Elliott’s Program

The Elliott invariant of a C*-algebra A is the 4-tuple 4.1 $$\mathrm{Ell}(\mathit{A})\,\, : = \,\, \left(\left(\mathrm{K}_0\mathit{A}, \mathrm{K}_0\mathit{A}^{+}, \sum{}_{\mathit{A}}\right), \mathrm{K}_1\mathit{A}, \,\mathrm{T}^{+}\,(\mathit{A}), \rho\mathit{A} \right),$$ where the K-groups are the Banach algebra ones, K₀A+ is the image of the Murray–von-Neumann semigroup V(A) under the Grothendieck map, $$\sum{}_\mathit{A}$$ is the subset of K₀A corresponding to projections in A, T+(A) is the space of positive tracial linear functionals on A, and ρA is the natural pairing of T+(A) and K₀A given by evaluating a trace at a K₀-class. The reader is referred to Rørdam’s monograph [27] for a detailed treatment of this invariant. In the case of a unital C*-algebra, the invariant becomes $$\left(\left(\mathrm{K}_0{\mathit{A}}, \mathrm{K}_0{\mathit{A}^{+}}, [{1}_{\mathit{A}}]\right), {\mathrm{K}_1}\mathit{A}, \, \mathrm{T}\,(\mathit{A}), \rho\mathit{A}\right), $$ , where [1_A] is the K₀-class of the unit, and T(A) is the (compact convex) space of tracial states. We will concentrate on unital C*-algebras in the sequel in order to limit technicalities.