Purely infinite simple $C^{\ast }$-algebras arising from free product constructions

Examples of simple, separable, unital, purely infinite $C^{\ast }$–algebras are constructed, including: (1) some that are not approximately divisible; (2) those that arise as crossed products of any of a certain class of $C^{\ast }$–algebras by any of a certain class of non–unital endomorphisms; (3) those that arise as reduced free products of pairs of $C^{\ast }$–algebras with respect to any from a certain class of states.