A survey of bivariant K-theories

Part of the Oberwolfach Seminars book series (OWS, volume 36)


In this chapter, we briefly survey a number of alternative bivariant K-theories. Each one has its own advantages and disadvantages. While we will not give complete details (especially when it comes to Kasparov’s KK-theory, which deserves, and has gotten, whole books by itself: [10, 61, 62, 67, 119]), it is helpful to know what each theory is good for and how the various theories differ from each other and from the bivariant theory developed elsewhere in these notes. The theories are:
  1. 1.

    Gennadi Kasparov’s KK — constructed from “generalised elliptic operators.” This was the first bivariant K-theory to be developed and works for C*-algebras [71]. Kasparov’s theory has been adapted to take into account symmetries such as group actions [73] and groupoid actions [77].

  2. 2.

    BDF-Kasparov Ext — constructed from extensions of C*-algebras by a stable C*-algebra, modulo split extensions. The original BDF (Brown-Douglas-Fillmore) one-variable version of [23] is constructed from C*-algebra extensions by \( \mathcal{K} \) .

  3. 3.

    Algebraic Dual K-Theory — an algebraic analogue of one-variable Ext. This is the easiest of these theories to define.

  4. 4.

    Homotopy-Theoretic KK — an analogue of KK constructed using homotopy theory, with a “built-in UCT.”

  5. 5.

    Connes-Higson E-Theory — A simpler replacement for KK, devised by Alain Connes and Nigel Higson [30], designed to eliminate certain technical difficulties that arise when working with non-nuclear C*-algebras. This often agrees with Kasparov’s theory and is somewhat easier to define; this theory also admits equivariant versions for groups and groupoids.



Exact Sequence Universal Property Inductive Limit Mapping Cone Hilbert Module 
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© Birkhäuser Verlag AG 2007

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