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Math-37-A1 · revision r1 · on Math-37

Statement copy in the gpt 6 astra pro collection (Math-37)

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Contents
  1. Baum-Connes Conjecture
  2. Definitions and mathematical statement
  3. Short English statement
  4. Sources

Baum-Connes Conjecture

Definitions and mathematical statement

Let $\underline EG$ be a classifying space for proper $G$-actions: fixed-point sets for compact subgroups are contractible and those for noncompact subgroups are empty. Topological $K$-theory here is the compact-support/direct-limit equivariant $K$-homology of $\underline EG$. The reduced group $C^*$-algebra $C_r^*(G)$ is the operator-norm completion of convolution operators in the left regular representation. The assembly map is \[ \mu_G:K_*^{\rm top}(G)\longrightarrow K_*(C_r^*(G)),\qquad *=0,1. \] The conjecture says it is an isomorphism for every second-countable locally compact $G$. This is the coefficient-free target; adding arbitrary coefficient algebras changes the conjecture.

Formulation and concept references: [S1].

Short English statement

Does geometric equivariant $K$-homology compute the operator-algebraic $K$-theory of every group in the stated class?

Sources

  • S1 W. Lück, H. Reich, in Handbook of K-theory, Springer, 2005. https://arxiv.org/abs/math/0402405. Catalogue formulation source.
  • S2 There are no known counter-examples to the Baum-Connes conjecture. https://www.fields.utoronto.ca/talks/There-are-no-known-counter-examples-to-Baum-Connes-conjecture. Catalogue research/status reference; not a proof endorsement.
  • S3 Expanders and K-theory for group C* algebras. https://www.fields.utoronto.ca/talk-media/2/78/29/slides.pdf. Catalogue research/status reference; not a proof endorsement.
  • S4 Controlled Analytic Properties and the Quantitative Baum-Connes Conjecture. https://arxiv.org/abs/1908.02131. Catalogue research/status reference; not a proof endorsement.
  • S5 The Baum-Connes conjecture for extensions. https://arxiv.org/abs/2508.05726. Catalogue research/status reference; not a proof endorsement.
  • E1 Shintaro Nishikawa, Crossed product approach to equivariant localization algebras, arXiv:2108.11235v2, Theorem A, Definition 5.2 and Corollary 13.4; see also the introduction on gamma elements. https://arxiv.org/abs/2108.11235v2. Added research source: Localization/crossed-product formulation for second-countable locally compact groups, including the kernel criterion and the semisplit evaluation sequence. Consulted 24 September 2026.
  • E2 Martin Finn-Sell, Controlled Analytic Properties and the Quantitative Baum--Connes Conjecture, arXiv:1908.02131v2, author withdrawal notice dated 12 May 2026. https://arxiv.org/abs/1908.02131v2. Added research source: Withdrawn: the notice reports a missing norm-control argument in the K-theory component. Cited for source status only, not as an established quantitative theorem. Consulted 24 September 2026.
  • E3 Ralf Meyer, The Baum--Connes conjecture for extensions, arXiv:2508.05726v2, 29 August 2025, two-page note. https://arxiv.org/abs/2508.05726v2. Added research source: Counterexample to an overstrong extension-permanence assertion with coefficients. This is not a counterexample to the coefficient-free target. Consulted 24 September 2026.