If Sn denotes the (2n + 1)-dimensional operator system spanned in the group C*-algebra C*(Fn) by the n generators of the free group Fn and their inverses, then the identity map in : Sn -> Sn is shown to be a pure completely positive map. Similarly, the identity map jn : NC(n) -> NC(n) on the noncommutative n-cube NC(n) in the group C_-algebra of the free product of n copies of Z2 is also shown to be pure. Further results on the purity of the reductions of the tensor product of pure completely positive maps are given. Some previously unrecorded generic features of pure completely positive linear maps are also presented, including a result on the pure extendibility of pure completely positive linear maps on operator systems with values in an injective von Neumann algebra.