Brown–Douglas–Fillmore theory

Remarks

BDF proved that $\operatorname{Ext}(X)$ is an abelian group, with neutral element the class of the trivial extensions, those of the form $\pi \circ \rho$ for a unital $*$-homomorphism $\rho \colon C(X) \to B(H)$. For $X \subseteq \C$ compact, $\operatorname{Ext}(X)$ is isomorphic to the group of integer-valued functions on the bounded components of $\C \setminus X$, through $[\tau] \mapsto \big(\lambda \mapsto \operatorname{ind}(T - \lambda)\big)$, where $T \in B(H)$ is any lift of $\tau(z)$. Example: the unilateral shift $S$ gives the extension $\tau(z) = \pi(S)$ of $C(\mathbb{T})$; the Fredholm index of $S - 0$ is $-1$, so $[\tau] \ne 0$ in $\operatorname{Ext}(\mathbb{T}) \cong \Z$. Example of the zero class: for a normal operator $N$ with $\sigma(N) = \sigma_{\mathrm{ess}}(N) = X$, the extension $f \mapsto \pi(f(N))$ is of the form $\pi \circ \rho$ with $\rho(f) = f(N)$, so it is trivial. Call $T$ essentially normal if $T^* T - T T^*$ is compact. Consequence: such a $T$ is a compact perturbation of a normal operator if and only if $\operatorname{ind}(T - \lambda) = 0$ for all $\lambda$ outside its essential spectrum. The group $\operatorname{Ext}(X)$ is the odd K-homology group $K_1(X) = K^1(C(X))$.