Math 2.0

Functional analysis Theorem

Connes–Thom isomorphism

Builds on Crossed productK0 of a C*-algebraK1 of a C*-algebraBott periodicity for C*-algebras

Statement

Let AA be a C*-algebra and let α\alpha be a strongly continuous action of the additive group ℝ\R on AA by automorphisms. Let A⋊αℝA \rtimes_\alpha \R be the crossed product. Then for i=0,1i = 0, 1 there are isomorphisms Ki(A⋊αℝ)≅K1−i(A),K_i(A \rtimes_\alpha \R) \cong K_{1-i}(A), where K0K_0 and K1K_1 are the groups of K0 and K1.

Remarks

Connes proved the theorem in 1981. The isomorphisms are natural with respect to equivariant ∗*-homomorphisms. Example: for the trivial action, A⋊ℝ≅C0(ℝ)⊗A=SAA \rtimes \R \cong C_0(\R) \otimes A = SA, and the theorem reduces to Ki(SA)≅K1−i(A)K_i(SA) \cong K_{1-i}(A), which is Bott periodicity together with K1(A)≅K0(SA)K_1(A) \cong K_0(SA). So the theorem says that, for K-theory, every action of ℝ\R behaves like the trivial action. The name refers to the analogy with the Thom isomorphism of topological K-theory.

Going deeper

For a student

The result is used to compute the K-theory of crossed products by ℝ\R, such as algebras of foliations and flows. Combined with Takai duality, it shows again that crossing twice by ℝ\R gives back K∗(A)K_*(A), consistent with A⋊ℝ⋊ℝ^≅A⊗𝒦A \rtimes \R \rtimes \hat{\R} \cong A \otimes \mathcal{K}. In KK-theory, the isomorphism is given by an invertible element of KK1(A,A⋊αℝ)KK^1(A, A \rtimes_\alpha \R).

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
C*-algebra, concept, no evidence levelC*-algebraconceptCrossed product, concept, no evidence levelCrossed productconceptK0 of a C*-algebra, concept, no evidence levelK0 of a C*-algebraconceptK1 of a C*-algebra, concept, no evidence levelK1 of a C*-algebraconceptTopological K-theory, concept, no evidence levelTopological K-theoryconceptUnitary element, concept, no evidence levelUnitary elementconceptBott periodicity for C*-algebras, statement, no evidence levelBott periodicity for …statementConnes–Thom isomorphism, statement, no evidence levelConnes–Thom isomorphi…statement
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. C*-algebra concept
  2. Crossed product concept
  3. K0 of a C*-algebra concept
  4. K1 of a C*-algebra concept
  5. Topological K-theory concept
  6. Unitary element concept
  7. Bott periodicity for C*-algebras statement
  8. Connes–Thom isomorphism statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Operator_K-theory
About this page
TagPF3ZN28M2A
URIm2://PF3ZN28M2A
Kindstatement, theorem
WritingW0 stub
MSC46L80, 46L55, 19K99
StatusThis copy of the site has no build data. Run m2 status PF3ZN28M2A.
AboutCrossed product, K0 of a C*-algebra, K1 of a C*-algebra
UsesCrossed product, K0 of a C*-algebra, K1 of a C*-algebra, Bott periodicity for C*-algebras
Changed3 November 2026

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