ -compact space and an even cycle in \(K_0^\Gamma \left( X \right)\).The goal is to define, out of these data, an element in \({K_0}\left( {C_r^ * \Gamma } \right)\). A naive approach would be to consider the kernel and co-kernel of P: these are indeed modules over \(C_r^ * \Gamma \), but these modules are in general not projective of finite type, as shown in the following example:Let Γ = Z, X = Z. Consider the cycle (U, π, F) where U is the left regular representation on H=l² Z, π the representation of C_o(Z) by pointwise multiplication and P = 0. By Example 4.2.3 (1), (U, π, F) is an even cycle, but ker(P)=l² Z is not projective of finite type as a \(C_r^ * \Gamma \)-module. Indeed, via Fourier transform, it gives L²(S¹) as a module over C(S¹) acting by pointwise multiplication.

Let Γ denote a countable group. Consider the following diagram:

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where i = 0,1 and FΓ is as in Theorem 4.2.12.

A length function on a group Γ is a function L:Γ→R ⁺ such that:

Let A be a Γ-C*-algebra. We say that A is proper if there exists a locally compact proper Γ-space X and a Γ-equivariant homomorphism such that: - for all f ∈ C ₀(X)and a, b ∈ A (that is, we view A as a bi-module over itself and require C ₀(X) to act on A by endomorphisms of bi-modules); if a net {f _n } in C ₀(X)converges to 1 uniformly on compact subsets of X, then for all a ∈ A. Note that if A is unital, σ defines a homomorphism .

The basic idea of KK ^Ban-theory is that Hilbert C*-modules over C*-algebras must be replaced by pairs of Banach modules in duality, over more general Banach algebras.