Math 2.0

K-theory Operator algebras and K-theory Conjecture

Baum–Connes conjecture

Builds on Group C*-algebraK-homologyK0 of a C*-algebraK1 of a C*-algebraKK-theory

Statement

Let GG be a second countable, locally compact group. Let E‾G\underline{E}G be a universal proper GG-space, the classifying space for proper actions of GG. Let K∗G(E‾G)K_*^G(\underline{E}G), ∗=0,1* = 0, 1, be its equivariant K-homology with GG-compact supports. Let Cr∗(G)C_r^*(G) be the reduced group C*-algebra of GG, with K-groups K0 and K1. The conjecture states that the assembly map μ ⁣:K∗G(E‾G)→K∗(Cr∗(G)),∗=0,1,\mu \colon K_*^G(\underline{E}G) \to K_*(C_r^*(G)), \qquad * = 0, 1, is an isomorphism.

Remarks

The assembly map sends an equivariant elliptic operator on a proper cocompact GG-space to its higher index in the K-theory of Cr∗(G)C_r^*(G); it is built with KK-theory. The conjecture in this form is due to Baum, Connes and Higson (1994), refining the 1982 formulation of Baum and Connes. It is open in general. It is proved for a-T-menable groups, which include amenable groups, by Higson and Kasparov, and for hyperbolic groups, by Lafforgue and by Mineyev and Yu. The version with coefficients in an arbitrary GG-C*-algebra is false: Higson, Lafforgue and Skandalis gave counterexamples using groups that contain expanders. For discrete torsion-free GG, injectivity of μ\mu implies the Novikov conjecture on higher signatures, and surjectivity implies the Kadison–Kaplansky conjecture that Cr∗(G)C_r^*(G) has no projections other than 00 and 11.

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Group C*-algebra, concept, no evidence levelGroup C*-algebraconceptK-homology, concept, no evidence levelK-homologyconceptK0 of a C*-algebra, concept, no evidence levelK0 of a C*-algebraconceptK1 of a C*-algebra, concept, no evidence levelK1 of a C*-algebraconceptKK-theory, concept, no evidence levelKK-theoryconceptBaum–Connes conjecture, conjecture, no evidence levelBaum–Connes conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group C*-algebra concept
  2. K-homology concept
  3. K0 of a C*-algebra concept
  4. K1 of a C*-algebra concept
  5. KK-theory concept
  6. Baum–Connes conjecture conjecture

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Baum%E2%80%93Connes_conjecture
About this page
Tag14CB5N0FPW
URIm2://14CB5N0FPW
Kindconjecture
WritingW0 stub
MSC19K35, 46L80, 58B34
StatusThis copy of the site has no build data. Run m2 status 14CB5N0FPW.
AboutGroup C*-algebra, K-homology
UsesGroup C*-algebra, K-homology, K0 of a C*-algebra, K1 of a C*-algebra, KK-theory
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Changed3 November 2026

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