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

Absorbing representations with respect to closed operator convex cones