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E-theory

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EE-theory

Idea

EE-Theory is the name of a category whose objects are C*-algebras and whose hom-sets are homotopy classes of slightly generalized C*C*-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.

This construction may be understood as the universal improvement of KK-theory under excision (Higson 90). Accordingly, the EE-groups behave like groups of a K-theory-like generalized cohomology 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).

Definition

First some notation and terminology.

For AA \in C*Alg, we write

ΣAC 0((0,1),A) \Sigma A \coloneqq C_0((0,1),A)

for the C *C^\ast-algebra of continuous AA-valued functions on the open inverval vanishing at infinity. This is also called the suspension of AA.

For A,BA,B \in C*Alg, write [A,B][A,B] for the set of homotopy-equivalence classes of asymptotic C*-homomorphisms ABA \to B. As discussed there

  1. there is a natural composition operation [A,B]×[B,C][A,C][A,B] \times [B,C] \to [A,C];

  2. [A,ΣB][A,\Sigma B] is naturally an abelian group.

Finally, write 𝒦\mathcal{K} \in C*Alg for the C *C^\ast-algebra of compact operators on an infinite-dimensional separable Hilbert space. For AC *AlgA \in C^\ast Alg the tensor product of C*-algebras A𝒦A \otimes \mathcal{K} is also called the stabilization of AA.

Definition

For A,BA,B \in C*Alg, the E-group of AA with coefficients in BB is

E(A,B)[(ΣA)𝒦,(ΣB)𝒦]Ab. E(A,B) \coloneqq [(\Sigma A )\otimes \mathcal{K}, (\Sigma B) \otimes \mathcal{K}] \in Ab \,.

Under the induced composition operation this yields an additive category EE whose objects are C*-algebras, and whose hom-objects are E(,)E(-,-).

Properties

Universal characterization

E-theory is the universal localization C*Alg E\to E which is homotopy-invariant, stable and preserves exact sequences in the middle.

(…)

Relation to KK-theory

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

KKE KK \to E

from the KK-theory homotopy category to that of EE-theory.

Restricted to nuclear C*-algebras this is a full and faithful functor. (Higson 90) (…)

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 KKEKK \to E is then just the corresponding forgetful functor.

It follows that the Kasparov product in KK-theory is equivalently given by the composition of the corresponding completely positive asymptotic C*-homomorphisms.

geometric contextuniversal additive bivariant (preserves split exact sequences)universal localizing bivariant (preserves all exact sequences in the middle)universal additive invariantuniversal localizing invariant
noncommutative algebraic geometrynoncommutative motives Mot addMot_{add}noncommutative motives Mot locMot_{loc}algebraic K-theorynon-connective algebraic K-theory
noncommutative topologyKK-theoryE-theoryoperator K-theory

References

General

The idea of E-theory was introduced in

  • Nigel Higson, Categories of fractions and excision in KK-theory J. Pure Appl. Algebra, 65(2):119–138, (1990) (pdf)

Reviews and surveys include

  • Introduction to KK-theory and E-theory, Lecture notes (Lisbon 2009) (pdf slides)

A standard textbook account is in section 25 of

The stable homotopy theory aspects are further discussed in

See also

Relation to shape theory

Relation to shape theory is discussed in

  • Vladimir Manuilov, Klaus Thomsen, Shape theory and extensions of C *C^\ast-algebras, (arxiv/1007.1663)

Revised on August 14, 2013 14:02:06 by Urs Schreiber (24.131.18.91)