In this chapter we take a closer look at injectivity and introduce injective envelopes and C*-envelopes of operator systems, operator algebras, and operator spaces. Loosely speaking, the injective envelope of an object is a “minimal” injective object that contains the original object. The C*-envelope of an operator algebra is a generalization of the Silov boundary of a uniform algebra. The C*-envelope of an operator algebra A is the “smallest” C*-algebra that contains A as a subalgebra, up to completely isometric isomorphism. These ideas will be made precise in this chapter. Many of the ideas of this chapter are derived from the work of M. Hamana [112].

Injectivity is really a categorical concept. Suppose that we are given some category C consisting of objects and morphisms. Then an object I is called injective in C provided that for every pair of objects E ⊆ F and every morphism φ: E → I, there exists a morphism ψ: F → I that extends φ, i.e., such that ψ(e) = φ(e) for every e in E.

If we let denote the collection of operator systems and define the morphisms between operator systems to be the completely positive maps, then since the composition of completely positive maps is again completely positive, we shall have a category, which we call the category of operator systems.