Absorbing representations with respect to closed operator convex cones

We initiate the study of absorbing representations of $C^\ast$-algebras with respect to closed operator convex cones. We completely determine when such absorbing representations exist, which leads to the question of characterising when a representation is absorbing, as in the classical Weyl-von Neumann type theorem of Voiculescu. In the classical case, this was proven by Elliott and Kucerovsky who proved that a representation is nuclearly absorbing if and only if it induces a purely large extension. By considering a related problem for extensions of $C^\ast$-algebras, which we call the purely large problem, we ask when a purely largeness condition similar to the one defined by Elliott and Kucerovsky, implies absorption with respect to some given closed operator convex cone. We solve this question for a special type of closed operator convex cone induced by actions of finite topological spaces on $C^\ast$-algebras. As an application of this result, we give $K$-theoretic classification for certain $C^\ast$-algebras containing a purely infinite, two-sided, closed ideal for which the quotient is an AF algebra. This generalises a similar result by the second author, S. Eilers and G. Restorff in which all extensions had to be full.