Topological K-theory is a generalised cohomologyPlanetmathPlanetmath theory on the categoryMathworldPlanetmath of compactPlanetmathPlanetmath Hausdorff spaces. It classifies the vector bundlesMathworldPlanetmath over a space X up to stable equivalences. Equivalently, via the Serre-Swan theorem, it classifies the finitely generated projective modules over the C*-algebraPlanetmathPlanetmath C(X).

Let A be a unital C*-algebra over and denote by M(A) the algebraic direct limitMathworldPlanetmath of matrix algebras Mn(A) under the embeddingsMathworldPlanetmathPlanetmath Mn(A)Mn+1(A):a(a000). Identify the completion of M(A) with the stable algebra A𝕂 (where 𝕂 is the compact operatorsMathworldPlanetmath on l2()), which we will continue to denote by M(A). The K0(A) group is the Grothendieck group (abelian groupMathworldPlanetmath of formal differencesPlanetmathPlanetmath) of the homotopy classes of the projections in M(A). Two projections p and q are homotopicMathworldPlanetmathPlanetmath if there exists a norm continuousPlanetmathPlanetmath path of projections from p to q. Let pMm(A) and qMn(A) be two projections. The sum of their homotopy classes [p] and [q] is the homotopy class of their direct sumMathworldPlanetmathPlanetmathPlanetmath: [p]+[q]=[pq] where pq=diag(p,q)Mm+n(A). Alternatively, one can consider equivalence classesMathworldPlanetmathPlanetmath of projections up to unitary transformations. Unitary equivalence coincides with homotopy equivalenceMathworldPlanetmathPlanetmath in M(A) (or Mn(A) for n large enough).

Denote by U(A) the direct limit of unitary groups Un(A) under the embeddings Un(A)Un+1(A):u(u001). Give U(A) the direct limit topologyMathworldPlanetmath, i.e. a subset U of U(A) is open if and only if UUn(A) is an open subset of Un(A), for all n. The K1(A) group is the Grothendieck group (abelian group of formal differences) of the homotopy classes of the unitaries in U(A). Two unitaries u and v are homotopic if there exists a norm continuous path of unitaries from u to v. Let uUm(A) and vUn(A) be two unitaries. The sum of their homotopy classes [u] and [v] is the homotopy class of their direct sum: [u]+[v]=[uv] where uv=diag(u,v)Um+n(A). Equivalently, one can work with invertibles in GL(A) (an invertible g is connected to the unitary u=g|g|-1 via the homotopyMathworldPlanetmath tg|g|-t).

Higher K-groups can be defined through repeated suspensionsMathworldPlanetmath,

Kn(A)=K0(SnA). (1)

But, the Bott periodicity theorem means that

K1(SA)K0(A). (2)

The main properties of Ki are:

Ki(AB) = Ki(A)Ki(B), (3)
Ki(Mn(A)) = Ki(A)(Morita invariance), (4)
Ki(A𝕂) = Ki(A)(stability), (5)
Ki+2(A) = Ki(A)(Bott periodicity). (6)

There are three flavours of topological K-theory to handle the cases of A being complex (over ), real (over ) or Real (with a given real structure).

Ki(C(X,)) = 𝐾𝑈-i(X)(complex/unitary), (7)
Ki(C(X,)) = 𝐾𝑂-i(X)(real/orthogonal), (8)
𝐾𝑅i(C(X),J) = 𝐾𝑅-i(X,J)(Real). (9)

Real K-theory has a Bott period of 8, rather than 2.


  • 1 N. E. Wegge-Olsen, K-theory and C*-algebras. Oxford science publications. Oxford University Press, 1993.
  • 2 B. Blackadar, K-Theory for Operator Algebras. Cambridge University Press, 2nd ed., 1998.
  • 3 M. Rørdam, F. Larsen and N. J. Laustsen, An Introduction to K-Theory for C*-Algebras. Cambridge University Press, 2000.
Title K-theory
Canonical name Ktheory
Date of creation 2013-03-22 12:58:06
Last modified on 2013-03-22 12:58:06
Owner mhale (572)
Last modified by mhale (572)
Numerical id 17
Author mhale (572)
Entry type Topic
Classification msc 19-00
Synonym Topological K-theory
Related topic KHomology
Related topic AlgebraicKTheory
Related topic GrothendieckGroup