Let A be a C*-algebra, and let be a subspace. Then we shall call an operator space. Clearly, Mn can be regarded as a subspace of Mn(A), and we let Mn have the norm structure that it inherits from the (unique) norm structure on the C*-algebra Mn(A). We make no attempt at this time to define a norm structure on Mn without reference to A. Thus, one thing that distinguishes from an ordinary normed space is that it comes naturally equipped with norms on Mn for all n ≥ 1. Later in this book we shall give a more axiomatic definition of operator spaces, at which time we shall begin to refer to subspaces of C*-algebras as concrete operator spaces. For now we simply stress that by an operator space we mean a concrete subspace of a C*-algebra, together with this extra “baggage” of a well-defined sequence of norms on Mn. Similarly, if S ⊆ A is an operator system, then we endow Mn(S) with the norm and order structure that it inherits as a subspace of Mn(A).

As before, if B is a C*-algebra and ϕ: S → B is a linear map, then we define ϕn: Mn(S) → Mn(B) by ϕn((ai, j)) = (ϕ(ai, j)). We call ϕ n-positive if ϕn is positive, and we call ϕ completely positive if ϕ is n-positive for all n.