# KK-Theory

## History of the characterization of $K$-homology and $\operatorname{KK}$.

• Atiyah tried to construct $K$-homology via elliptic differential operators.
• Brown-Douglas-Fillmore related $K$-homology to $C^\ast$-algebra extensions.
• Kasparov defined $\operatorname{KK}$ via generalised elliptic operators and related it to extensions.
• Cuntz described $\operatorname{KK}(A,B)$ as the set of homotopy classes of $\ast$-homomorphisms $qA\to B\otimes \mathcal{K}$ or $q(A\otimes\mathcal{K})\to q(B\otimes \mathcal{K})$ where $qA = \operatorname{Ker}(A\sqcup A \xrightarrow{(\operatorname{id},\operatorname{id})} A)$ ($\sqcup$ is the coproduct), and almost stated its universal property.
• Connes-Higson realised $\operatorname{E}(A,B)$ using asymptotic morphisms $A\otimes C_0(\mathbb{R},\mathcal{K}) \to B\otimes C_0(\mathbb{R},\mathcal{K})$.