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K-theory and \(C^*\)-algebras. (English) Zbl 0548.46056

Algebraic \(K\)-theory, number theory, geometry and analysis, Proc. int. Conf., Bielefeld/Ger. 1982., Lect. Notes Math. 1046, 55-79 (1984).
[For the entire collection see Zbl 0518.00003.]
This paper contains a very elegant and lucid account of K-theory and Kasprov’s more general KK-theory for \(C^*\)-algebras. It is shown that any homotopy invariant half exact and stable functor E from \(C^*\)- algebras to abelian groups necessarily has Bott periodicity: \(E(A\otimes C_ 0(R^ 2))\cong E(A)\) for all \(C^*\)-algebras A. A detailed discussion is included of Kasparov’s KK groups KK(A,B), these being the sets of all homotopy classes of quasi-homomorphisms A to \(k\otimes B\), where k denotes the algbra of compact operators on countably infinite dimensional Hilbert space. Also Kasparov’s product \(KK(A,B)\times KK(B,C)\to KK(A,C)\) is discussed in detail. This culminates in a simple characterisation of the K-functor on a large class of \(C^*\)-algebras by means of five axioms: homotopy invariance; half exactness; stability; continuity and normalization.
Reviewer: A.K.Seda

MSC:

46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Citations:

Zbl 0518.00003