Brown–Douglas–Fillmore theory

Summary

Brown–Douglas–Fillmore theory, or BDF theory, classifies extensions of the algebra $C(X)$ of continuous functions on a compact metric space $X$ by the compact operators. Such an extension is the same as an injective unital $*$-homomorphism from $C(X)$ into the Calkin algebra. Their classes form an abelian group $\operatorname{Ext}(X)$. BDF showed that $\operatorname{Ext}$ is a homology theory, now identified with odd K-homology. It solved the question of which essentially normal operators are compact perturbations of normal operators: for $X \subseteq \C$ the obstruction is the Fredholm index.