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

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

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

Definition

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

$$E(A,B)≔[(\Sigma A)\otimes 𝒦,(\Sigma B)\otimes 𝒦]\in \mathrm{Ab}.$$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 $E$ whose objects are C*-algebras, and whose hom-objects are $E(-,-)$.

Properties

Relation to KK-theory

There is a universal functor $\mathrm{KK}\to E$ 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 $\mathrm{KK}\to E$ 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

• operator K-theory

  • KK-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)

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

• Snigdahayan Mahanta, Higher nonunital Qullen $K\prime$-theory, KK-dualities and applications to topological $𝕋$-duality pdf

Relation to shape theory

Relation to shape theory is discussed in

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

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