Elliott classification program

Definition

Let $A$ be a unital, separable, simple, nuclear C*-algebra. Let $T(A)$ be its set of tracial states, a compact convex set in the weak* topology. The Elliott invariant of $A$ is $$\operatorname{Ell}(A) = \big(K_0(A), K_0(A)^+, [1_A], K_1(A), T(A), r_A\big).$$ Here $K_0(A)$ is K0, with positive cone $K_0(A)^+$ given by classes of projections, and $[1_A]$ is the class of the unit. The group $K_1(A)$ is K1. The map $r_A \colon T(A) \times K_0(A) \to \R$ is the pairing $r_A(\tau, [p] - [q]) = \tau(p) - \tau(q)$, with $\tau$ extended to matrices by the trace. An isomorphism of Elliott invariants has three parts, compatible with the pairings. They are an isomorphism of ordered groups preserving the unit class, an isomorphism of the $K_1$-groups, and an affine homeomorphism of the trace spaces. A class $\mathcal{C}$ of such algebras is classified if, for all $A, B \in \mathcal{C}$, $A \cong B$ if and only if $\operatorname{Ell}(A) \cong \operatorname{Ell}(B)$. The Elliott classification program is the problem of finding classified classes and of determining which invariants occur in them.