group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
AQFT and operator algebra
(geometry $\leftarrow$ Isbell duality $\to$ algebra)
$E$-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).
First some notation and terminology.
For $A \in$ C*Alg, we write
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
there is a natural composition operation $[A,B] \times [B,C] \to [A,C]$;
$[A,\Sigma B]$ 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 $A$.
For $A,B \in$ C*Alg, the E-group of $A$ with coefficients in $B$ is
Under the induced composition operation this yields an additive category $E$ whose objects are C*-algebras, and whose hom-objects are $E(-,-)$.
E-theory is the universal localization C*Alg $\to E$ which is homotopy-invariant, stable and preserves exact sequences in the middle.
(…)
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
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.
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 | … |
The idea of E-theory was introduced in
Reviews and surveys include
A standard textbook account is in section 25 of
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
Relation to shape theory is discussed in