-Theory is the name of a category whose objects are C*-algebras and whose hom-sets are homotopy classes of slightly generalized -homomorphisms, called asymptotic C*-homomorphisms. These hom-sets have the structure of an abelian group and are also called the E-groups of their arguments. Since under Gelfand duality C*-algebras may be thought of as exhibiting noncommutative topology, one also speaks of noncommutative stable homotopy theory.
In terms of noncommutative topology (regarding, in view of Gelfand duality, noncommutative C*-algebras as algebras of functions on “noncommutative topological spaces”) one may understand this as dealing with “locally badly behaved space” such as certain quotients of foliations (Connes-Higson 90) in a way that resembles a noncommutative version of shape theory (Dādārlat 94).
First some notation and terminology.
For C*Alg, we write
for the -algebra of continuous -valued functions on the open inverval vanishing at infinity. This is also called the suspension of .
Because KK-theory is the universal split exact (stable and homotopy-invariant) localization of C*Alg, and E-theory the universal half-exact localization, and since every split exact sequence is in particular exact, there is a universal functor
If in the definition of E-theory by asymptotic C*-homomorphisms one restricts to those which take values in contractive completely positive maps, then the results is isomorphic to KK-theory again. (K. Thomsen, Introduction, p. 34). The above universal functor is then just the corresponding forgetful functor.
|geometric context||universal additive bivariant (preserves split exact sequences)||universal localizing bivariant (preserves all exact sequences in the middle)||universal additive invariant||universal localizing invariant|
|noncommutative algebraic geometry||noncommutative motives||noncommutative motives||algebraic K-theory||non-connective algebraic K-theory|
|noncommutative topology||KK-theory||E-theory||operator K-theory||…|
The idea of E-theory was introduced in
Reviews and surveys include
A standard textbook account is in section 25 of
Martin Grensing, Noncommutative stable homotopy theory (arXiv:1302.4751)
Rasmus Bentmann, Homotopy-theoretic E-theory and n-order (arXiv:1302.6924)
web page of a project Noncommutative topology - homotopy functors and E-theory
Relation to shape theory is discussed in