Conjecture: the assembly map μ: \(K^{G}\)_*(EG) → K_*(C*_r(G)) is an isomorphism, where EG is the classifying space for proper G-actions. With coefficients: replace \(K^{G}\)_*(EG) by \(KK^{G}\)(C₀(X), ...) for appropriate coefficient algebras.
Baum and Connes formulated the map in the early 1980s connecting index theory to representation theory. Proven for: torsion-free hyperbolic groups (Connes–Moscovici), word-hyperbolic groups with coefficients (Lafforgue), linear algebraic groups over local fields, discrete subgroups of Lie groups with torsion-free finite-index subgroups. Counterexamples WITH COEFFICIENTS were found by Mislin–Valette for specific groups, showing the coefficient version is strictly harder than the original formulation. The surjectivity half implies the Kadison–Kaplansky idempotent conjecture and connects to the Novikov conjecture.
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