Baum–Connes conjecture

A symmetry group can be studied by moving points in a space or by operators that move entries in a list. Do those two descriptions always give exactly the same stable invariant information? For the integers, adding one moves every point one step along a line. The same move shifts a square-summable list whose positions are labelled by integers. Sums and limits of these shift operators form the group’s reduced operator algebra. K-theory records stable information about an operator algebra rather than each individual operator. Baum–Connes proposes an exact translation from the geometry of proper actions to that K-theory for every countable discrete group, without an additional algebra of coefficients.[1]

A group acting by motions of a space also acts by shifting square-summable functions on itself. Can the geometric description recover exactly the invariant information of the resulting operator algebra, with no loss? For $G=\mathbb Z$, translation by $k$ sends the basis vector $\delta_n$ to $\delta_{n+k}$; the same group acts properly on the real line. For any countable discrete $G$, its left-regular representation generates the reduced group C*-algebra $C_r^*G$ after taking operator-norm closure. Proper-action geometry supplies $K_*^{\mathrm{top}}(G)$. The coefficient-free conjecture requires $\mu_G:K_*^{\mathrm{top}}(G)\to K_*(C_r^*G)$ to be an isomorphism for $*=0,1$. Geometric cycles and their operator indices give the candidate translation; asserting an isomorphism asks both that every target class arise and that distinct source classes not be identified.[1]

Integer translations and free groups are among the known cases; more generally, groups admitting a proper action by affine isometries on a Hilbert space are covered. Not every countable discrete group is assumed to have this property. The arbitrary-group statement without coefficients is unresolved. Counterexamples when an extra coefficient algebra is allowed address a stronger statement and do not disprove this one.[1]

Groups with the Haagerup property, meaning a proper affine isometric action on a Hilbert space, satisfy even the coefficient version in the surveyed theorem. Free groups and hyperbolic groups are also covered by the stated known results. The Hecke-pair paper supplies further cases under co-Haagerup and finite-index-overgroup hypotheses. These assumptions do not cover every countable discrete group. Failure of the general arbitrary-coefficient version is distinct from failure of the coefficient-free assembly requested here.[1] [2]

The group’s operators may be difficult to distinguish by inspecting individual formulas. Stable invariants give another way to compare them: for free groups, the cited survey describes a K-theory group that records the number of free generators and hence distinguishes their reduced operator algebras. The geometric description also connects to the shape of manifolds: suitable injectivity of assembly implies that higher signatures stay unchanged under homotopy equivalence. These are concrete uses of the translation, rather than a claim that K-theory completely classifies all operator algebras.[1] [2]

The known calculation $K_0(C_r^*F_n)=\mathbb Z$ and $K_1(C_r^*F_n)=\mathbb Z^n$ makes the second invariant distinguish the reduced algebras of free groups of different finite ranks. Assembly supplies a geometric organizing principle for such analytic invariants. Its injectivity also controls a different kind of information: the cited survey explains that rational injectivity of the relevant map from a group’s classifying space implies Novikov’s higher-signature invariance, and assembly injectivity supplies that implication. This connects index classes of operators to manifold quantities preserved by homotopy equivalence, rather than merely adding another formula for the assembly map.[1] [2]