noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
(geometry $\leftarrow$ Isbell duality $\to$ algebra)
The equivariant cohomology version of KK-theory.
Let $G$ be a locally compact topological group.
$KK_G$ is the category…
(Blackadar, section 20.2) The composition is in (Blackadar, theorem 20.3.1) .
The KK-theoretic representation ring of $G$ is the ring
The Green-Julg theorem:
(Green-Julg theorem)
Let $G$ be a topological group acting on a C*-algebra $A$.
If $G$ is a compact topological group then the descent map
is an isomorphism, identifying the equivariant operator K-theory of $A$ with the ordinary operator K-theory of the crossed product C*-algebra $G \ltimes A$.
if $G$ is a discrete group then the descent map
is an isomorphism, identifying the equivariant K-homology of $A$ with the ordinary K-homology of the crossed product C*-algebra $G \ltimes A$.
If $G$ is a compact topological group, then the KK-theoretic representation ring, def. coincides with the ordinary representation ring of $G$.
Section 20 of
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