nLab E-theory

Idea

$E$-Theory is the name of a category whose objects are C*-algebras and whose hom-sets are homotopy classes of slightly generalized $C*$-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 $E$-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 $A \in$ C*Alg, we write

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

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

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

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

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

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

Definition

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

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

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

Properties

Universal characterization

E-theory is the universal localization C*Alg $\to E$ 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

$$KK \to E$$

from the KK-theory homotopy category to that of $E$-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 $KK \to E$ 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.

Related concepts

• KK-bootstrap category

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 $Mot_{add}$	noncommutative motives $Mot_{loc}$	algebraic K-theory	non-connective algebraic K-theory
noncommutative topology	KK-theory	E-theory	operator K-theory	…

References

General

The idea of E-theory was introduced in

• Alain Connes, Nigel Higson, Déformations, morphismes asymptotiques et $K$-théorie bivariante, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 2, 101–106, MR91m:46114, pdf

• 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

• Bruce Blackadar, K-Theory for Operator Algebras

The stable homotopy theory aspects are further discussed in

• Martin Grensing, Noncommutative stable homotopy theory (arXiv:1302.4751)

• Rasmus Bentmann, Homotopy-theoretic E-theory and n-order (arXiv:1302.6924)

See also

• web page of a project Noncommutative topology - homotopy functors and E-theory

• Snigdhayan Mahanta, Higher nonunital Quillen $K'$-theory, KK-dualities and applications to topological $\mathbb{T}$-duality pdf

• pdf

Relation to shape theory

Relation to shape theory is discussed in

• Marius Dādārlat, Shape theory and asymptotic morphisms for $C^\ast$-algebras, Duke Math. J. 73 (3):687-711, 1994, MR95c:46117, pdf

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