Purely infinite simple $C^*$-algebras arising from free product   constructions
  Examples of simple, separable, unital, purely infinite $C^*$--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 $C^*$--algebras by any of a certain class of non--unital endomorphisms;   (3) those that arise as reduced free products of pairs of $C^*$--algebras with respect to any from a certain class of states. 
Kenneth J. Dykema, Mikael Rordam